notes_15_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 Doors • I have posted my solutions to HW #4 and HW #5. • HW #6 is due on Thursday @ 4pm. + The final is in class on 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 was discussed, in detail, in lecture last week. • Today: Chapters 7 & 8 — L2PL – I will only be covering (some of) the L2PL parts of Chapters 7 & 8. – I will say something about what will be on the final at the end of today’s lecture (which, alas, will be my last lecture at Berkeley). UCB Philosophy Chapters 7&8 (L2PL) 06/30/10 Branden Fitelson Philosophy 12A Notes 2 ' & Why ( † ) is So Important — L2PL vs LMPL: Infinite Domains • In LMPL, if p is true on any interpretation I , then it is true on a finite interpretation. Indeed, p will be true on an interpretation of size no greater than 2 k , where k is the # of monadic predicate letters in p . • In L2PL, some statements are true only on infinite interpretations. It is for this reason that there is no general decision procedure for validity (or logical truth) in L2PL. ( † ) on the last slide is a good example of this. ( † ) ( ∀ x)( ∃ y)Rxy,( ∀ x)( ∀ y)( ∀ z)[(Rxy & Ryz) → Rxz] ( ∃ x)Rxx • Fact . Only infinite interpretations I can be counterexamples to the validity in ( † ). To see why, try to construct such an interpretation. • We start by showing that no interpretation I 1 with a 1-element domain can be an interpretation on which the premises of ( † ) are > and its conclusion is ⊥ . Then, we will repeat this argument for I 2 and I 3 . • This reasoning can, in fact, be shown correct for all (finite) n . So, only I ’s with infinite domains will work [ e.g. , D = N , Rxy : x < y ]. • Begin with a 1-element domain { α } . For the conclusion of (4) to be ⊥ , no UCB Philosophy Chapters 7&8 (L2PL) 06/30/10 Branden Fitelson Philosophy 12A Notes 3 ' & $ % object can be related to itself: ( ∀ x) ∼ Rxx . Thus, we must have ∼ Raa : α • But, to make the first premise > , we need there to be some y such that Ray is > . That means we need another object β to allow Rab . Thus: α β • Now, because we need the conclusion to remain ⊥ , we must have ∼ Rbb . And, because we need the first premise to remain > , we need there to be some y such that Rby is > . We could try to make Rba > , as follows: α β UCB Philosophy Chapters 7&8 (L2PL) 06/30/10 Branden Fitelson Philosophy 12A Notes 4 ' & • But, this picture is not consistent with the second premise being > and (at the same time) the conclusion being ⊥ . If R is transitive, then Rab & Rba (as pictured) entails Raa , which makes the conclusion > ....
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This note was uploaded on 11/26/2011 for the course PHILOSOPHY 101 taught by Professor Buechner during the Fall '06 term at Rutgers.

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notes_15_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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