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Lecture 1

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University of Georgia

Philosophy

PHIL 2020

Sean Meslar

Spring

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PHIL 2020 Unit 5: Lecture 1 Subproofs We have two or more rules of inference to introduce to our system. These rules will involve creating subproofs, proofs within a proof, so their citation and application will need to work differently. Indirect Proof (IP) Formal equivalent: reduction ad absurdum o Begins by assuming opposite of what it tries to prove o Assume what you want to disprove Assumption within Proofs There is an important rule to remember about how to proceed with a proof once you make an assumption: support only works one way Ex: o 1. (A v B) (asm) o 2. A (asm) o 3.1. B (asm for IP) o 3.2. A (3.1 and 1, DS) o 3.3. (A A) (2 and 3.3, Conj) o 4. B (3.13.3, IP) Support only works one way You can use propositions shown earlier in your proof if you move them into a subproof, but not if you move them out of your subproof. The rules for IP The major operator of an inference justified by IP will always be negation The first step for IP is to begin a new sub proof (with corresponding subnumeration) with the assumption of what you want to prove, only without the negation. The proof continues until you produce a formal contradiction (i.e., something of the form (p ~p). The inferences cites the whole range of steps involved in the subproof. Conditional Proof (CP) Conditional proof functions similarly to indirect proof, but the major operator of a proposition inferred from CP will always be a conditional. One begins a conditional proof by assuming the antecedent of the conditional one wishes to prove. Ex: o 1. (A B) (asm) o 2.1. C (asm for CP) o 2.2. B (1, Simp) o 3. (C B) (2.12.2, CP)

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