# Copyright 2008, Scott Gray1 Propositional Logic 4) If.

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Copyright 2008, Scott Gray1 Propositional Logic 4) If

Copyright 2008, Scott Gray 2 Compound Statements Simple statements coupled by the sentence connectives we have already learned: if, and, not, if and only if, or Simple statements cannot be further divided and studied by logic; this isnt a grammatical issue

Copyright 2008, Scott Gray 3 Review We have already introduced the connectives and the representation of simple statements Parentheses eliminate ambiguity does A B C mean (A B) C or A (B C)

Copyright 2008, Scott Gray 4 English & Logical Representations Consider this sentence: If the bottle is open, then I take another Logical representation: O A English representations: I take another if the bottle is open Provided that the bottle is open I take another I take another provided that the bottle is open Should the bottle be open, I will take another

Copyright 2008, Scott Gray 5 Representing if One must understand the antecedent and consequent In the previous example, O A, O (bottle is open) is then antecedent and A (I open another) is the consequent The antecedent is always to the left of the arrow

Copyright 2008, Scott Gray 6 Solving Strategies Most of this course will be spent learning and applying strategies for solving the types of problems we face: evaluating the validity of arguments; this is called a proof in your text For now, this informal definition of what we are talking about will suffice

Copyright 2008, Scott Gray 7 Solving Strategies, cont. We already have a tool for determining validity, the truth table Truth tables become unwieldy for anything but a simple argument Our strategies allow us to break a complex argument into discrete pieces and manipulate them, knowing at each step that we arent injecting invalidity

Copyright 2008, Scott Gray 8 Solving Strategies, cont. Is this argument valid? if P then Q P Q Could you do a truth table for this? How else could we determine if it is valid?

Copyright 2008, Scott Gray 9 Solving Strategies, cont. What about this argument, is it valid? if P then Q Q P Do you think so when you see the previous argument at the same time: if P then Q P Q

Copyright 2008, Scott Gray 10 Arrow Out Arrow out is build on the understanding that the modus ponens argument is valid We proved this last week with truth tables Since it was proven valid once, we dont need to always reprove it

Copyright 2008, Scott Gray 11 Arrow Out, cont. Here is the rule: If you have a sentence of the P Q form and a sentence of P, then you may write down Q Examples: A B(A & B) C A(A & B) …… BC

Copyright 2008, Scott Gray 12 Proofs You are give a set of premises and a conclusion Your proof will be a table of three columns The first column numbers the row The second column is the content column (your text never really labels it) The third column is the justification column

Copyright 2008, Scott Gray 13 Proofs, cont. Write down the premises, each one on a row, with an A (for assumption) in the justification column Then, add rows where you apply our strategies, moving toward the conclusion At every point, you are maintaining validity if you justifications are correct

Copyright 2008, Scott Gray 14 Proof Example A university has a duty to disseminate knowledge of the cultures of the people who inhabit the section. If this is true, then provided that many Jews live in South Florida, the University of Miami should institute a Jewish studies program. Many Jews do live in South Florida. It follows that the University should institute this program.

Copyright 2008, Scott Gray 15 Proof Example, cont. Develop a dictionary D = the Universitys duty to disem… L = many Jews live in… I = should institute a Jewish studies… Symbolize the argument D, D (L I), L I

Copyright 2008, Scott Gray 16 Proof Example, cont. Begin the proof by writing out the premises as assumptions 1 D A 2 D (L I)A 3 LA

Copyright 2008, Scott Gray 17 Proof Example, cont. Now apply the strategies where you can. We currently have only Arrow Out 1 D A 2 D (L I)A 3 LA 4 L I2,1 O 5 I4,3 O

Copyright 2008, Scott Gray 18 Arguments Regarding Christianity Many (most?) in the field of philosophy are not Christians This doesnt make their work in logic wrong An argument advanced by a Christian isnt automatically right either See the argument on p.16

Copyright 2008, Scott Gray 19 Assignments Read Chapter 2 Do the exercises at the end of the chapter