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Propositional Logic

4) If

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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 isn’t a grammatical issue

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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)

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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

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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

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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

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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 aren’t injecting invalidity

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Solving Strategies, cont.

□Is this argument valid?if P then QP∴ Q

□Could you do a truth table for this?□How else could we determine if it is

valid?

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Solving Strategies, cont.

□What about this argument, is it valid?if P then QQ∴ P

□Do you think so when you see the previous argument at the same time:

if P then QP∴ Q

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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 don’t need to always reprove it

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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) → CA (A & B)… …B C

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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

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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

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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.

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Proof Example, cont.

□Develop a dictionary□D = the University’s duty to disem…

L = many Jews live in…I = should institute a Jewish studies…

□Symbolize the argument□D, D → (L → I) , L ∴ I

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Proof Example, cont.

□Begin the proof by writing out the premises as assumptions

1 D A2 D → (L → I) A3 L A

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Proof Example, cont.

□Now apply the strategies where you can. We currently have only Arrow Out1 D A2 D → (L → I) A3 L A4 L → I 2,1 →O5 I 4,3 →O

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Arguments Regarding Christianity

□Many (most?) in the field of philosophy are not Christians

□This doesn’t make their work in logic wrong

□An argument advanced by a Christian isn’t automatically right either

□See the argument on p.16

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Assignments

□ Read Chapter 2□ Do the exercises at the end of the

chapter