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notes_13_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements and Such • Today’s Music: The Rolling Stones • I have posted my solutions to HW #4 (with the shortest proofs I know). • HW #5 is due today @ 4pm. – I’ve posted a handout entitled “Working with LMPL Interpretations”, which contains model answers for LMPL semantics problems. • HW #6 has been posted, and will be due next Thursday @ 4pm. + The final is in class next Thursday. You’ll be given 3 hours to do it. • I’ve posted two important handouts concerning the final exam: – The (Complete) Natural Deduction Rules Handout (provided at final). – A sample final exam, which has the same structure as the actual final. This sample will be discussed, in detail, in lecture tomorrow. • Today: Chapter 6 — Natural deduction proofs in LMPL UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 2 ' & The Rule of ∃-Introduction Rule of ∃-Introduction : For any sentence φτ , if φτ has been inferred at line j in a proof, then at line k we may infer [ ( ∃ ν)φν , labeling the line ‘j ∃ I’ and writing on its left the numbers that occur on the left of j. a 1 ,. . . , a n (j) φτ . . . a 1 ,. . . , a n (k) ( ∃ ν)φν j ∃ I Where [ ( ∃ ν)φν is obtained syntactically from φτ by: • Replacing one or more occurrences of τ in φτ by a single variable ν . • Note: the variable ν must not already occur in the expression φτ . [This prevents double-binding , e.g. , ‘ ( ∃ x)( ∃ x)(Fx & Gx) ’.] • And, finally, prefixing the quantifier [ ( ∃ ν) in front of the resulting expression (which may now have both [ ν s and [ τ s occurring in it). UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 3 ' & $ % The Rule of ∀-Elimination Rule of ∀-Elimination : For any sentence [ ( ∀ ν)φν and constant τ , if [ ( ∀ ν)φν has been inferred at a line j, then at line k we may infer φτ , labeling the line ‘j ∀ E’ and writing on its left the numbers that appear on the left of j. a 1 ,. . . , a n (j) ( ∀ ν)φν . . . a 1 ,. . . , a n (k) φτ j ∀ E Where φτ is obtained syntactically from [ ( ∀ ν)φν by: • Deleting the quantifier prefix [ ( ∀ ν) . • Replacing every occurrence of ν in the open sentence φν by one and the same constant τ . [This prevents fallacies , e.g. , ( ∀ x)(Fx → Gx) Fa → Gb .] • Note: since ‘ ∀ ’ means everything , there are no restrictions on which individual constant may be used in an application of ∀ E. UCB Philosophy Chapter 6 06/23/10 Branden Fitelson Philosophy 12A Notes 4 ' & The Rule of ∀-Introduction: Some Background • It is useful to think of a universal claim [ ( ∀ ν)φν as a conjunction which asserts that the predicate expression φ is satisfied by all objects in the domain of discourse ( i.e. , the conjunction [ φa & (φb & (φc & ...)) is true)....
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notes_13_2x2 - Branden Fitelson Philosophy 12A Notes 1&...

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