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Homework/forallxHomework/homework1/homework1.tex
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\documentclass{article}
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\usepackage[shortlabels]{enumitem}
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\usepackage{xcolor}
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\usepackage{soul}
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\usepackage{cancel}
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\usepackage{amsmath}
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\usepackage{listings}
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\title{Homework 1\\\large PHIL-205-01: Symbolic Logic}
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\date{08/30/2021}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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\section{Section 1: Arguments}
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Highlight the phrase which expresses the conclusion of each of these arguments.\\
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\begin{enumerate}
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\item It is sunny. So \hl{I should take my sunglasses.}
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\item \hl{It must have been sunny.} I did wear my sunglasses, after all.
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\item No one but you has had their hands in the cookie-jar. And the scene of the crime is littered with cookie-crumbs. \hl{You're the culprit!}
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\item Miss Scarlett and Professor Plum were in the study at the time of the murder. And Reverend Green had the candlestick in the ballroom, and we know that there is no blood on his hands. Hence, \hl{Colonel Mustand did it in the kitchen with the lead-piping.} Recall, after all, that the gun had not been fired.
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\end{enumerate}
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\section{Section 2: Valid Arguments}
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\begin{enumerate}[A.]
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\item Which of the following arguments are valid? Which are invalid?\\
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\begin{enumerate}[1.]
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\item Socrates is a man.\\All men are carrots.\\ Therefore, Socrates is a carrot. \\ \\ This argument is valid.
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\item Abe Lincoln was either born in Illinois or he was once president.\\Abe Lincoln was never president.\\Abe Lincoln was born in Illinois.\\ \\This argument is valid.
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\item If I pull the trigger, Abe Lincoln will die.\\I do not pull the trigger.\\Abe Lincoln will not die.\\ \\This argument is invalid.
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\item Abe Lincoln was either from France or from Luxemborg.\\ Abe Lincoln was not from Luxemborg.\\Abe Lincoln was from France.\\ \\This argument is valid.
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\item If the world were to end today, then I would not need to get up tomorrow morning.\\I will need to get up tomorrow morning.\\The world will not end today.\\ \\This argument is invalid.
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\item Joe is now 19 years old.\\Joe is now 87 years old.\\Bob is now 20 years old.\\ \\This argument is invalid. (This is, in fact, a series of statements, rather than a complete argument.)
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\end{enumerate}
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\item In each of the following cases, give an example of its truth, or explain its falsehood.\\
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Can there be:
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\begin{enumerate}[1.]
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\item A valid argument that has one false premise and one true premise?\\ Yes. See Section 2, Question A, part 1.
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\item A valid argument that has only false premises?\\ Yes. See Section 2, Question A, part 2.
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\item A valid argument with only false premises and a false conclusion?\\Yes. See Section 2, Question A, part 3.
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\item A sound argument with a false conclusion?\\No. A sound argument is defined as being a valid argument with all true premises. By definition, a valid argument requires that if all premises are true, the conclusion cannot be false.\newpage
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\item An invalid argument that can be made valid by the addition of a new premise?\\Yes. (In the below demonstration, the highlighted lines show the added premises.)\\
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\\
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If my computer is plugged in, it will stay on.\\
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\hl{If my computer loses power, but remains plugged in, it will turn off.}\\
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My computer is plugged in.\\
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\hl{My computer lost power.}\\
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My computer has been turned off.
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\item A valid argument that can be made invalid by the addition of a new premise.\\ Yes. (In the below demonstration, the highlighted lines show the added premises.)\\
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All vegetables are tasty.\\
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\hl{Not all vegetables are green.}\\
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All tasty food is green.\\
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All vegetables are green.
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\end{enumerate}
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\end{enumerate}
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\section{Section 3: Other Logical Notations}
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\begin{enumerate}[A.]
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\item For each of the following: Is it necessarily true, necessarily false, or contingent?
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\begin{enumerate}[1.]
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\item Caesar crossed the Rubicon.\\
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Contingent.
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\item Someone once crossed the Rubicon.\\
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Necessarily true.
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\item No one has ever crossed the Rubicon.\\
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Necessarily false.
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\item If Caesar crossed the Rubicon, then someone has.\\
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Necessarily true.
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\item Even through Caesar crossed the Rubicon, no one has ever crossed the Rubicon.\\
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Necessarily false.
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\item If anyone has ever crossed the Rubicon, it was Caesar.\\
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Contingent.
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\end{enumerate}\newpage
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\item Look at the following sentences.
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\begin{enumerate}[G1.]
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\item There are at least 4 giraffes at the wild animal park.
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\item There are exactly 7 gorillas at the wild animal park.
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\item There are not more than 2 martians at the wild animal park.
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\item Every giraffe at the wild animal park is a martian.
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\end{enumerate}
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Consider the following combinations of the above, and determine which are jointly consistent, and which are inconsistent.
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\begin{enumerate}[1.]
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\item G2, G3, G4\\
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Jointly consistent
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\item G1, G3, G4\\
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Jointly inconsistent
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\item G1, G2, G4\\
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Jointly consistent
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\item G1, G2, G3\\
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Jointly consistent
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\end{enumerate}
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\item Could there be:
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\begin{enumerate}[1.]
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\item A valid argument whose conclusion is necessarily false?\\Yes. See Section 2, Question A, part 1.
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\item An invalid argument whose conclusion is necessarily true?\\ Yes.\\ \\
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If a I pull the trigger, Abe Lincoln will die.\\
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I do not pull the trigger.\\
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Abe Lincoln dies.
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\item jointly consistent sentences who are necessarily false?\\ Yes.\\
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Vegetarians eat meat.\\
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Rabbits eat meat.
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\item jointly inconsistent sentences who are necessarily true?\\No. Jointly inconsistent means that by definition, the sentences must disagree. Necessarily true requires that the sentence be a truth of the world, something that MUST be true. As such, necessary truths cannot contradict each other.
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\end{enumerate}
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\end{enumerate}
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\end{document}
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% File: homework1.tex
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% Created: 12:43:04 Wed, 01 Sep 2021 EDT
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% Last Change: 12:43:04 Wed, 01 Sep 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{cancel}
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\usepackage{amssymb}
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\usepackage{listings}
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\usepackage[shortlabels]{enumitem}
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\usepackage{lipsum}
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\usepackage{soul}
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\usepackage{geometry}
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\geometry{portrait, margin=1in}
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\date{09/01/2021}
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\title{%
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Homework 1\\
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\large PHIL-205-01: Symbolic Logic}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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Describe something that you now firmly understand as a result of the exercises in \emph{forallx}.\\
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I now better understand the logical meaning behind $\implies$ vs $\iff$, and can understand the difference between them.\\
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Describe something you want to know more about.\\
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The ways of ducking around truth tables. Having taken a class previously that focused on truth tables, I'm curious to see what similarities and differences exist.
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\begin{enumerate}[I.]
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\item Are the following arguments valid or invalid?
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\begin{enumerate}[1.]
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\item All vegans eat tofu.\\
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All vegans take vitamins.\\
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All tofu-eaters take vitamins.\\
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Invalid. Not all tofu-eaters are vegan.\\
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\item If you take this medicine then you'll get over your cold.\\
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You get over your cold.\\
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Therefore, you must have taken the medicine.\\
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Invalid. You can get over your cold without taking the medicine.\\
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\item Most Independents voted for Biden.\\
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Susanna is an Independent.\\
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Therefore, Susanna voted for Biden.\\
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Invalid. Susanna may fall outside of the group "Most Independents".\\
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\item All tofu-eaters take vitamins.\\
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All vegans take vitamins.\\
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Therefore, some vegans are tofu-eaters.\\
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Valid.
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\end{enumerate}
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\newpage
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\item Using the symbolization key below, please symbolize the following sentences.\\
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$C$: We treat ourselves to a large cup of coffee.\\
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$H$: We are happy.\\
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$J$: We have joy in our hearts.\\
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$L$: Life will be complete.\\
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$Q$: We take quizzes.\\
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\begin{enumerate}[1.]
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\item If we are happy and have joy in our hearts, then life will be complete.\\
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$(H\land J)\implies L$
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\item Life will be complete only if we take quizzes.\\
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$L\implies Q$
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\item If we treat ourselves to a large cup of coffee, then we will be happy, unless we take quizzes.\\
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$(C\implies H)\lor Q$
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\item Life will be complete if and only if we have joy in our hearts and we take quizzes.\\
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$L\iff (J \land Q)$
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\item Either we will be happy or we will take quizzes, but not both.\\
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$(H\lor Q)\land\neg(H\land Q)$, $(H\lor Q)\land (H|Q)$, or $\neg(H\iff Q)$
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\item We treat ourselves to a large cup of coffee, if we have joy in our hearts.\\
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$J\implies C$
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\item Either we will treat ourselves to a large cup of coffee and are happy, or we take quizzes and life is complete.\\
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$\neg((C\land H)\iff (Q\land L))$
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\end{enumerate}\newpage
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\item Using the symbolization key below, please symbolize the following sentences.\\
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$F$: Frege was anti-Semitic, but not insane.\\
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$G$: G\"{o}del was insane, but not anti-Semitic.\\
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$P$: Peirce was the greatest American logician.\\
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$Q$: Quine removed the "?" key from his typewriter.\\
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$R$: Russell was British.\\
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\begin{enumerate}[1.]
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\item Quine did not remove the "?" key from his keyboard.\\
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$\neg Q$
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\item Peirce was the greatest American logician, Quine removed the "?" key from his typewriter, and G\"{o}del was insane, but not anti-Semitic.\\
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$P\land Q\land G$
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\item If G\"{o}del was insane, but not anti-Semitic, and Frege was anti-Semitic, but not insane, then Quine removed the "?" key from his typewriter.\\
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$(G\land F)\implies Q$
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\item Peirce was not the greatest American logician if, and only if, Russell was not British.\\
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$(\neg P)\iff (\neg R)$ (This is functionally equivalent to $P\iff R$)
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\item Peirce was the greatest American logician only if Quine removed the "?" key from his typewriter.\\
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$Q\implies P$
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\item Quine removed the "?" key from his typewriter unless G\"{o}del was insane, but not anti-Semitic.\\
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$Q\lor G$
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\item If Frege was anti-Semitic, but not insane, then either G\"{o}del was insane, but not anti-Semitic, or Russell was British.\\
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$F\implies (\neg[G\lor R])$
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\item Either Russell was British or Peirce was not the greatest American logician, but not both.\\
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$\neg(R|P)$
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\item If G\"{o}del was insane, but not anti-Semitic, and Frege was anti-Semitic, but not insane, then Russell was British only if Peirce was the greatest American logician.\\
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$(G\land F)\implies(R\iff P)$
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\item G\"{o}del was insane, but not anti-Semitic if, and only if: Quine removed the "?" key from his typewriter and either Russell was British or Peirce was the greatest American logician.\\
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$G\iff(Q\land [R\lor P])$
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\end{enumerate}
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\end{enumerate}
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\end{document}
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% File: homework2.tex
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% Created: 23:34:38 Thu, 16 Sep 2021 EDT
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\date{09/16/2021}
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\title{%
|
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Homework 2\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
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\author{Blizzard MacDougall}
|
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\begin{document}
|
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\maketitle
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\pagenumbering{arabic}
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\section{Section 1}
|
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Using the following symbolization key:\\
|
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domain: songs\\
|
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Bx: x was written by Joan Baez.\\
|
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Dx: x is about Bob Dylan.\\
|
||||
Px: x was covered by Judas Priest.\\
|
||||
d: "Diamonds and Rust".\\
|
||||
|
||||
Symbolize the following English sentences in First Order Logic.
|
||||
\begin{enumerate}
|
||||
\item "Diamonds and Rust " was written by Joan Baez, is about Bob Dylan, and was covered by Judas Priest.
|
||||
\begin{equation}
|
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Bd\land Dd\land Pd
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\end{equation}
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\item Judas Priest covered a song that Joan Baez wrote.
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\begin{equation}
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\exists x(Px\implies Bx)
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\end{equation}
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\item If there is no song that was written by Joan Baez and covered by Judas Priest, then "Diamonds and Rust" either wasn't written by Joan Baez or it isn't about Bob Dylan.
|
||||
\begin{equation}
|
||||
(\neg\exists x(Bx\land Px))\implies(\neg Bd\lor\neg Dd)
|
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\end{equation}
|
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\item Judas Priest covered every song that Joan Baez wrote about Bob Dylan.
|
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\begin{equation}
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\forall x((Bx\land Dx)\implies Px)
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\end{equation}
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\item If "Diamonds and Rust" is about Bob Dylan, then Judas Priest covered a song about Bob Dylan.
|
||||
\begin{equation}
|
||||
Dd\implies (\exists x [Px\land Dx])
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\end{equation}
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\end{enumerate}
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\newpage
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\section{Section 2}
|
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Determine whether the following argument is valid (preferably using shortcuts).\\
|
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\begin{equation}
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\begin{split}
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\hline
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$A$ & $B$ & $C$ & $D$ & \multicolumn{3}{c||}{$(A\lor B)\land(C\lor D)$} & \multicolumn{3}{c||}{$B\implies D$} & $\neg D$ & $A\land C$\\
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To make solving this easier, I set $A$ and $C$ such that $A\land C$ would be false. I then set $D$ such that $\neg D$ would be true. This leaves the remaining lines being pertinent and necessary to determine the validity of the argument. The rows removed were removed as they do not change whether the argument is valid, as they either contain a false premise or a true conclusion. Solving $(A\lor B)\land (C\lor D)$ next showed only one line in the table where said premise was true. This line is when $B$ is true, and D is false, meaning that $B\implies D$ is false. This means that there are no false conclusions with all-true premises. \\
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\end{document}
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Homework \#4\\
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\large PHIL-205-01:Symbolic Logic}
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\author{Blizzard MacDougall}
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\begin{document}
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Consider the following interpretation:\\
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Determine whether each of the following sentences are true or false, given the above interpretation.\\
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% File: homework5.tex
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% Created: 19:10:30 Fri, 22 Oct 2021 EDT
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% Last Change: 19:10:30 Fri, 22 Oct 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\geometry{portrait, margin=1in}
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\date{10/22/2021}
|
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\title{%
|
||||
Homework \#5\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section I}
|
||||
Construct an interpretation to show that the following is invalid:
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\exists x(Px\land\neg Qx)\\
|
||||
\forall x(Qx\rightarrow Rx)\\
|
||||
\therefore\exists x\neg Rx
|
||||
\end{split}
|
||||
\end{equation}
|
||||
English translation: There is someone who both is P and is not Q. For everyone, if they are Q, then they are R. Therefore, there is someone who is not R.
|
||||
|
||||
Domain:all right-handed individuals\\
|
||||
$Px$:$x$ pays taxes.\\
|
||||
$Qx$:$x$ has lost their left hand.\\
|
||||
$Rx$:$x$ is right-handed.\\
|
||||
|
||||
\newpage
|
||||
\section{Section II}
|
||||
Add annotations to the following proofs:
|
||||
\begin{enumerate}
|
||||
\item Proof 1\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\lnot P\lor\lnot Q}
|
||||
}
|
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{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P\land Q}
|
||||
}
|
||||
{
|
||||
\pline[3.]{P}[\lande{2}]\\
|
||||
\pline[4.]{Q}[\lande{2}]\\
|
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\subproof
|
||||
{
|
||||
\pline[5.]{\lnot P}
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}
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{
|
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\pline[6.]{\lfalse}[\lnote{3, 5}]
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}
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\subproof
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{
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}
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{
|
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\pline[8.]{\lfalse}[\lnote{4, 7}]
|
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}
|
||||
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|
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}
|
||||
\pline[10.]{\lnot(P\land Q)}[\lnoti{\cancel{1,} 2--9}]
|
||||
}
|
||||
|
||||
\item Proof 2\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{P\lif Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P}
|
||||
}
|
||||
{
|
||||
\pline[3.]{Q}[\life{1}{2}]\\
|
||||
\pline[4.]{\lnot P\lor Q}[\lori{3}]
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{\lnot P}
|
||||
}
|
||||
{
|
||||
\pline[6.]{\lnot P\lor Q}[\lori{5}]
|
||||
}
|
||||
\pline[7.]{\lnot P\lor Q}[\textbf{TND:} 2--4, 5--6]
|
||||
}
|
||||
|
||||
\item Proof 3\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\lnot P\lor Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P}
|
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}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{Q}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Q\land Q}[\landi{3}]\\
|
||||
\pline[5.]{Q}[\lande{4}]
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[6.]{\lnot P}
|
||||
}
|
||||
{
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% File: homework6.tex
|
||||
% Created: 10:24:35 Wed, 03 Nov 2021 EDT
|
||||
% Last Change: 10:24:35 Wed, 03 Nov 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{amssymb}
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\usepackage{listings}
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|
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%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
||||
%\usepackage{minted}
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|
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|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{11/03/2021}
|
||||
\title{%
|
||||
Homework \#6\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
Prove the following theorems:
|
||||
\begin{enumerate}
|
||||
\item $(P\rightarrow Q)\leftrightarrow(\neg Q\rightarrow \neg P)$ (Contrapositive)\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P\lif Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{\lnot Q}
|
||||
}
|
||||
{
|
||||
\pline[3.]{\lnot P}[\textbf{MT:} 1, 2]
|
||||
}
|
||||
\pline[4.]{\lnot Q\lif\lnot P}[\lifi{2--3}]
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{\lnot Q\lif\lnot P}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[6.]{P}
|
||||
}
|
||||
{
|
||||
\pline[7.]{Q}[\textbf{MT: }5, 6]
|
||||
}
|
||||
\pline[8.]{P\lif Q}[\lifi{6--7}]
|
||||
}
|
||||
\pline[9.]{(P\lif Q)\liff(\lnot Q\lif\lnot P)}[\liffi{1--4}{5--8}]
|
||||
}
|
||||
\item $(P\land \neg Q)\rightarrow\neg(P\leftrightarrow Q)$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P\land\lnot Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P\liff Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{P}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Q}[\liffe{2}{3}]\\
|
||||
\pline[5.]{\lnot Q}[\lande{1, 3}]\\
|
||||
\pline[6.]{\lfalse}[\lfalsei{4}{5}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[7.]{\lnot P}
|
||||
}
|
||||
{
|
||||
\pline[8.]{\lnot Q}[\liffe{2}{7}]\\
|
||||
\pline[9.]{P}[\lande{1, 8}]\\
|
||||
\pline[10.]{\lfalse}[\lfalsei{7}{9}]
|
||||
}
|
||||
\pline[11.]{\lfalse}[\textbf{TND:} 3--6, 7--10]
|
||||
}
|
||||
\pline[12.]{\lnot (P\liff Q)}[\lnoti{2--11}]
|
||||
}
|
||||
\pline[13.]{(P\land\lnot Q)\lif \neg(P\liff Q)}[\lifi{1--12}]
|
||||
}
|
||||
\newpage
|
||||
\item $[P\lor (Q\land R)]\leftrightarrow [(P\lor Q)\land (P\lor R)]$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P\lor (Q\land R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{\neg P}
|
||||
}
|
||||
{
|
||||
\pline[3.]{Q\land R}[\textbf{DS:} 1, 2]\\
|
||||
\pline[4.]{Q}[\lande{3}]\\
|
||||
\pline[5.]{R}[\lande{3}]\\
|
||||
\pline[6.]{P\lor Q}[\lori{4}]\\
|
||||
\pline[7.]{P\lor R}[\lori{5}]\\
|
||||
\pline[8.]{(P\lor Q)\land (P\lor R)}[\landi{6, 7}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[9.]{P}
|
||||
}
|
||||
{
|
||||
\pline[10.]{P\lor Q}[\lori{9}]\\
|
||||
\pline[11.]{P\lor R}[\lori{9}]\\
|
||||
\pline[12.]{(P\lor Q)\land (P\lor R)}[\landi{10--11}]
|
||||
}
|
||||
\pline[13.]{(P\lor Q)\land (P\lor R)}[\textbf{TND:}1--8, 9--12]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[14.]{(P\lor Q)\land (P\lor R)}
|
||||
}
|
||||
{
|
||||
\pline[15.]{P\lor Q}[\lande{14}]\\
|
||||
\pline[16.]{P\lor R}[\lande{14}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[17.]{\lnot P}
|
||||
}
|
||||
{
|
||||
\pline[18.]{Q}[\textbf{DS:} 15, 17]\\
|
||||
\pline[19.]{R}[\textbf{DS:} 16, 17]\\
|
||||
\pline[20.]{Q\land R}[\landi{18, 19}]\\
|
||||
\pline[21.]{P\lor(Q\land R)}[\lori{20}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[22.]{P}
|
||||
}
|
||||
{
|
||||
\pline[23.]{P\lor(Q\land R)}[\lori{22}]
|
||||
}
|
||||
\pline[24.]{P\lor (Q\land R)}[\textbf{TND:} 17--21, 22--23]
|
||||
}
|
||||
\pline[25.]{[P\lor (Q\land R)]\liff [(P\lor Q)\land (P\lor R)]}[\liffi{1--13}{14--24}]
|
||||
}
|
||||
\item $[(P\leftrightarrow Q)\land (Q\leftrightarrow R)]\rightarrow (P\leftrightarrow R)$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{(P\liff Q)\land (Q\liff R)}
|
||||
}
|
||||
{
|
||||
\pline[2.]{P\liff Q}[\lande{1}]\\
|
||||
\pline[3.]{Q\liff R}[\lande{1}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{P}
|
||||
}
|
||||
{
|
||||
\pline[5.]{Q}[\liffe{2}{4}]\\
|
||||
\pline[6.]{R}[\liffe{3}{5}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[7.]{R}
|
||||
}
|
||||
{
|
||||
\pline[8.]{Q}[\liffe{3}{7}]\\
|
||||
\pline[9.]{P}[\liffe{2}{8}]
|
||||
}
|
||||
\pline[10.]{P\liff R}[\liffi{4--6}{7--9}]
|
||||
}
|
||||
\pline[11.]{[(P\liff Q)\land (Q\liff R)]\lif(P\liff R)}[\lifi{1--10}]
|
||||
}
|
||||
\newpage
|
||||
\item $\neg(P\rightarrow Q)\leftrightarrow (P\land\neg Q)$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{\lnot(P\lif Q)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{P}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Q}[\reit{2}]
|
||||
}
|
||||
\pline[5.]{P\lif Q}[\lifi{3--4}]\\
|
||||
\pline[6.]{\lfalse}[\lfalsei{1}{5}]
|
||||
}
|
||||
\pline[7.]{\lnot Q}[\lnoti{2--6}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[8.]{\lnot P}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[9.]{P}
|
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}
|
||||
{
|
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\pline[10.]{\lfalse}{\lfalsei{8}{9}}\\
|
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\pline[11.]{Q}[\textbf{X:} 10]
|
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}
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\pline[12.]{P\lif Q}[\lifi{9--11}]\\
|
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\pline[13.]{\lfalse}[\lfalsei{1}{12}]
|
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}
|
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\pline[14.]{\lnot\lnot P}[\lnoti{8--13}]\\
|
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\pline[15.]{P}[\textbf{DNE:} 14]\\
|
||||
\pline[16.]{P\land\neg Q}[\landi{15, 7}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[17.]{P\land \neg Q}
|
||||
}
|
||||
{
|
||||
\pline[18.]{P}[\lande{17}]\\
|
||||
\pline[19.]{\neg Q}[\lande{17}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[20.]{P\lif Q}
|
||||
}
|
||||
{
|
||||
\pline[21.]{Q}[\life{20}{18}]\\
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\pline[22.]{\lfalse}[\lfalsei{19}{21}]
|
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}
|
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|
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}
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\pline[24.]{\lnot(P\lif Q)\liff (P\land\lnot Q)}[\liffi{1--16}{17--23}]
|
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}
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\end{enumerate}
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\end{document}
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|
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$
|
||||
l.57 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
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|
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|
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|
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l.57 }
|
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|
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I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework7.tex:57: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.57 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework7.tex:57: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.57 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
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|
||||
Overfull \hbox (74.94438pt too wide) in paragraph at lines 57--57
|
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[]| []|
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[]
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|
||||
<inserted text>
|
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$
|
||||
l.57 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework7.tex:57: Extra }, or forgotten $.
|
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<template> \unskip \hfil }
|
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|
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l.57 }
|
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|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework7.tex:57: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.57 }
|
||||
|
||||
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||||
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|
||||
|
||||
./homework7.tex:57: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.57 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
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||||
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||||
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$
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||||
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||||
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./homework7.tex:69: Extra }, or forgotten $.
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l.69 }
|
||||
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||||
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|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework7.tex:69: Missing $ inserted.
|
||||
<inserted text>
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$
|
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||||
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|
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./homework7.tex:69: Missing } inserted.
|
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<inserted text>
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}
|
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l.69 }
|
||||
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||||
I've inserted something that you may have forgotten.
|
||||
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|
||||
With luck, this will get me unwedged. But if you
|
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really didn't forget anything, try typing `2' now; then
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my insertion and my current dilemma will both disappear.
|
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./homework7.tex:69: Missing $ inserted.
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<inserted text>
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$
|
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./homework7.tex:69: Extra }, or forgotten $.
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l.69 }
|
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I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
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deleted material, e.g., by typing `I$}'.
|
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|
||||
./homework7.tex:69: Missing $ inserted.
|
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<inserted text>
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$
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||||
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||||
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|
||||
you left one out. Proceed, with fingers crossed.
|
||||
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./homework7.tex:69: Missing } inserted.
|
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<inserted text>
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||||
}
|
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l.69 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
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./homework7.tex:91: Missing $ inserted.
|
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<inserted text>
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$
|
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||||
I've inserted a begin-math/end-math symbol since I think
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you left one out. Proceed, with fingers crossed.
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./homework7.tex:91: Extra }, or forgotten $.
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l.91 }
|
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I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework7.tex:91: Missing $ inserted.
|
||||
<inserted text>
|
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$
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l.91 }
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||||
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||||
I've inserted a begin-math/end-math symbol since I think
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you left one out. Proceed, with fingers crossed.
|
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./homework7.tex:91: Missing } inserted.
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<inserted text>
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}
|
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l.91 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (57.55554pt too wide) in paragraph at lines 91--91
|
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<inserted text>
|
||||
$
|
||||
l.91 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework7.tex:91: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
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l.91 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework7.tex:91: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.91 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework7.tex:91: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.91 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
[1
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% File: homework7.tex
|
||||
% Created: 10:02:43 Wed, 10 Nov 2021 EST
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% Last Change: 10:02:43 Wed, 10 Nov 2021 EST
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%
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|
||||
|
||||
\date{11/10/2021}
|
||||
\title{%
|
||||
Homework \# 7\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section 1}
|
||||
Prove the following arguments:
|
||||
\begin{enumerate}
|
||||
\item
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\exists x(Pb\lif Gx)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{Pb}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{Pb \lif Ga}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Ga}[\life{3}{2}]\\
|
||||
\pline[5.]{\exists xGx}[\exists\textbf{Intro:} 4]
|
||||
}
|
||||
\pline[6.]{\exists xGx}[\exists\textbf{Elim:} 1,3-5]
|
||||
}
|
||||
\pline[7.]{Pb\lif \exists xGx}[\lifi{1--6}]
|
||||
}
|
||||
\item
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\lnot\exists x(Px\lor Qx)}
|
||||
}
|
||||
{
|
||||
\pline[2.]{\forall x\lnot(Px\lor Qx)}[\textbf {CQ:} 1]\\
|
||||
\pline[3.]{\lnot(Pa\lor Qa)}[\forall \textbf{Elim:} 2]\\
|
||||
\pline[4.]{\lnot Pa\land \lnot Qa}[\textbf{DeM:} 3]\\
|
||||
\pline[5.]{\lnot Pa}[\lande{4}]\\
|
||||
\pline[6.]{\exists x\lnot Px}[\exists\textbf{Intro:} 5]
|
||||
}
|
||||
\item
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\lnot\forall xPx}\\
|
||||
\pline[2.]{\lnot\exists x\lnot(Rx\lif Px)}\\
|
||||
\pline[3.]{\forall x(Rx\lor Qd)}
|
||||
}
|
||||
{
|
||||
\pline[4.]{\exists x\lnot Px}[\textbf{CQ:} 1]\\
|
||||
\pline[5.]{\forall x\lnot\lnot(Rx\lif Px)}[\textbf{CQ:} 2]\\
|
||||
\pline[6.]{\lnot\lnot(Rd\lif Pd)}[\forall\textbf{Elim:} 5]\\
|
||||
\pline[7.]{(Rd\lif Pd)}[\textbf{DNE:} 6]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[8.]{\lnot Pd}
|
||||
}
|
||||
{
|
||||
\pline[9.]{\lnot Rd}[\textbf{MT:} 7, 8]\\
|
||||
\pline[10.]{Rd\lor Qd}[\forall \textbf{Elim:} 3]\\
|
||||
\pline[11.]{Qd}[\textbf{DS:} 10, 9]
|
||||
}
|
||||
\pline[12.]{Qd}[\exists \textbf{Elim:}1, 8-11]
|
||||
}
|
||||
\end{enumerate}
|
||||
\end{document}
|
||||
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|||
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|
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<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Extra }, or forgotten $.
|
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<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.60 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.60 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.60 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.60 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (72.36107pt too wide) in paragraph at lines 60--60
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.60 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.60 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (84.94434pt too wide) in paragraph at lines 60--60
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.60 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:60: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.60 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:60: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.60 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
[1
|
||||
|
||||
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.111 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:111: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.111 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.111 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:111: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.111 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.111 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:111: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.111 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.111 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:111: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.111 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.111 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./homework8.tex:111: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.111 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
./homework8.tex:111: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
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|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
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||||
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||||
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./homework8.tex:111: Missing } inserted.
|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
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||||
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||||
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|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
||||
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|
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||||
}
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l.111 }
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||||
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||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
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./homework8.tex:111: Extra }, or forgotten $.
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|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
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<inserted text>
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$
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}
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l.111 }
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I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
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|
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$
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|
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I've inserted a begin-math/end-math symbol since I think
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you left one out. Proceed, with fingers crossed.
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./homework8.tex:111: Extra }, or forgotten $.
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<template> \unskip \hfil }
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|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./homework8.tex:111: Missing $ inserted.
|
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<inserted text>
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$
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I've inserted a begin-math/end-math symbol since I think
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you left one out. Proceed, with fingers crossed.
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./homework8.tex:111: Missing } inserted.
|
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<inserted text>
|
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}
|
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l.111 }
|
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|
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I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
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Homework/otherHomeworks/homework8/homework8.pdf
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|
|||
% File: homework8.tex
|
||||
% Created: 20:28:23 Wed, 17 Nov 2021 EST
|
||||
% Last Change: 20:28:23 Wed, 17 Nov 2021 EST
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
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\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{lipsum}
|
||||
\usepackage{soul}
|
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\usepackage{geometry}
|
||||
\usepackage{lplfitch}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
\date{11/17/2021}
|
||||
\title{%
|
||||
Homework \# 8\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section 1}
|
||||
Annotate the following proof.\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\exists xJx}\\
|
||||
\pline[2.]{\exists x\lnot Jx}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{Ja}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{\lnot Jb}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{a=b}
|
||||
}
|
||||
{
|
||||
\pline[6.]{Jb}[=\textbf{Elim:} 5, 3]\\
|
||||
\pline[7.]{\lfalse}[\lfalsei{4}{6}]
|
||||
}
|
||||
\pline[8.]{\lnot a=b}[\lnoti{5--7}]\\
|
||||
\pline[9.]{\exists y\lnot a=y}[\exists\textbf{Intro:} 8]\\
|
||||
\pline[10.]{\exists x\exists y\lnot x=y}[\exists\textbf{Intro:} 9]
|
||||
}
|
||||
\pline[11.]{\exists x\exists y\lnot x=y}[\exists\textbf{Elim:} 2,4-10]
|
||||
}
|
||||
\pline[12.]{\exists x\exists y\lnot x=y}[\exists\textbf{Elim:} 1,3-11]
|
||||
}
|
||||
\newpage
|
||||
\section{Section 2}
|
||||
Annotate the following proof.\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\exists x\exists y(\lnot x=y\land Gx\land Gy)}\\
|
||||
\pline[2.]{\forall x(Gx\lif Hx)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{\exists x\forall y(Hy\liff y=x)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{\exists y(\lnot a=y\land Ga\land Gy)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{\lnot a=b\land Ga\land Gb}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[6.]{\forall y(Hy\liff y=c)}
|
||||
}
|
||||
{
|
||||
\pline[7.]{Ga\lif Ha}[\forall\textbf{Elim: } 2]\\
|
||||
\pline[8.]{Ga}[\lande{5}]\\
|
||||
\pline[9.]{Ha}[\life{7}{8}]\\
|
||||
\pline[10.]{Gb\lif Hb}[\forall\textbf{Elim:} 2]\\
|
||||
\pline[11.]{Gb}[\lande{5}]\\
|
||||
\pline[12.]{Hb}[\life{7}{8}]\\
|
||||
\pline[13.]{Ha\liff a=c}[\forall\textbf{Elim:} 6]\\
|
||||
\pline[14.]{a=c}[\liffe{13}{9}]\\
|
||||
\pline[15.]{Hb\liff b=c}[\forall\textbf{Elim:} 6]\\
|
||||
\pline[16.]{b=c}[\liffe{15}{12}]\\
|
||||
\pline[17.]{a=b}[=\textbf{Elim:} 14,16]\\
|
||||
\pline[18.]{\lnot a=b}[\lande{5}]\\
|
||||
\pline[19.]{\lfalse}[\lfalsei{17}{18}]
|
||||
}
|
||||
\pline[20.]{\lfalse}[\exists\textbf{Elim:} 6-19]
|
||||
}
|
||||
\pline[21.]{\lfalse}[\exists\textbf{Elim:} 5-20]
|
||||
}
|
||||
\pline[22.]{\lfalse}[\exists\textbf{Elim:} 4-21]
|
||||
}
|
||||
\pline[23.]{\lnot\exists x\forall y(Hy\liff y=x)}[\lnoti{3--22}]
|
||||
}
|
||||
|
||||
\end{document}
|
||||
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quizzes/quiz1/quiz1.tex
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|
|
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|
|||
% File: quiz1.tex
|
||||
% Created: 11:47:23 Sat, 04 Sep 2021 EDT
|
||||
% Last Change: 11:47:23 Sat, 04 Sep 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{lipsum}
|
||||
\usepackage{soul}
|
||||
\usepackage{geometry}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
\date{09/04/2021}
|
||||
\title{%
|
||||
Quiz 1\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\begin{enumerate}[I.]
|
||||
\item Are the following arguments valid or invalid. (Optional: Explain your answer.)
|
||||
\begin{enumerate}[1.]
|
||||
\item If you can't do the time, then you shouldn't do the crime.\\
|
||||
You can't do the time.\\
|
||||
Therefore, you shouldn't do the crime.\\
|
||||
|
||||
Valid.
|
||||
\item All RIT students are residents of Monroe County.\\
|
||||
Some residents of Monroe County may borrow library books.\\
|
||||
Therefore, Some RIT students may borrow library books.\\
|
||||
|
||||
Valid.
|
||||
\item Some vegetarians are vegans.\\
|
||||
Some vegetarians are teetotallers.\\
|
||||
Therefore, some vegans are teetotallers.\\
|
||||
|
||||
Valid.
|
||||
\item No decent singers are rock drummers.\\
|
||||
Ginger Baker was a rock drummer.\\
|
||||
Therefore, he wasn't a decent singer.\\
|
||||
|
||||
Valid.
|
||||
\item If you have the flu, then you'll feel sick.\\
|
||||
You don't have the flu.\\
|
||||
Therefore, you won't feel sick.\\
|
||||
|
||||
Invalid. You can be sick of a different disease.
|
||||
\item If you don't drink the water, you won't get giardia.\\
|
||||
You got giardia.\\
|
||||
Therefore, you must have drunk the water.\\
|
||||
|
||||
Invalid. You could have gotten giardia from another source.
|
||||
\item Donald Trump is a very stable genius.\\
|
||||
Donald Trump has been married only three times.\\
|
||||
Therefore, Donald Trump would get along well with Ginger Baker.\\
|
||||
|
||||
Invalid. The conclusion is completely unrelated to the premises.
|
||||
\item Shimano makes the best bike parts.\\
|
||||
Shimano does not make the best bike parts.\\
|
||||
Therefore, you're better off with vintage SunTour parts.\\
|
||||
|
||||
Invalid. The premises are self-contradictory, so the conclusion can never be true.
|
||||
\end{enumerate}
|
||||
\item Are the following true, or false. (Optional: Explain your answer.)
|
||||
\begin{enumerate}[1.]
|
||||
\item A deductively valid argument must have at least two premises.\\
|
||||
True.
|
||||
\item All inductive arguments are invalid.\\
|
||||
True.
|
||||
\item Some inductive arguments are strong.\\
|
||||
True. An argument can be invalid, but still strong.
|
||||
\item If an argument has true premises and a true conclusion then it must be valid.\\
|
||||
True. When an arguments premises are true, and the conclusion cannot be false, then the argument is valid.
|
||||
\item If an argument is valid then it must have a true conclusion.\\
|
||||
False. If the premises of the argument are false, the argument can still be valid, but the conclusion does not have to be true.
|
||||
\end{enumerate}
|
||||
\item Use the following symbolization key to symbolize the sentences.\\
|
||||
$K$: Kleene pronounced his name idiosyncratically.\\
|
||||
$N$: Neurath preferred to spend his time on Isotype.\\
|
||||
$R$: Russell spent time in prison during WWI.\\
|
||||
$T$: Tarski narrowly escaped the Holocaust.\\
|
||||
$W$: Wittgenstein was a failure as an elementary school teacher.
|
||||
\begin{enumerate}[1.]
|
||||
\item If Wittgenstein was a failure as an elementary school teacher, then either Kleene pronounced his name idiosyncratically or Russell did \emph{not} spend time in prison during WWI.
|
||||
\begin{equation}
|
||||
W\implies (K\lor [\neg R])
|
||||
\end{equation}
|
||||
|
||||
\item Russell spent time in prison during WWI but Neurath preferred to spend his time on Isotype.
|
||||
\begin{equation}
|
||||
R\land (\neg N)
|
||||
\end{equation}
|
||||
|
||||
\item Kleene neither pronounced his name idiosyncratically nor was Wittgenstein a failure as an elementary school teacher.
|
||||
\begin{equation}
|
||||
(\neg K)\land (\neg W)
|
||||
\end{equation}
|
||||
|
||||
\item If Neurath preferred to spend his time on Isotype and Wittgenstein was a failure as an elementary school teacher, then Kleene pronounced his name idiosyncratically only if Tarsi narrowly escaped the Holocaust.
|
||||
\begin{equation}
|
||||
(N\land W)\implies (K\implies T)
|
||||
\end{equation}
|
||||
|
||||
\item Kleene pronounced his name idiosyncratically if and only if both Russell spent time in prison during WWI and Neurath did not prefer the spend his time on Isotype.
|
||||
\begin{equation}
|
||||
K\iff (R\land [\neg N])
|
||||
\end{equation}
|
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\item Russell spent time in prison during WWI unless both Kleene pronounced his name idiosyncratically and Neurath preferred to spend his time on Isotype.
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\date{11/30/2021}
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\title{%
|
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Quiz \#10\\
|
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\large PHIL-205-01: Symbolic Logic}
|
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\author{Blizzard MacDougall}
|
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\begin{document}
|
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\maketitle
|
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\pagenumbering{arabic}
|
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Annotate the following proof.\\
|
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(Note that because the proof is rather large, it has been moved to the next page to ensure that it fits properly.)\\
|
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\fitchprf
|
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{
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\pline[1.]{\exists x\exists y(Fx\land Fy\land \lnot x=y)}\\
|
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\pline[2.]{\exists x\exists y(Gx \land Gy\land \lnot x=y)}
|
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}
|
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{
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||||
\subproof
|
||||
{
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\pline[3.]{\lnot\exists x\exists y(Fx\land Gy\land \lnot x=y)}
|
||||
}
|
||||
{
|
||||
\pline[4.]{\forall x\lnot\exists y(Fx\land Gy\land \lnot x=y)}[\textbf{CQ: }3]\\
|
||||
\pline[5.]{\forall x\forall y\lnot (Fx\land Gy\land \lnot x=y)}[\textbf{CQ: }4]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[6.]{\exists y(Fa\land Fy\land \lnot a=y)}
|
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}
|
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{
|
||||
\subproof
|
||||
{
|
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\pline[7.]{Fa\land Fb \land \lnot a=b}
|
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}
|
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{
|
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\subproof
|
||||
{
|
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\pline[8.]{\exists y(Gc\land Gy\land \lnot c=y)}
|
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}
|
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{
|
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\subproof
|
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{
|
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\pline[9.]{Gc\land Gd\land \lnot c=d}
|
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}
|
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{
|
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\pline[10.]{\forall y\lnot(Fa\land Gy\land \lnot a=y)}[\lalle{5}]\\
|
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\pline[11.]{\lnot(Fa\land Gc\land \lnot a=c)}[\lalle{10}]\\
|
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\pline[12.]{Fa}[\lande{9}]\\
|
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\pline[13.]{Gc}[\lande{9}]\\
|
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\pline[14.]{Fa\land Gc}\landi{12, 13}\\
|
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\subproof
|
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{
|
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\pline[15.]{\lnot a=c}
|
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}
|
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{
|
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\pline[16.]{Fa\land Gc\land \lnot a=c}[\landi{14, 15}]\\
|
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|
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}
|
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|
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\pline[19.]{a=c}[\textbf{DNE: }18]\\
|
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\pline[20.]{\forall y\lnot (Fb\land Gy\land \lnot b=y)}[\lalle{5}]\\
|
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\pline[21.]{\lnot(Fb\land Gc\land \lnot b=c)}[\lalle{20}]\\
|
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\pline[22.]{Fb}[\lande{7}]\\
|
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\pline[23.]{Gc}[\reit{13}]\\
|
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\pline[24.]{Fb\land Gc}[\landi{22--23}]\\
|
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\subproof
|
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{
|
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\pline[25.]{\lnot b=c}
|
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}
|
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{
|
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\pline[26.]{Fb\land Gc\land \lnot b=c}[\landi{24--25}]\\
|
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|
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}
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|
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}
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\pline[37.]{\lnot\lnot\exists x\exists y(Fx\land Gy\land \lnot x=y)}[\lnoti{3--36}]\\
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\pline[38.]{\exists x\exists y(Fx\land Gy\land \lnot x=y)}[\textbf{DNE: }37]
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}
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\end{document}
|
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% Created: 15:57:17 Sat, 18 Sep 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\date{09/18/2021}
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\title{%
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Quiz \#3\\
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\large PHIL-205-01:Symbolic Logic}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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\section{Section 1: Translations in Truth Functional Logic}
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Using the following symbolization key, please symbolize the following sentences.
|
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$B$: Your shifters aren't compatible with 12 speeds in the back.
|
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$C$: Campagnolo doesn't offer a 52T cog.
|
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$F$: Bikes really are "machines for freedom".
|
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$M$: You need a monster cog in the back.
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$O$: You ought to install a 1x drivetrain on your bike.
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$S$: You're better off with a Shimano drivetrain.
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$V$: You can find a vintage SunTour derailleur on eBay.
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\begin{enumerate}
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\item You ought to install a 1x drivetrain on your bike only if you need a monster cog in the back.\\
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\begin{equation}
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O\implies M
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\end{equation}
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\item If you need a monster cog in the back but Compagnolo doesn't offer a 52T cog, then you're better off with a Shimano drivetrain.\\
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\begin{equation}
|
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(M\land C)\implies S
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\end{equation}
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\item Either your shifters aren't compatible with 12 speeds in the back and you ought to install a 1x drivetrain on your bike, or Campagnolo doesn't offer a 52T cog and you're better off with a Shimano drivetrain.\\
|
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\begin{equation}
|
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(B\land O)\lor(C\land S)
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\end{equation}
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\item You ought to install a 1x drivetrain on your bike if either Campagnolo doesn't offer a 52T cog or your shifters aren't compatible with 12 speeds in the back (but not both).\\
|
||||
\begin{equation}
|
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(\neg[C\iff B])\implies O
|
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\end{equation}
|
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\item If a) you're better off with a Shimano drivetrain only if your shifters aren't compatible with 12 speeds in the back, and b) you ought to install a 1x drivetrain on your bike unless you cannot find a vintage SunTour derailleur on eBay, then bikes really are "machines of freedom" only if you don't need a monster cog in the back.
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\begin{equation}
|
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([B\implies S]\land[O\lor V])\implies(M\implies F)
|
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\end{equation}
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\end{enumerate}
|
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\newpage
|
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|
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\section{Section 2: Translation in First Order Logic}
|
||||
Using the following symbolization key symbolize the following sentences.
|
||||
|
||||
Domain: Songs
|
||||
|
||||
$Cx$: x was written by Tracy Chapman
|
||||
|
||||
$Sx$: x is, without a doubt, one of the best songs of the 1980s
|
||||
|
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$Bx$: x was recently covered by Black Pumas
|
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|
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$f$: "Fast Car"
|
||||
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\begin{enumerate}
|
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\item "Fast Car" is, without a doubt, one of the best songs of the 1980s if and only if it was recently covered by Black Pumas.\\
|
||||
\begin{equation}
|
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Sf\iff Bf
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\end{equation}
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|
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\begin{equation}
|
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\exists x(Cx\land Sx)
|
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\end{equation}
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\exists x(Cx\land Sx\land Bx)
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\neg(\exists x[Bx\land Cx])
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|
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\item If Black Pumas recently covered a song that is, without a doubt, one of the best songs of the 1980s, then "Fast Car" is, without a doubt, one of the best songs of the 1980s.\\
|
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|
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\exists x(Bx\land Sx)\implies Sf
|
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|
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\begin{equation}
|
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Bf\lor(\exists x[(\neg Sx)\land(\neg Cx)])
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\end{equation}
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\newpage
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\section{Section 3: Truth Table}
|
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Is the following a valid argument?
|
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\begin{equation}
|
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|
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P\land R\\
|
||||
\neg R\lor Q\\
|
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|
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\hline
|
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$P$ & $Q$ & $R$ & \multicolumn{3}{c||}{$P\land R$} & \multicolumn{3}{c||}{$\neg R\lor Q$} & \multicolumn{3}{c|}{$P\implies Q$}\\
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0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1\\
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0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0\\
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0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0\\
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\hline
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\end{tabular}
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\end{center}
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There are two lines in the truth table where the conclusion is false. In both of these lines, the premises are not both true. Therefore, this argument is valid.
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\end{document}
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% Created: 16:13:51 Thu, 30 Sep 2021 EDT
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\date{09/30/2021}
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\title{%
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Quiz 4\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section I}
|
||||
Use the following symbolization key to symbolize the English sentences in FOL:\\
|
||||
Domain: People\\
|
||||
$Hx$: $x$ is the holdout.\\
|
||||
$Mx$: $x$ has the most to gain.\\
|
||||
$Nx$: $x$ has nothing to lose.\\
|
||||
$Px$: $x$ is a politician.\\
|
||||
$m$: Senator Joe Manchin\\
|
||||
$s$: Senator Kyrsten Sinema
|
||||
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||||
\begin{enumerate}
|
||||
\item If Joe Manchin has the most to gain, then he's the holdout.
|
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\begin{equation}
|
||||
Mm\implies Hm
|
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\end{equation}
|
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\item Those who have the most to gain have nothing to lose.
|
||||
\begin{equation}
|
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\begin{split}
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\cancel{\forall x(Mx\land Nx)}\\
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\forall x(Mx\implies Nx)\\
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||||
\end{split}
|
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\end{equation}
|
||||
The submission with $\land$ says \say{Everyone has the most to gain and nothing to lose.}
|
||||
\item Some holdouts have nothing to lose and not all politicians have the most to gain.
|
||||
\begin{equation}
|
||||
\exists x(Hx\land Nx)\land \neg\forall y(Py\implies My)
|
||||
\end{equation}
|
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\item If anyone is a politician, then Kyrsten Sinema is.
|
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\begin{equation}
|
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\exists x(Px\implies Ps)
|
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\end{equation}
|
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\end{enumerate}
|
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\newpage
|
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\section{Section II}
|
||||
Using the following symbolization key to symbolize the English sentences in FOL:\\
|
||||
Domain: People\\
|
||||
$Cxy$: $x$ confuses $y$\\
|
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$Jx$: $x$ is a journalist.\\
|
||||
$Px$: $x$ is a politician.\\
|
||||
$Wxy$: $x$ writes about $y$.\\
|
||||
$m$: Senator Joe Manchin\\
|
||||
$s$: Senator Kyrsten Sinema
|
||||
|
||||
\begin{enumerate}
|
||||
\item There's a journalist who writes about Senator Kyrsten Sinema.
|
||||
\begin{equation}
|
||||
\exists x(Jx\land Wxs)
|
||||
\end{equation}
|
||||
\item Senator Joe Manchin confuses every journalist.
|
||||
\begin{equation}
|
||||
\forall x(Jx\implies Cmx)
|
||||
\end{equation}
|
||||
\item No politician is confused by Senator Kyrsten Sinema.
|
||||
\begin{equation}
|
||||
\neg\exists x(Px\land Cxs)
|
||||
\end{equation}
|
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\item Every journalist is confused by someone or other. (Different people confuse different journalists.)
|
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\begin{equation}
|
||||
\begin{split}
|
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\end{equation}
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|
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|
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\begin{split}
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\end{split}
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\end{equation}
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The submission with $\land$ says \say{Everyone is both a journalist and confused by someone.}
|
||||
\item Every journalist who writes about politicians is confused by Senator Joe Manchin. [\emph{BONUS}]
|
||||
\begin{equation}
|
||||
\forall x\exists y[(Jx\land Py\land Wxy)\implies Cmx]
|
||||
\end{equation}
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\end{enumerate}
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\end{document}
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% Created: 18:22:13 Mon, 11 Oct 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\date{10/11/2021}
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\title{%
|
||||
Quiz \#5\\
|
||||
\large PHIL-205-01: Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section I}
|
||||
Using the following symbolization key to connect the sentences in english to the sentences in FOL:\\
|
||||
domain: people\\
|
||||
$Hx$: $x$ testified before the House Select Committee to investigate the January 6th Attack.\\
|
||||
$Px$: $x$ is President of the United States.\\
|
||||
$Sx$: $x$ is a septuagenarian.\\
|
||||
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||||
\begin{enumerate}[1.]
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||||
\item \hl{C.}At least two people have testified before the House Select Committee.
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||||
\item $\cancel{I}$\hl{G} Exactly two people have testified before the House Select Committee.\\
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I is \say{If two people have testified, then everyone else hasn't.} This can be true if no one has testified.
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\item \hl{B.} Somebody has testified before the House Select Committee.
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\item \hl{E.} At most two people have testified before the House Select Committee.
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\item \hl{H.} Everyone has testified before the House Select Committee.
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\item $\cancel{D}$\hl{F.} The President of the United States is a septuagenarian.\\
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||||
D says all presidents are septuagenarians.
|
||||
\end{enumerate}
|
||||
\begin{enumerate}[(A)]
|
||||
\item $\forall x\forall y (Hx\land Hy\land \forall z(x=y\lor x=z\lor y=z))$
|
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\item $\exists xHx$
|
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\item $\exists x\exists y(Hx\land Hy\land \neg x=y)$
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\item $\forall x(Px\implies Sx)$
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\item $\forall x\forall y\forall z[(Hx\land Hy\land Hz)\implies(x=y\lor x=z\lor y=z)]$
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\item $\exists x(Px\land\forall y(Py\implies x=y)\land Sx)$
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\item $\exists x\exists y[Hx\land Hy\land \neg x=y\land\forall z(Hz\implies x=z\lor y=z)]$
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\item $\forall xHx$
|
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\item $\exists x\exists y[(Hx\land Hy\land \neg x=y)\implies\forall z(Hz\implies [x=z\lor y=z])]$
|
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\end{enumerate}
|
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\newpage
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\section{Section 2}
|
||||
Use the following symbolization key to symbolize the sentences into FOL.\\
|
||||
domain:people\\
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$Sx$:$x$ is a septuagenarian.\\
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||||
$Gx$:$x$ attended a Grateful Dead concert in 1984.\\
|
||||
$Wx$:$x$ lives in the White House.\\
|
||||
\begin{enumerate}
|
||||
\item A septuagenarian who lives in the White House attended a Grateful Dead concert in 1984.
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\begin{equation}
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\begin{split}
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\cancel{\forall x(Sx \implies (Gx\land Wx))}\\
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\end{equation}
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\end{split}
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\end{equation}
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The [] brackets give the additional information that there is only one septuagenarian who lives in the White House.
|
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\item There are at most two septuagenarians living in the White House.
|
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\begin{equation}
|
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\cancel{\exists x\exists y\forall z[Sx\land Wx\land Sy\land Wy\land ([Sz\land Wz]\implies[x=z\lor y=z])]}\\
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\end{equation}
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.\\
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\begin{equation}
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\end{equation}
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\item There is exactly one septuagenarian living in the White House.
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\begin{equation}
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\end{equation}
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\exists x(Sx\land Wx\land\forall y[(Sy\land Wy)\implies x=y])
|
||||
\end{equation}
|
||||
Remember to define where that one exists in the first place.
|
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\item The septuagenarian who lives in the White House is not the septuagenarian who attended a Grateful Dead concert in 1984.
|
||||
\begin{equation}
|
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\cancel{\exists x\exists y(Sx\land Wx \land Sy\land Gy\land \neg x=y)}
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\end{equation}
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\begin{equation}
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\exists x\exists z[Sx\land Wx\land \forall y([Sy\land Wy)\implies x=y])\land Sz\land Gz\land \forall y([Sy\land Gy]\implies z=y)\land\neg x=z]
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||||
\end{equation}
|
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You need to define the septuagenarian who lives in the White House is the only one.
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\end{enumerate}
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\end{document}
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% File: quiz6.tex
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% Created: 14:31:35 Sun, 17 Oct 2021 EDT
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% Last Change: 14:31:35 Sun, 17 Oct 2021 EDT
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%
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\documentclass[letterpaper]{article}
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{cancel}
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\usepackage{amssymb}
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\geometry{portrait, margin=1in}
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\date{10/17/2021}
|
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\title{%
|
||||
Quiz \#6\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
Consider the following interpretation:\\
|
||||
The domain comprises Hasiasson-Lindenbaum (hereafter, HL), Ramsey, and Stebbing.\\
|
||||
$A$ is a one place predicate, true of and only of HL.\\
|
||||
$B$ is a one place predicate, true of only Ramsey and Stebbing.\\
|
||||
$C$ is a one place predicate, true of only HL and Stebbing.\\
|
||||
$D$ is a one place predicate, true of no one.\\
|
||||
$E$ is a one place predicate, true of HL, Ramsey, and Stebbing.\\
|
||||
$h$ refers to HL.\\
|
||||
$r$ refers to Ramsey.\\
|
||||
$s$ refers to Stebbing.\\
|
||||
|
||||
Determine whether each of the following sentences are true or false, given the above interpretation.
|
||||
\NumTabs{4}
|
||||
\begin{enumerate}
|
||||
\item $Br$ \tab\tab\tab 1. \ul{True}
|
||||
\item $\neg Dh\iff \neg(Ar\lor Bh)$ \tab\tab\tab 2. \ul{True}
|
||||
\item $\forall x(Ax\implies Cx)$ \tab\tab\tab 3. \ul{True}
|
||||
\item $\neg\forall x(Ax\implies Cx)$ \tab\tab\tab 4. \ul{False}
|
||||
\item $\exists xBx\implies \neg\exists xDx$ \tab\tab\tab 5. \ul{True}
|
||||
\item $\forall x(Bx\land Cx)$ \tab\tab\tab6. \ul{False}
|
||||
\item $\forall x\neg(Ax\land Dx)$ \tab\tab\tab7. \ul{True}
|
||||
\item $\forall x(Ax\implies Bx)\implies \exists y(Ay\land By)$ \tab\tab 8. \ul{True}
|
||||
\item $\exists x(Dx\implies Cx)$ \tab\tab\tab 9. \ul{True}
|
||||
\item $\forall xEx\implies \exists xDx$ \tab\tab\tab 10.\ul{False}
|
||||
\item $\exists xCx\land \exists x\neg Cx$ \tab\tab\tab 11.\ul{False}\\
|
||||
This is dependent upon the assumption that $x$ refers to the same person in both instances. If $x$ can mean different people in the different terms, then it would be true.
|
||||
\item $\exists xCx\land \exists x\neg Ex$ \tab\tab\tab 12.\ul{False}
|
||||
\item $\exists x\exists y[Ex\land Ey\land\neg x=y\land \forall z(Ex\implies [z=x\lor z=y])]$ \tab13.\ul{False}\\
|
||||
This means there are exactly two $E$s. This, however is incorrect, as there are 3 $E$s in the domain.
|
||||
\item $\exists x\exists y(Bx\land By\land\neg x=y)$ \tab\tab\tab14.\ul{True}\\
|
||||
This means that there is at most two $B$s, which is true in this interpretation.
|
||||
\item $\forall x\forall y\forall z[(Ex\land Ey\land Ez)\implies (x=y\lor y=z\lor x=z)]$ \tab15.\ul{False}
|
||||
\end{enumerate}
|
||||
|
||||
\end{document}
|
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|||
% File: quiz7.tex
|
||||
% Created: 16:14:41 Sun, 31 Oct 2021 EDT
|
||||
% Last Change: 16:14:41 Sun, 31 Oct 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
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\usepackage{graphicx}
|
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\usepackage{cancel}
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\usepackage{amssymb}
|
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\usepackage{listings}
|
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\usepackage[shortlabels]{enumitem}
|
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\usepackage{soul}
|
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%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
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\usepackage{minted}
|
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\usepackage{geometry}
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\usepackage{dirtytalk}
|
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\usepackage{lplfitch}
|
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\geometry{portrait, margin=1in}
|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{10/31/2021}
|
||||
\title{%
|
||||
Quiz 7\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section 1}
|
||||
Please annotate the following proofs.
|
||||
\begin{enumerate}
|
||||
\item
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{T}\\
|
||||
\pline[2.]{(P\land T )\lif S}\\
|
||||
\pline[3.]{S\lif Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{P}
|
||||
}
|
||||
{
|
||||
\pline[5.]{P\land T}[\landi{1, 4}]\\
|
||||
\pline[6.]{S}[\life{2}{5}]\\
|
||||
\pline[7.]{Q}[\life{3}{6}]\\
|
||||
\pline[8.]{Q\lor R}[\lori{7}]
|
||||
}
|
||||
\pline[9.]{P\lif (Q\land R)}[\lifi{4--8}]
|
||||
}
|
||||
\item
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{P\land (Q\lor R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{\lnot [(P\land Q)\lor (P\land R)]}
|
||||
}
|
||||
{
|
||||
\pline[3.]{P}[\lande{1}]\\
|
||||
\pline[4.]{Q\lor R}[\lande{1}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{Q}
|
||||
}
|
||||
{
|
||||
\pline[6.]{P\land Q}[\landi{3, 5}]\\
|
||||
\pline[7.]{(P\land Q)\lor (P\land R)}[\lori{6}]
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[8.]{R}
|
||||
}
|
||||
{
|
||||
\pline[9.]{P\land R}[\landi{3, 8}]\\
|
||||
\pline[10.]{(P\land Q)\lor (P\land R)}[\lori{9}]
|
||||
}
|
||||
\pline[11.]{(P\land Q)\lor (P\land R)}[\lore{4}{5--7}{8--10}]\\
|
||||
\pline[12.]{\lfalse}[\lfalsei{2}{11}]
|
||||
}
|
||||
\pline[13.]{\lnot\lnot [(P\land Q)\lor(P\land R)]}[\lnoti{2--12}]\\
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\pline[14.]{[(P\land Q)\lor(P\land R)]}[\textbf{DNE:} 13]
|
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}
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\end{enumerate}
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\newpage
|
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\section{Section 2}
|
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Complete one or both of the following proofs.
|
||||
\begin{enumerate}
|
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|
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\fitchprf
|
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{
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{
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{
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\end{document}
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% File: quiz8.tex
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% Created: 13:33:31 Fri, 05 Nov 2021 EDT
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% Last Change: 13:33:31 Fri, 05 Nov 2021 EDT
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%
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\documentclass[letterpaper]{article}
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%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
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||||
\date{11/05/2021}
|
||||
\title{%
|
||||
Quiz \#8\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section 1}
|
||||
Annotate the following proof:\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{(P \lif R)\lor (Q\lif R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P\land Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{P\lif R}
|
||||
}
|
||||
{
|
||||
\pline[4.]{P}[\lande{2}]\\
|
||||
\pline[5.]{R}[\life{3}{4}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[6.]{Q\lif R}
|
||||
}
|
||||
{
|
||||
\pline[7.]{Q}[\lande{2}]\\
|
||||
\pline[8.]{R}[\life{6}{7}]
|
||||
}
|
||||
\pline[9.]{R}[\lore{1}{3--5}{6--8}]
|
||||
}
|
||||
\pline[10.]{(P\land Q)\lif R}[\lifi{2--10}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[11.]{(P\land Q)\lif R}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[12.]{\lnot[(P\lif R)\lor (Q\lif R)]}
|
||||
}
|
||||
{
|
||||
\pline[13.]{\lnot(P\lif R)\land \lnot(Q\lif R)}[\textbf{DeM:}12]\\
|
||||
\pline[14.]{\lnot(P\lif R)}[\lande{13}]\\
|
||||
\pline[15.]{\lnot(Q\lif R)}[\lande{13}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[16.]{P}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[17.]{Q}
|
||||
}
|
||||
{
|
||||
\pline[18.]{P\land Q}[\landi{16, 17}]\\
|
||||
\pline[19.]{R}[\life{11}{18}]
|
||||
}
|
||||
\pline[20.]{Q\lif R}[\lifi{17--19}]\\
|
||||
\pline[21.]{\lfalse}[\lfalsei{15}{20}]\\
|
||||
\pline[22.]{R}[\textbf{X:}21]
|
||||
}
|
||||
\pline[23.]{P\lif R}[\lifi{16--22}]\\
|
||||
\pline[24.]{\lfalse}[\lfalsei{14}{23}]
|
||||
}
|
||||
\pline[25.]{\lnot\lnot[(P\lif R)\lor (Q\lif R)]}[\lnoti{12--24}]\\
|
||||
\pline[26.]{[(P\lif R)\lor (Q\lif R)]}[\textbf{DNE:}25]
|
||||
}
|
||||
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|
||||
}
|
||||
\section{Section 2}
|
||||
Prove one or both of the following theorems:
|
||||
\begin{enumerate}
|
||||
\item $\vdash [P\rightarrow (Q\rightarrow R)]\leftrightarrow [Q\rightarrow (P\rightarrow R)]$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P\lif (Q\lif R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{P}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Q\lif R}[\life{1}{3}]\\
|
||||
\pline[5.]{R}[\life{4}{2}]
|
||||
}
|
||||
\pline[6.]{P\lif R}[\lifi{3--5}]
|
||||
}
|
||||
\pline[7.]{Q\lif(P\lif R)}[\lifi{2--6}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[8.]{Q\lif(P\lif R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[9.]{P}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[10.]{Q}
|
||||
}
|
||||
{
|
||||
\pline[11.]{P\lif R}[\life{8}{10}]\\
|
||||
\pline[12.]{R}[\life{11}{9}]
|
||||
}
|
||||
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|
||||
}
|
||||
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|
||||
}
|
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|
||||
}
|
||||
\item $\vdash (Q\leftrightarrow S)\rightarrow [(P\land Q)\leftrightarrow (P\land S)]$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{Q\liff S}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P\land Q}
|
||||
}
|
||||
{
|
||||
\pline[3.]{Q}[\lande{2}]\\
|
||||
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|
||||
\pline[5.]{S}[\liffe{1}{3}]\\
|
||||
\pline[6.]{P\land S}[\landi{4, 5}]
|
||||
}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[7.]{P\land S}
|
||||
}
|
||||
{
|
||||
\pline[8.]{S}[\lande{7}]\\
|
||||
\pline[9.]{P}[\lande{7}]\\
|
||||
\pline[10.]{Q}[\liffe{1}]\\
|
||||
\pline[11.]{P\land Q}[\landi{9, 10}]
|
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}
|
||||
\pline[12.]{[(P\land Q)\liff (P\land S)]}[\liffi{2--6}{7--11}]
|
||||
}
|
||||
\pline[13.]{(Q\liff S)\lif [(P\land Q)\liff (P\land S)]}[\lifi{1--12}]
|
||||
}
|
||||
\end{enumerate}
|
||||
\end{document}
|
||||
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|
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|
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))
|
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|
||||
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|
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|
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|
||||
Package: lipsum 2019/01/02 v2.2 150 paragraphs of Lorem Ipsum dummy text
|
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|
||||
File: lipsum.ltd.tex 2019/01/02 v2.2 The Lorem ipsum dummy text
|
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))
|
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|
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|
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)
|
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|
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|
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|
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|
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|
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|
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|
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)
|
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|
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Package: lplfitch 2013/05/16 v0.9 LPL Fitch style
|
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Package: lplfitch 2013/05/07 -- Fitch Proofs a la LPL
|
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|
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\fitchctxwidth=\skip56
|
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|
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) (./quiz9.aux)
|
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[Loading MPS to PDF converter (version 2006.09.02).]
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|
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) (/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
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Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
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Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4
|
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|
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|
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File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
|
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e
|
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))
|
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|
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|
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*geometry* verbose mode - [ preamble ] result:
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LaTeX Font Info: Trying to load font information for U+msa on input line 25.
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|
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|
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LaTeX Font Info: Trying to load font information for U+msb on input line 25.
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|
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|
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)
|
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./quiz9.tex:93: Extra alignment tab has been changed to \cr.
|
||||
<recently read> \endtemplate
|
||||
|
||||
l.93 }
|
||||
|
||||
You have given more \span or & marks than there were
|
||||
in the preamble to the \halign or \valign now in progress.
|
||||
So I'll assume that you meant to type \cr instead.
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.93 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.93 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (56.73602pt too wide) in paragraph at lines 93--93
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
|
||||
Overfull \hbox (82.91663pt too wide) in paragraph at lines 93--93
|
||||
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|
||||
[]
|
||||
|
||||
|
||||
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|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.93 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.93 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (90.05542pt too wide) in paragraph at lines 93--93
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
|
||||
Overfull \hbox (87.91664pt too wide) in paragraph at lines 93--93
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.93 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.93 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (84.58331pt too wide) in paragraph at lines 93--93
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.93 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:93: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.93 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:93: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.93 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (63.38884pt too wide) in paragraph at lines 93--93
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
[1
|
||||
|
||||
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.118 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.118 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.118 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.118 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (60.24994pt too wide) in paragraph at lines 118--118
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.118 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:118: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.118 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:118: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.118 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (63.83325pt too wide) in paragraph at lines 147--147
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./quiz9.tex:147: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.147 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:147: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.147 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:147: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.147 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:147: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.147 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (75.6944pt too wide) in paragraph at lines 147--147
|
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|
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|
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|
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|
||||
<inserted text>
|
||||
$
|
||||
l.147 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:147: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.147 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./quiz9.tex:147: Missing $ inserted.
|
||||
<inserted text>
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||||
$
|
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l.147 }
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||||
|
||||
I've inserted a begin-math/end-math symbol since I think
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||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./quiz9.tex:147: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.147 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (75.61098pt too wide) in paragraph at lines 147--147
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[]
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quizzes/quiz9/quiz9.pdf
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quizzes/quiz9/quiz9.tex
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quizzes/quiz9/quiz9.tex
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|
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|||
% File: quiz9.tex
|
||||
% Created: 14:56:39 Sun, 14 Nov 2021 EST
|
||||
% Last Change: 14:56:39 Sun, 14 Nov 2021 EST
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
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\usepackage{listings}
|
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\usepackage[shortlabels]{enumitem}
|
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\usepackage{lipsum}
|
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\usepackage{soul}
|
||||
\usepackage{geometry}
|
||||
\usepackage{lplfitch}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
\date{11/14/2021}
|
||||
\title{%
|
||||
Quiz \# 9\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Section 1}
|
||||
Annotate the following proof :\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{\lnot\forall x(Fx\lif Gx)}
|
||||
}
|
||||
{
|
||||
\pline[2.]{\exists x\lnot(Fx\lif Gx)}[\textbf{CQ:} 1]
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{\lnot(Fa\lif Ga)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{\lnot\exists x(Fx\land\lnot Gx)}
|
||||
}
|
||||
{
|
||||
\pline[5.]{\forall x\lnot(Fx\land \lnot Gx)}[\textbf{CQ:} 4]\\
|
||||
\pline[6.]{\lnot(Fa\land\lnot Ga)}[\forall\textbf{Elim: }5]\\
|
||||
\pline[7.]{\lnot Fa\lor \lnot\lnot Ga}[\textbf{DeM: }6]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[8.]{Fa}
|
||||
}
|
||||
{
|
||||
\pline[9.]{\lnot\lnot Ga}[\textbf{DS: }7, 8]\\
|
||||
\pline[10.]{Ga}[\textbf{DNE: }9]
|
||||
}
|
||||
\pline[11.]{Fa\lif Ga}[\lifi{8--10}]\\
|
||||
\pline[12.]{\lfalse}[\lfalsei{3}{11}]
|
||||
}
|
||||
\pline[13.]{\lnot\lnot\exists x(Fx\land\lnot Gx)}[\lnoti{4--12}]\\
|
||||
\pline[14.]{\exists x(Fx\land\lnot Gx)}[\textbf{DNE: }13]
|
||||
}
|
||||
\pline[15.]{\exists x(Fx\land\lnot Gx)}[\exists\textbf{Elim: }2, 3-14]
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[16.]{\exists x(Fx\land\lnot Gx)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[17.]{Fa\land\lnot Ga}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[18.]{Fa\lif Ga}
|
||||
}
|
||||
{
|
||||
\pline[19.]{Fa}[\lande{17}]\\
|
||||
\pline[20.]{Ga}[\life{18}{19}]\\
|
||||
\pline[21.]{\lnot Ga}[\lande{17}]\\
|
||||
\pline[22.]{\lfalse}[\lfalsei{20}{21}]
|
||||
}
|
||||
\pline[23.]{\lnot(Fa\lif Ga)}[\lnoti{18-22}]\\
|
||||
\pline[24.]{\exists x\lnot(Fx\lif Gx)}[\exists\textbf{Intro: }23]
|
||||
}
|
||||
\pline[25.]{\exists x\lnot(Fx\lif Gx)}[\exists\textbf{Elim: }16, 17-24]\\
|
||||
\pline[26.]{\lnot\forall x(Fx\lif Gx)}[\textbf{CQ: }25]
|
||||
}
|
||||
\pline[27.]{\lnot\forall x(Fx\lif Gx)\liff \exists x(Fx\land\lnot Gx)}[\liffi{1--15}{16--26}]
|
||||
}
|
||||
|
||||
\newpage
|
||||
\section{Section 2}
|
||||
Construct a formal proof of one (or both) of the following argument(s).
|
||||
|
||||
\begin{enumerate}[A.]
|
||||
\item $[\forall x(Fx\lor Gx);\ \neg\forall xGx]\vdash\exists xFx$\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\forall x(Fx\lor Gx)}\\
|
||||
\pline[2.]{\lnot\forall xGx}
|
||||
}
|
||||
{
|
||||
\pline[3.]{\exists x\lnot Gx}[\textbf{CQ:} 2]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{\lnot Ga}
|
||||
}
|
||||
{
|
||||
\pline[5.]{Fa\lor Ga}[\forall\textbf{Elim: }1]\\
|
||||
\pline[6.]{Fa}[\textbf{DS: }5, 4]\\
|
||||
\pline[7.]{\exists xFx}[\exists\textbf{Intro: }6]
|
||||
}
|
||||
\pline[8.]{\exists xFx}[\exists\textbf{Elim: }3,4-7]
|
||||
}
|
||||
\item $\vdash [\forall x(Fx\rightarrow Gx)\land\exists x(Fx\land Hx)]\rightarrow\exists x(Gx\rightarrow Hx)$\\
|
||||
\fitchprf{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{\forall x(Fx\lif Gx)\land\exists x(Fx\land Hx)}
|
||||
}
|
||||
{
|
||||
\pline[2.]{\exists x(Fx\land Hx)}[\lande{1}]\\
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{Fa\land Ha}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[4.]{Ga}
|
||||
}
|
||||
{
|
||||
\pline[5.]{Ha}[\lande{3}]
|
||||
}
|
||||
\pline[6.]{Ga\lif Ha}[\lifi{4--5}]\\
|
||||
\pline[7.]{\exists x(Gx\lif Hx)}[\exists\textbf{Intro: }6]
|
||||
}
|
||||
\pline[8.]{\exists x(Gx\lif Hx)}[\exists\textbf{Elim: }2,3-7]
|
||||
}
|
||||
\pline[9.]{[\forall x(Fx\lif Gx)\land\exists x(Fx\land Hx)]\lif\exists x(Gx\lif Hx)}[\lifi{1--8}]
|
||||
}
|
||||
\end{enumerate}
|
||||
|
||||
\end{document}
|
||||
12
requiredTexts.txt
Normal file
12
requiredTexts.txt
Normal file
|
|
@ -0,0 +1,12 @@
|
|||
forallx: Cambridge (P.D. Magnus, 2019, with solutions)
|
||||
Logic For Law Students: How To Think Like A Lawyer (Aldisert et al.)
|
||||
Philosophical Papers (Austin, Third Edition, Chapter 10 "Performative Utterances")
|
||||
Metatheory (Tim Button, 2018, Chapter 5 "Soundness")
|
||||
On The Philosophy of Logic (Fisher, Chapter 12 "The Metaphysics of Logic", Chapter 9 "Two Logical Truths")
|
||||
Truth-Meaning-Reality (Horwich, 2010, Chapter 1 "What is Truth?")
|
||||
The Metaphysics of Logic (Rush, 2014, Chapter 5 "A Second Philosophy of Logic" by Penelope Maddy)
|
||||
Dilemmas: The Tarner Lectures (Ryle, 1954, Chapter 2 "It Was To Be")
|
||||
How The World Discovered Logic (Schenefelt & White, Chapter 9 "Symbolic Logic and the Digital Future")
|
||||
Logic for Philosophy (Sider, 2010, Chapter 2 Section 9 "Completeness of PL")
|
||||
Fatalism (Taylor, 1962 (The Philosophical Review, Vol. 71, No.1; Pages 56-66))
|
||||
Tracatus Logico-Philosophicus (Wittgenstein, composite)
|
||||
BIN
week1/23-08-2021/23-08-2021.pdf
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BIN
week1/23-08-2021/23-08-2021.pdf
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47
week1/23-08-2021/23-08-2021.tex
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47
week1/23-08-2021/23-08-2021.tex
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|
|
@ -0,0 +1,47 @@
|
|||
\documentclass{article}
|
||||
|
||||
\usepackage{enumitem}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\title{%
|
||||
Intro Course Notes\\
|
||||
\large PHIL-205-01: Symbolic Logic}
|
||||
\date{08/31/2021}
|
||||
\author{Blizzard MacDougall}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
|
||||
This is a description of the course on the whole.\\
|
||||
Symbolic logic is the study of formal methods of deductive interference. It is more about the content of the information, rather than the information itself. Example:\\
|
||||
All echidnas are monotremes.\\
|
||||
Lulu is an echidna.\\
|
||||
Lulu is a monotreme.\\
|
||||
\\
|
||||
The general idea is that you can take any argument, and logically analyze it, without knowing all of the key details. \\
|
||||
\\
|
||||
\\
|
||||
If logic's power derives from its formality, then the best way to use it powerfully is to remove language, and only use symbols. (ex. $\therefore,\ \neg$) There's two forms of logic. Deductive, and Inductive. Deductive is binary. Its an \verb|if(then)| statement. Induction is more human; it allows for wiggle room, allows for it to be incorrect.\\
|
||||
Logic is universal. Not only that, it can be seen as a game. Furthermore, it makes us better people, by pushing us to think and work logically. Additionally, its mental exercise.\\
|
||||
\\
|
||||
This class is significantly more clear cut than others. Its not a math class, there will be the fuzzy borders where we talk about philosophers, but it will mainly be about problem solving and logic.\\
|
||||
\\
|
||||
Obviously\ldots don't fall behind.\\
|
||||
There will be weekly quizzes. As such, there will not be a final exam. You can retake 2 quizzes sometime near the last day of classes (tbd). Textbook PDFs are fine. Slides will NOT be posted on MyCourses.\\
|
||||
\\
|
||||
\section{Logic}
|
||||
\subsection{Arguments}
|
||||
Logic is about arguments. Arguments aren't human "arguments". They're premises that lead to conclusions. A premise-less argument is an assertion. Next class will cover good and bad arguments, specifically on deductive arguments.\\
|
||||
There are not restrictions about arguments. If we can make arguments about ethics, aesthetics, science, \emph{anything}, \emph{and} if arguments consist of reasons for believing that a conclusion is true, then this is a reason to think that there are aesthetic, ethical, scientific, etc truth. \ldots This, in itself, is an argument.\\
|
||||
\\
|
||||
\\
|
||||
This class may get difficult\ldots Definitely a philosophy class. \\Also, slavery is bad (tm).\\
|
||||
\\
|
||||
\\
|
||||
There's a fine line between a bad argument and no argument at all. To convert English arguments into symbolic logic, you have to make sure there aren't any missing premises, or missing conclusions. \\
|
||||
|
||||
|
||||
\end{document}
|
||||
65
week1/25-08-2021/25-08-2021.tex
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65
week1/25-08-2021/25-08-2021.tex
Normal file
|
|
@ -0,0 +1,65 @@
|
|||
% File: 25-08-2021.tex
|
||||
% Created: 10:01:24 Wed, 25 Aug 2021 EDT
|
||||
% Last Change: 10:01:24 Wed, 25 Aug 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage{enumitem}
|
||||
|
||||
\date{08/25/2021}
|
||||
\title{%
|
||||
Day Two Notes\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Arguments}
|
||||
Logic is about arguments, which consist of premises, which provide epistemic support for conclusions. Modern symbolic logic is of the idea that if you put all the work in up-front, there won't be nearly as much labor later.\\
|
||||
\paragraph .We're focusing on deductive arguments; arguments where the conclusion is definite. A good deductive argument is valid, bad, invalid. Our focus is on validity, so how do we prove that an argument is valid? The goal is to build a common format to make arguments. \\All inductive arguments are invalid because by the definition of an inductive argument, it leaves room for it to be incorrect. An argument with true premises is both valid and sound. An argument can be valid without being sound, but an argument cannot be sound and invalid.\\
|
||||
\subsection{Examples of Arguments}
|
||||
Example of a valid, sound, argument:\\
|
||||
All women are mortal.\\
|
||||
Diotima was a woman.\\
|
||||
Therefore, Diotima was mortal.\\
|
||||
\\
|
||||
Example of an inductive argument:\\
|
||||
I am going to drop this very fragile egg, from such an astoundingly great height, directly onto this astoundingly hard surface.\\
|
||||
The egg is going to break.\\
|
||||
\\
|
||||
\\
|
||||
Example of a valid, unsound, argument:\\
|
||||
Beijing is in France.\\
|
||||
France is in Asia.\\
|
||||
Therefore, Beijing is in Asia.\\
|
||||
\\
|
||||
This holds a false premise, but a true conclusion.\\
|
||||
\\
|
||||
\\
|
||||
The truth of the premise and the conclusion is irrelevant when determining the validity of an argument.\\
|
||||
\\
|
||||
Example of an invalid argument:\\
|
||||
Joe Biden is President.\\
|
||||
Joe Biden was once Vice-President.\\
|
||||
Therefore, Joe Biden is from Delaware.\\
|
||||
\\
|
||||
\\
|
||||
\section{Truth Values}
|
||||
We distinguish sentences between sentences and propositions. Sentences are physical, propositions aren't. Sentences hold the same information, but are conveyed in different forms. (i.e. Writing the same information in different languages, speaking vs writing, etc) Utterances are sentences that are identical, but hold different information. (i.e. When I say "I'm tired" vs when someone else says "I'm tired")\\
|
||||
Sentences are not inherently true or false in a vacuum. To make them true, things have to be clarified and disambiguated. If a sentence doesn't have a truth value, then we might argue that it isn't really a statement, even if its structured as one. (i.e. "Your hair looks awful" might actually be an exclamation; "Lying is wrong" is perhaps an imperative.)
|
||||
n logic, we're interested in whats possible and what is \emph{necessary}. We're interested in statements which are necessarily true. There's several types of possibilities:\\
|
||||
\begin{itemize}
|
||||
\item
|
||||
practical possibility
|
||||
\item
|
||||
physical possibility
|
||||
\item
|
||||
Deontic (moral) possibility
|
||||
\item
|
||||
logical/metaphysical possibility
|
||||
\end{itemize}
|
||||
\end{document}
|
||||
35
week1/27-08-2021/27-08-2021.tex
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35
week1/27-08-2021/27-08-2021.tex
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|
|
@ -0,0 +1,35 @@
|
|||
% File: 27-08-2021.tex
|
||||
% Created: 10:17:18 Fri, 27 Aug 2021 EDT
|
||||
% Last Change: 10:17:18 Fri, 27 Aug 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage{enumitem}
|
||||
|
||||
\date{08/27/2021}
|
||||
\title{%
|
||||
Day 3 Notes\\
|
||||
\large PHIL-205-01: Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\section{Axiomatic Methods}
|
||||
If we focus on the starting points, the rest of the argument becomes mechanical. This frees up our brain to think on the bigger questions, rather than making sure an argument is valid.\\
|
||||
There are statements which are intrinsically true, are true. There are others that are contingently true. \\
|
||||
Something we'll get back to is "Do truth and provability overlap?" (I think this is p vs np).\\
|
||||
\\
|
||||
\section{Truth Functional Logic}
|
||||
Truth Functional Logic (hereafter, TFL), is the language of logic we are going to start with.\\
|
||||
\subsection {Atomic Sentences} Atomic sentences that express a single, complete thought (She is Elizabeth, Dogs are smelly, etc etc.) Each one of these will be abbreviated with a singular uppercase italic character.\\
|
||||
\subsection{Connectives} Connectives are ways of changing atomic sentences in simple ways. $\neg$ negates; $\land$ is the logical and; $\lor$ is the logical or. To further clarify what is going on, we will use parentheses. Remember that logic has no temporal order. \\
|
||||
\subsection{Conditionals} Arrows $\Rightarrow\ or\ \implies$ are logical \verb|if..then..| statements. The flat side of an arrow is the antecedent, the pointy side of the arrow is consequent. Conditionals \emph{DO NOT} have to be causal.\\
|
||||
A sufficient condition is one that guarantees that something else will obtain. This is conveyed as (in the instance of $Q$ meaning passing quizzes and $P$ meaning passing the class.)$Q\implies P$ A necessary condition is one that must be present for something else to obtain. This is conveyed as $P\implies Q$. Bidirectional dependence is shown as $\iff$\\
|
||||
\\
|
||||
An expression is a string of symbols. It doesn't have to make any type of sense, its just shoving symbols together. A sentence is a well-formed expression.
|
||||
|
||||
\end{document}
|
||||
42
week10/25-10-2021/25-10-2021.tex
Normal file
42
week10/25-10-2021/25-10-2021.tex
Normal file
|
|
@ -0,0 +1,42 @@
|
|||
% File: 25-10-2021.tex
|
||||
% Created: 10:01:24 Mon, 25 Oct 2021 EDT
|
||||
% Last Change: 10:01:24 Mon, 25 Oct 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{soul}
|
||||
%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
||||
\usepackage{minted}
|
||||
\usepackage{geometry}
|
||||
\usepackage{dirtytalk}
|
||||
\usepackage{lplfitch}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{10/25/2021}
|
||||
\title{%
|
||||
Homework \\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
(See edit of homework for corrections)
|
||||
|
||||
\section{Natural Deduction for TFL}
|
||||
Only having a few tools makes for an elegant system, but it limits your capacity to communicate easily.
|
||||
|
||||
To repeat a sentence, we previously took 4 lines. We can now repeat a sentence in one line, using the reiterate rule.
|
||||
|
||||
We can also reduce double negation in one line now, rather than using 8 lines.
|
||||
|
||||
If we have a disjunction, and a negation of one of the disjuncts, then the other disjunct must be true.
|
||||
|
||||
\end{document}
|
||||
230
week11/03-11-2021.tex
Normal file
230
week11/03-11-2021.tex
Normal file
|
|
@ -0,0 +1,230 @@
|
|||
% File: 11-03-2021.tex
|
||||
% Created: 10:16:20 Wed, 03 Nov 2021 EDT
|
||||
% Last Change: 10:16:20 Wed, 03 Nov 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{soul}
|
||||
%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
||||
\usepackage{minted}
|
||||
\usepackage{geometry}
|
||||
\usepackage{dirtytalk}
|
||||
\usepackage{lplfitch}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{11/03/2021}
|
||||
\title{%
|
||||
Expanded TFL Proofs\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
Below is a complete list of available rules for TFL Proofs.
|
||||
\section{Conjunction $\land$ Rules}
|
||||
|
||||
\subsection{$\land$ Elim}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{A\land B}\\
|
||||
\pline[n.]{A}[\lande{m}]
|
||||
}
|
||||
\subsection{$\land$ Intro}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{A}\\
|
||||
\pline[n.]{B}\\
|
||||
\pline[r.]{A\land B}[\landi{m, n}]
|
||||
|
||||
}
|
||||
\section{Disjunction $\lor$ Rules}
|
||||
\subsection{$\lor$ Intro}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{A}\\
|
||||
\pline[n.]{A\lor B}[\lori{m}]
|
||||
|
||||
}
|
||||
\subsection{$\lor$ Elim}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{A\lor B}\\
|
||||
\subproof
|
||||
{
|
||||
\pline[i.]{A}
|
||||
}
|
||||
{
|
||||
\pline[j.]{C}
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[k.]{B}
|
||||
}
|
||||
{
|
||||
\pline[l.]{C}
|
||||
}
|
||||
\pline[r.]{C}[\lore{m}{i--j}{k--l}]
|
||||
}
|
||||
|
||||
\section{Conditional $\rightarrow$ Rules}
|
||||
\subsection{$\rightarrow$ Elim (\textit{Modus Ponens})}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{P\lif Q}\\
|
||||
\pline[n.]{P}\\
|
||||
\pline[r.]{Q}[\life{m}{n}]
|
||||
}
|
||||
Note that order in the reference is important. You must first reference the conditional before referencing the anticedent.
|
||||
|
||||
\subsection{$\rightarrow$ Intro}
|
||||
\fitchctx
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[i.]{A}
|
||||
}
|
||||
{
|
||||
\pline[j.]{B}
|
||||
}
|
||||
\pline[r.]{A\lif B}[\lifi{i--j}]
|
||||
}
|
||||
|
||||
\section{Biconditional $\leftrightarrow$ Rules}
|
||||
\subsection{$\leftrightarrow$ Intro}
|
||||
\fitchctx
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[i.]{A}
|
||||
}
|
||||
{
|
||||
\pline[j.]{B}
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[k.]{B}
|
||||
}
|
||||
{
|
||||
\pline[l.]{A}
|
||||
}
|
||||
\pline{A\liff B}[\liffi{i--j}{k--l}]
|
||||
}
|
||||
\subsection{$\leftrightarrow$ Elim}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{A\liff B}\\
|
||||
\pline[n.]{A}\\
|
||||
\pline[r.]{B}[\liffe{m}{n}]
|
||||
}
|
||||
|
||||
As with $\rightarrow$'s reference, first list the biconditional, then the condition.
|
||||
|
||||
\section{Negation $\neg$ Rules}
|
||||
\subsection{$\neg$ Elim}
|
||||
Any double negation can be eliminated.
|
||||
|
||||
\subsection{$\bot$ Intro}
|
||||
This is proven by showing a contradiction.\\
|
||||
\fitchctx
|
||||
{
|
||||
\pline[1.]{P}\\
|
||||
\pline[2.]{\neg P}\\
|
||||
\pline[3.]{\lfalse}
|
||||
}
|
||||
\subsection{$\neg$ Intro}
|
||||
We have to prove this by proof by contradiction. (Shown below)\\
|
||||
\fitchctx
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P}
|
||||
}
|
||||
{
|
||||
\pline[2.]{\lfalse}
|
||||
}
|
||||
\pline[3.]{\neg P}[\lfalsei{1--2}]
|
||||
}
|
||||
|
||||
\subsection{Explosions}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{\lfalse}\\
|
||||
\pline[r.]{A}[\textbf{X:} m]
|
||||
}
|
||||
Anything can be proven after a contradiction.
|
||||
|
||||
\subsection{Tertium non datur}
|
||||
Latin for \say{no third way}.\\
|
||||
\fitchctx
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[i.]{A}
|
||||
}
|
||||
{
|
||||
\pline[j.]{B}
|
||||
}
|
||||
\subproof
|
||||
{
|
||||
\pline[k.]{\lnot A}
|
||||
}
|
||||
{
|
||||
\pline[l.]{B}
|
||||
}
|
||||
\pline[r.]{B}[\textbf{TND:} i--j, k--l]
|
||||
}
|
||||
|
||||
\section{New Additions}
|
||||
\subsection{Disjunctive Syllogism}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{P\lor Q}\\
|
||||
\pline[n.]{\lnot Q}\\
|
||||
\pline[r.]{P}[\textbf{DS:} 1, 2]
|
||||
}
|
||||
\subsection{Modus Tollens}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{P\lif Q}\\
|
||||
\pline[n.]{\lnot Q}\\
|
||||
\pline[r.]{\lnot P}[\textbf{MT:} 1, 2]
|
||||
}
|
||||
\section{Example Theorem Proofs}
|
||||
\subsection{Frege's Theorem}
|
||||
\fitchprf
|
||||
{}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[1.]{P\lif (Q\lif R)}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[2.]{P\lif Q}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{P}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Q\lif R}[\life{1}{3}]\\
|
||||
\pline[5.]{Q}[\life{2}{3}]\\
|
||||
\pline[6.]{R}[\life{4}{5}]
|
||||
}
|
||||
\pline[7.]{P\lif R}[\lifi{3--6}]
|
||||
}
|
||||
\pline[8.]{(P\lif Q)\lif(P\lif R)}[\lifi{2--7}]
|
||||
}
|
||||
\pline[9.]{(P\lif (Q\lif R))\lif((P\lif Q)\lif(P\lif R))}[\lifi{1--8}]
|
||||
}
|
||||
\end{document}
|
||||
25
week11/11-03-2021.aux
Normal file
25
week11/11-03-2021.aux
Normal file
|
|
@ -0,0 +1,25 @@
|
|||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1}Conjunction $\land $ Rules}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}$\land $ Elim}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}$\land $ Intro}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {2}Disjunction $\lor $ Rules}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}$\lor $ Intro}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}$\lor $ Elim}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {3}Conditional $\rightarrow $ Rules}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}$\rightarrow $ Elim (\textit {Modus Ponens})}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}$\rightarrow $ Intro}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {4}Biconditional $\leftrightarrow $ Rules}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}$\leftrightarrow $ Intro}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}$\leftrightarrow $ Elim}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {5}Negation $\neg $ Rules}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}$\neg $ Elim}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}$\bot $ Intro}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}$\neg $ Intro}{2}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Explosions}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Tertium non datur}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {6}New Additions}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Disjunctive Syllogism}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Modus Tollens}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {7}Example Theorem Proofs}{3}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Frege's Theorem}{3}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{3}
|
||||
556
week11/11-03-2021.log
Normal file
556
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|
|
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|
|||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.9.12) 3 NOV 2021 20:35
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||||
entering extended mode
|
||||
\write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**11-03-2021.tex
|
||||
(./11-03-2021.tex
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LaTeX2e <2020-10-01> patch level 4
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L3 programming layer <2021-01-09> xparse <2020-03-03>
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|
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|
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|
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|
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LaTeX Info: Redefining \cdots on input line 620.
|
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|
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LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
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l.48 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:48: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.48 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./08-11-2021.tex:48: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.48 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:48: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.48 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./08-11-2021.tex:55: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.55 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:55: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.55 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./08-11-2021.tex:55: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.55 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:55: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.55 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 57--59
|
||||
|
||||
[]
|
||||
|
||||
./08-11-2021.tex:67: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.67 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:67: Extra }, or forgotten $.
|
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<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.67 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./08-11-2021.tex:67: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.67 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:67: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.67 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./08-11-2021.tex:67: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.67 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:67: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.67 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./08-11-2021.tex:67: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.67 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:67: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.67 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./08-11-2021.tex:83: Extra alignment tab has been changed to \cr.
|
||||
<recently read> \endtemplate
|
||||
|
||||
l.83 }
|
||||
|
||||
You have given more \span or & marks than there were
|
||||
in the preamble to the \halign or \valign now in progress.
|
||||
So I'll assume that you meant to type \cr instead.
|
||||
|
||||
|
||||
Overfull \hbox (223.81006pt too wide) in paragraph at lines 83--83
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./08-11-2021.tex:83: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.83 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:83: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.83 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./08-11-2021.tex:83: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.83 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./08-11-2021.tex:83: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.83 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
[1
|
||||
|
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{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] (./08-11-2021.aux) )
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BIN
week12/08-11-2021/08-11-2021.pdf
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week12/08-11-2021/08-11-2021.tex
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week12/08-11-2021/08-11-2021.tex
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@ -0,0 +1,84 @@
|
|||
% File: 11-08-2021.tex
|
||||
% Created: 10:13:49 Mon, 08 Nov 2021 EST
|
||||
% Last Change: 10:13:49 Mon, 08 Nov 2021 EST
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{soul}
|
||||
%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
||||
%\usepackage{minted}
|
||||
\usepackage{geometry}
|
||||
\usepackage{dirtytalk}
|
||||
\usepackage{lplfitch}
|
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|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{11/08/2021}
|
||||
\title{%
|
||||
FOL Proofs\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
TFL is fun and all, but it doesn't allow us to prove truths. So, lets take the rules of TFL and go to FOL
|
||||
|
||||
\section{Conversion of Quantifiers}
|
||||
This allows us to move a negation into or out of a sentence:\\
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{\forall x\lnot Dx}\\
|
||||
\pline[n.]{\lnot\exists xDx}[\textbf{CQ:} m]
|
||||
}\\
|
||||
This can be done in either direction, and with either quantifier.
|
||||
Note that if a sentence has multiple quantifiers, this rule must be done for each quantifier.
|
||||
|
||||
\section{Universal Elimination}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{\forall xFx}\\
|
||||
\pline[n.]{Fa}[\forall\textbf{Elim:} 1]
|
||||
}
|
||||
|
||||
\section{Existential Introduction}
|
||||
\fitchctx
|
||||
{
|
||||
\pline[m.]{Fa}\\
|
||||
\pline[n.]{\exists x Fx}[\exists\textbf{Intro:} m]
|
||||
}
|
||||
\section{Universal Introduction}
|
||||
This is incredibly hard, because we have to prove that \textit{any} value is true.
|
||||
That being said, it will basically never be used, because its not a tool we need very often.\\
|
||||
|
||||
\fitchprf
|
||||
{
|
||||
\pline[m.]{\forall xFx}
|
||||
}
|
||||
{
|
||||
\pline[n.]{Fb}[\forall \textbf{Elim:} m]\\
|
||||
\pline[o.]{\forall yFy}[\forall \textbf{Intro:} n]
|
||||
}
|
||||
\section{Existential Elimination}
|
||||
The goal of this rule isn't actually to remove the existential quantifier.
|
||||
Its as part of a subproof, generally.
|
||||
|
||||
\fitchprf{}
|
||||
{
|
||||
\pline[m.]{\exists xAx}
|
||||
\subproof
|
||||
{
|
||||
\pline[i.]{Ac}
|
||||
}
|
||||
{
|
||||
\pline[j.]{\exists xBx}
|
||||
}
|
||||
\pline[r.]{\exists xBx}[\exists\textbf{Elim:} m, i-j]
|
||||
}
|
||||
\end{document}
|
||||
2
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|||
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||||
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|
||||
649
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|
|||
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.9.12) 30 NOV 2021 04:55
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file:line:error style messages enabled.
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**17-11-2021.tex
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||||
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<inserted text>
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$
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l.55 }
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I've inserted a begin-math/end-math symbol since I think
|
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you left one out. Proceed, with fingers crossed.
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I've deleted a group-closing symbol because it seems to be
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spurious, as in `$x}$'. But perhaps the } is legitimate and
|
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you forgot something else, as in `\hbox{$x}'. In such cases
|
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the way to recover is to insert both the forgotten and the
|
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deleted material, e.g., by typing `I$}'.
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./17-11-2021.tex:55: Missing $ inserted.
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<inserted text>
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$
|
||||
l.55 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./17-11-2021.tex:55: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.55 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
|
||||
Overfull \hbox (74.94438pt too wide) in paragraph at lines 55--55
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./17-11-2021.tex:84: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.84 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./17-11-2021.tex:84: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.84 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./17-11-2021.tex:84: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.84 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./17-11-2021.tex:84: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.84 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
./17-11-2021.tex:84: Extra alignment tab has been changed to \cr.
|
||||
<recently read> \endtemplate
|
||||
|
||||
l.84 }
|
||||
|
||||
You have given more \span or & marks than there were
|
||||
in the preamble to the \halign or \valign now in progress.
|
||||
So I'll assume that you meant to type \cr instead.
|
||||
|
||||
|
||||
Overfull \hbox (276.22661pt too wide) in paragraph at lines 84--84
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
|
||||
Overfull \hbox (79.02773pt too wide) in paragraph at lines 84--84
|
||||
[]| []|
|
||||
[]
|
||||
|
||||
./17-11-2021.tex:84: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.84 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./17-11-2021.tex:84: Extra }, or forgotten $.
|
||||
<template> \unskip \hfil }
|
||||
\hskip \tabcolsep \endtemplate
|
||||
l.84 }
|
||||
|
||||
I've deleted a group-closing symbol because it seems to be
|
||||
spurious, as in `$x}$'. But perhaps the } is legitimate and
|
||||
you forgot something else, as in `\hbox{$x}'. In such cases
|
||||
the way to recover is to insert both the forgotten and the
|
||||
deleted material, e.g., by typing `I$}'.
|
||||
|
||||
./17-11-2021.tex:84: Missing $ inserted.
|
||||
<inserted text>
|
||||
$
|
||||
l.84 }
|
||||
|
||||
I've inserted a begin-math/end-math symbol since I think
|
||||
you left one out. Proceed, with fingers crossed.
|
||||
|
||||
./17-11-2021.tex:84: Missing } inserted.
|
||||
<inserted text>
|
||||
}
|
||||
l.84 }
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> above.)
|
||||
With luck, this will get me unwedged. But if you
|
||||
really didn't forget anything, try typing `2' now; then
|
||||
my insertion and my current dilemma will both disappear.
|
||||
|
||||
[1
|
||||
|
||||
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./17-11-2021.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
8726 strings out of 479304
|
||||
141361 string characters out of 5869779
|
||||
505250 words of memory out of 5000000
|
||||
25774 multiletter control sequences out of 15000+600000
|
||||
412218 words of font info for 61 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
85i,26n,88p,277b,339s stack positions out of 5000i,500n,10000p,200000b,80000s
|
||||
</usr/
|
||||
share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/
|
||||
texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texliv
|
||||
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf
|
||||
-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texlive/texmf-dist/f
|
||||
onts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texlive/texmf-dist/fonts/ty
|
||||
pe1/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/pu
|
||||
blic/amsfonts/cm/cmsy10.pfb>
|
||||
Output written on 17-11-2021.pdf (1 page, 74510 bytes).
|
||||
PDF statistics:
|
||||
36 PDF objects out of 1000 (max. 8388607)
|
||||
25 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
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1 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
BIN
week13/17-11-2021/17-11-2021.pdf
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BIN
week13/17-11-2021/17-11-2021.pdf
Normal file
Binary file not shown.
BIN
week13/17-11-2021/17-11-2021.synctex.gz
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week13/17-11-2021/17-11-2021.synctex.gz
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85
week13/17-11-2021/17-11-2021.tex
Normal file
85
week13/17-11-2021/17-11-2021.tex
Normal file
|
|
@ -0,0 +1,85 @@
|
|||
% File: 17-11-2021.tex
|
||||
% Created: 10:01:33 Wed, 17 Nov 2021 EST
|
||||
% Last Change: 10:01:33 Wed, 17 Nov 2021 EST
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{soul}
|
||||
%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
||||
\usepackage{minted}
|
||||
\usepackage{geometry}
|
||||
\usepackage{dirtytalk}
|
||||
\usepackage{lplfitch}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
||||
|
||||
\date{11/17/2021}
|
||||
\title{%
|
||||
Identity\\
|
||||
\large PHIL--205--01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
Example identity:\\
|
||||
|
||||
I see a President and no one else.\\
|
||||
I see Joe Biden.\\
|
||||
Therefore, Biden is President.\\
|
||||
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\forall x[(Px\land Six)\land\forall y(Siy\lif y=x)]}\\
|
||||
\pline[2.]{Sib}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{Pa\land Sia\land \forall y(Siy\lif y=a)}
|
||||
}
|
||||
{
|
||||
\pline[4.]{\forall y(Siy\lif y=a)}[\lande{3}]\\
|
||||
\pline[5.]{Sib\lif b=a}[\forall\textbf{Elim:} 4]\\
|
||||
\pline[6.]{b=a}[\life{5}{2}]\\
|
||||
\pline[7.]{Pa}[\lande{3}]\\
|
||||
\pline[8.]{Pb}[=\textbf{Elim:} 6, 7]
|
||||
}
|
||||
\pline[9.]{\exists\textbf{Elim:}1,3-8}
|
||||
}
|
||||
|
||||
The F is a G. a is not a G. Therefore a is not an F.\\
|
||||
\fitchprf
|
||||
{
|
||||
\pline[1.]{\exists x\forall y(Fx\land(Fy\lif x=y)\land Gx)}\\
|
||||
\pline[2.]{\lnot Ga}
|
||||
}
|
||||
{
|
||||
\subproof
|
||||
{
|
||||
\pline[3.]{\forall y(Fb\land(Fy\lif b=y)\land Gb)}
|
||||
}
|
||||
{
|
||||
\pline[4.]{Fb\land(Fa\lif b=a)\land Gb}[\forall\textbf{Elim:}3]
|
||||
\subproof
|
||||
{
|
||||
\pline[5.]{Fa}
|
||||
}
|
||||
{
|
||||
\pline[6.]{Fa\lif b=a}[\lande{4}]\\
|
||||
\pline[7.]{b=a}[\life{6}{5}]\\
|
||||
\pline[8.]{Gb}[\lande{4}]\\
|
||||
\pline[9.]{Ga}[=\textbf{Elim:}7,8]\\
|
||||
\pline[10.]{\lfalse}[\lfalsei{9}{2}]
|
||||
}
|
||||
\pline[11.]{\lnot Fa}[\lnoti{5-10}]
|
||||
}
|
||||
\pline[12.]{\lnot Fa}[\exists\textbf{Elim:}1,3-11]
|
||||
}
|
||||
\end{document}
|
||||
49
week2/01-09-2021/01-09-2021.tex
Normal file
49
week2/01-09-2021/01-09-2021.tex
Normal file
|
|
@ -0,0 +1,49 @@
|
|||
% File: 01-09-2021.tex
|
||||
% Created: 10:07:52 Wed, 01 Sep 2021 EDT
|
||||
% Last Change: 10:07:52 Wed, 01 Sep 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage{enumitem}
|
||||
\usepackage{soul}
|
||||
\usepackage{multirow}
|
||||
|
||||
\date{09/01/2021}
|
||||
\title{%
|
||||
Continued Truth Tables\\
|
||||
\large PHIL-205-01: Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
Important note from the reading: "Unless". $P$ unless $Q$ translates to \\$\neg P\implies Q == P\lor Q$.\\
|
||||
|
||||
\section{Truth Tables}
|
||||
|
||||
We can use truth tables to determine truth or falsity of a complex sentence. We can find when a complex sentence is always true (a tautology), or always false (necessarily false contradiction). It can also help us find if two sentences are equivalent, and we can determine if an argument is valid.\\
|
||||
|
||||
Truth tables are tedious, but\dots thats kinda the point. Example of a complete truth table:\\ $(P\iff Q)\lor[(Q\implies\neg P)\land P]$\\
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|c|c c c c c c c c c c| }
|
||||
\hline
|
||||
P & Q & \multicolumn{10}{|c|}{$(P\iff Q)\lor[(Q\implies\neg P)\land P]$}\\
|
||||
\hline
|
||||
\hline
|
||||
1&1&1&1&1&1&1&1&0&1&1&1\\
|
||||
\hline
|
||||
1&0&1&0&0&1&0&1&0&1&1&1\\
|
||||
\hline
|
||||
0&1&0&0&1&0&1&1&1&0&0&0\\
|
||||
\hline
|
||||
0&0&0&1&0&1&0&0&1&0&0&0\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
We can prove an argument is valid by showing where the premises and the arguments differ. [$\neg P$, $P\lor (Q\land \neg P)$, and $\therefore\neg(\neg Q\lor P)$ make a valid argument]
|
||||
|
||||
\end{document}
|
||||
51
week2/30-08-2021/30-08-2021.tex
Normal file
51
week2/30-08-2021/30-08-2021.tex
Normal file
|
|
@ -0,0 +1,51 @@
|
|||
% File: 30-08-2021.tex
|
||||
% Created: 09:59:41 Mon, 30 Aug 2021 EDT
|
||||
% Last Change: 09:59:41 Mon, 30 Aug 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage{enumitem}
|
||||
|
||||
\date{08/30/2021}
|
||||
\title{%
|
||||
Truth Functional Logic\\
|
||||
\large PHIL-205-01: Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
We have the raw materials of language in logic, as defined last class. (Atomic Sentence Letters, Connectives, and Parentheses).\\
|
||||
\section{Some Asides}
|
||||
As established last week, an expression is just a string of symbols, while a sentence is a well-formed string of symbols. Another important clarification is the difference between referring to the object/idea that a word names, and referring to the word itself. (i.e. Gold is a precious metal. [use] vs "Gold" has four letters. [mention]) \\
|
||||
Finally, the object language is the language we're using (TFL), while the metalanguage is the language we use to talk about the object language (English). As an example, the word "truth" might seem to be a part of the object language, but this results in paradoxes (i.e. "This sentence is not true". [This is the liar's paradox.]) Rather, the word "truth" should be a part of the metalanguage. So, the sentence "It's true that snow is white" should be reworded as "'Snow is White' is true." This clarification removes the possibility of liar paradoxes. (Note that this may be overly restrictive, and tends to only work for formal languages, rather than natural languages.)\\
|
||||
|
||||
\section{Truth Tables}
|
||||
We have the establishment of the sentences. Now, we need to show how to evaluate these sentences. We're starting with the most mind-numbing way of doing things: the truth table. (\emph{Engineers across the globe groan.}) These are your standard Boolean truth tables. The only available (traditional Boolean) modifiers are $\neg$ (\verb|NOT|), $\land$ (\verb|AND|), $\lor$ (\verb|OR|), $|$ (\verb|NAND|). \verb|XOR| and \verb|NOR| must be manually created.\\
|
||||
\newpage
|
||||
Non-Boolean functions:\\
|
||||
$\neg P\lor Q = P\implies Q$\\
|
||||
Bi-conditionals $(\iff)$ are true when they are both true OR both false. The Boolean \verb|XOR| is the same as $\neg\iff$.\\\\
|
||||
Full Truth Table explanation:
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|c||c|c|c|c|c|c|}
|
||||
\hline
|
||||
$P$ & $Q$ & $\neg P$ & $P\lor Q$ \verb|(P OR Q)| & $P\land Q$ \verb|(P AND Q)| & $P\iff Q$ & $\neg P\lor Q$ & $P\implies Q$\\
|
||||
\hline\hline
|
||||
T & T & F & T & T & T & T & T\\
|
||||
\hline
|
||||
T & F & F & T & F & F & F & F\\
|
||||
\hline
|
||||
F & T & T & T & F & F & T & T\\
|
||||
\hline
|
||||
F & F & T & F & F & T & T & T\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
For every statement in this course, we can fill out a truth table to prove or disprove its truth.
|
||||
|
||||
\end{document}
|
||||
119
week3/08-09-2021/08-09-2021.tex
Normal file
119
week3/08-09-2021/08-09-2021.tex
Normal file
|
|
@ -0,0 +1,119 @@
|
|||
% File: 08-09-2021.tex
|
||||
% Created: 10:13:01 Wed, 08 Sep 2021 EDT
|
||||
% Last Change: 10:13:01 Wed, 08 Sep 2021 EDT
|
||||
%
|
||||
\documentclass[letterpaper]{article}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{cancel}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{listings}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage{lipsum}
|
||||
\usepackage{soul}
|
||||
\usepackage{geometry}
|
||||
|
||||
\geometry{portrait, margin=1in}
|
||||
|
||||
\date{09/08/2021}
|
||||
\title{%
|
||||
Truth Table Examples\\
|
||||
\large PHIL-205-01:Symbolic Logic}
|
||||
\author{Blizzard MacDougall}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\pagenumbering{arabic}
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|c|c|c|c|c|}
|
||||
\hline
|
||||
$A$ & $B$ & $C$ & $\neg A\land \neg B$ & $\neg B\implies \neg C$ & $\neg(A\land\neg C)$\\
|
||||
\hline \hline
|
||||
1 & 1 & 1 & 0 & 1 & 1\\
|
||||
\hline
|
||||
1 & 1 & 0 & 0 & 1 & 0\\
|
||||
\hline
|
||||
1 & 0 & 1 & 0 & 0 & 1\\
|
||||
\hline
|
||||
1 & 0 & 0 & 0 & 1 & 0\\
|
||||
\hline
|
||||
0 & 1 & 1 & 0 & 1 & 1\\
|
||||
\hline
|
||||
0 & 1 & 0 & 0 & 1 & 1\\
|
||||
\hline
|
||||
0 & 0 & 1 & 1 & 0 & 1\\
|
||||
\hline
|
||||
0 & 0 & 0 & 1 & 1 & 1\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
This argument is valid. All places where premises are true, conclusions are also true. (only the last line)
|
||||
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|c|c|c|c|c|}
|
||||
\hline
|
||||
$A$ & $B$ & $C$ & $\neg A\lor \neg B$ & $\neg B\implies \neg C$ & $\neg(A\land\neg C)$\\
|
||||
\hline \hline
|
||||
1 & 1 & 1 & 0 & 1 & 1\\
|
||||
\hline
|
||||
1 & 1 & 0 & 0 & 1 & 0\\
|
||||
\hline
|
||||
1 & 0 & 1 & 1 & 0 & 1\\
|
||||
\hline
|
||||
1 & 0 & 0 & 1 & 1 & 0\\
|
||||
\hline
|
||||
0 & 1 & 1 & 1 & 1 & 1\\
|
||||
\hline
|
||||
0 & 1 & 0 & 1 & 1 & 1\\
|
||||
\hline
|
||||
0 & 0 & 1 & 1 & 0 & 1\\
|
||||
\hline
|
||||
0 & 0 & 0 & 1 & 1 & 1\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
This argument is invalid, because there is a line where all premises are true, and the conclusion is false.
|
||||
|
||||
|
||||
\begin{center}
|
||||
\begin{tabular}{|c|c|c|c|c|c|}
|
||||
\hline
|
||||
$A$ & $B$ & $C$ & $A\land (B\implies C)$ & $\neg[(\neg B\land C)\lor \neg A]$ & $\neg(\neg A\land B)$\\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
Rather than working through the truth table, set the conclusion to be false, then use that to determine if the letters are true or false. Then, plug those letters back into the premises, and see whether the premises can be true.
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\begin{center}
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\begin{tabular}{|c|c|c|c|c|c|}
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\hline
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$A$ & $B$ & $C$ & $A\land (B\implies C)$ & $\neg[(\neg B\land C)\lor \neg A]$ & $\neg(\neg A\land B)$\\
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\section{Digression}
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There's times where things are vague. This could lead to something being neither true nor false (this is paracomplete), or being neither true nor false (paraconsistent).\\
|
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End of digression
|
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\section{Truth Table Shortcuts}
|
||||
Truth tables are mechanical. That sucks. We need a system of natural deduction. TFL (or sentential logic) also is a bit too coarse. So we need a system of First Order Logic, or predicate logic.\\
|
||||
|
||||
We can't prove the following with TFL:\\
|
||||
All men are mortal.\\
|
||||
Bill is a man.\\
|
||||
Bill is mortal.\\
|
||||
|
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Keeping the old systems, we're adding a couple notations.
|
||||
\begin{itemize}
|
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\item $\forall$ is the universal quantifier (read as "For all\dots")
|
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\item $\exists$ is the existential quantifier (read as "There is some situation where\dots")
|
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\item Name letters. Lowercase letters refer to specific nouns.
|
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\item Predicate letters. Generally, $Xx$, where the $X$ is a relevant description, and $x$ is a specific name letter (or $x$ for "all")
|
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\end{document}
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\title{%
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Notes\\
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\large PHIL-205-01:Symbolic Logic}
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\author{Blizzard MacDougall}
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\begin{document}
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\maketitle
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\pagenumbering{arabic}
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If we keep the concept of truth minimal, all will be fine.\\
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There's three main theories on how to expand the definition of truth.
|
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\begin{enumerate}
|
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\item Correspondence Theory\\
|
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"p" is true iff it corresponds to the facts.
|
||||
\item Coherence Theory\\
|
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"p" is true iff p coheres with the rest of our beliefs.\\
|
||||
Don't mess with this one.
|
||||
\item Epistemic Theories/Provability Theory\\
|
||||
"p" is true iff p is provable, knowable.
|
||||
\item Pragmatic Theory\\
|
||||
"p" is true iff the belief that p works is durable, dependable, in the long run.
|
||||
\item Deflationism/Minimalism\\
|
||||
"p" is true iff p
|
||||
\end{enumerate}
|
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|
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Theories of truth don't go too far. It doesn't correspond to science, it can't. There's some things you just have to experiment with.\\
|
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|
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Picking up from where we left off\dots \\
|
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Translation time!\\
|
||||
|
||||
$\forall x(Dx\implies Kx)$ = For all x, if it's a doctor, then it is kind. = All doctors are kind.\\
|
||||
|
||||
Usually, the universal quantifier goes with a conditional $\implies$, and the existential quantifier goes with a conjunction $\land$. Not \emph{every} time, but this is where you should start.\\
|
||||
$\forall x(Sx\implies Px)$ (all students will pass the course)\\
|
||||
NOT $\forall x(Sx\land Px)$ (everything is a student who will pass the course)\\
|
||||
$\exists x(Sx\land Px)$ (some things, given that they are students, will pass the course)\\
|
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NOT $\exists x(Sx\implies Px)$ (there are some things such that if its a student, it will pass the course [This will sometimes be true at the same time as the other, but not always])\\
|
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|
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Parentheses are essential.\\
|
||||
$\forall x(Dx\implies Kx)$ = Every doctor is kind.\\
|
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$\forall xDx\implies Kx)$ = Everything is a doctor, and x is kind.\\
|
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|
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Domains of $\forall x$ need to be explicit. Otherwise, one can assume the domain is the universe.\\
|
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"Syncategorematic adjectives" are adjectives that functions differently in the context. i.e. "expectant" in "expectant mother" is syncategorematic.
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\end{document}
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usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/sh
|
||||
are/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/tex
|
||||
live/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texlive/te
|
||||
xmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texlive/texmf-dis
|
||||
t/fonts/type1/public/amsfonts/cm/cmsy10.pfb>
|
||||
Output written on 15-09-2021.pdf (2 pages, 60902 bytes).
|
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PDF statistics:
|
||||
31 PDF objects out of 1000 (max. 8388607)
|
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21 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
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1 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
BIN
week4/15-09-2021/15-09-2021.pdf
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week4/15-09-2021/15-09-2021.pdf
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week4/15-09-2021/15-09-2021.synctex.gz
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week4/15-09-2021/15-09-2021.synctex.gz
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