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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements and Such • Today’s Music: Big Star • I have posted my solutions to HW #4 (with the shortest proofs I know). • I’ve posted HW #5, which is due on Thursday @ 4pm. – A few LMPL symbolization problems (chapter 5). – Mostly, working with LMPL Interpretations (chapter 6). + I’ve posted a new handout entitled “Working with LMPL Interpretations”, which I will be going over in class today. • Today: Chapter 6 — LMPL Semantics – Working with given LMPL interpretations. – Constructing LMPL interpretations to establish claims. – LMPLvalidity is decidable (but infeasible). – Next Topic: Natural deduction proofs in LMPL. UCB Philosophy Chapter 6 06/22/10 Branden Fitelson Philosophy 12A Notes 2 ' & Working with LMPL Interpretations (Handout: Part I) • Consider the following LMPL interpretation: ( I 1 ) F G H I J α + + +  β + + γ + + • So, I 1 is such that: D = { α,β,γ } , Ext (F) = { α,γ } , Ext (G) = { α } , Ext (H) = ∅ ( ∅ is the null set ), Ext (I) = { α,β } , and Ext (J) = { β,γ } . • What are the Itruthvalues of the following LMPL sentences? (5) ∼ Ja (8) ( ∀ x)[Jx → (Gx ∨ Fx)] (6) Fc → Ic (9) ( ∃ x)Gx → ( ∀ y)(Fy ∨ Gy) (7) ( ∃ x)(Jx ↔ Hx) (10) ( ∃ y)( ∀ x)[Gy & (Jx → (Ix ∨ Fx))] • These are solved on page 1 of my “Working with LMPL Interpretations”. UCB Philosophy Chapter 6 06/22/10 Branden Fitelson Philosophy 12A Notes 3 ' & $ % Constructing LMPL Interpretations to Prove Claims • The notion of semantic consequence ( ) in LMPL is defined in the usual way. We say that p 1 ,...,p n q in LMPL iff there is no LMPL interpretation on which all of p 1 ,...,p n are true, but q is false. • In HW #5, you are asked to prove that p 1 ,...,p n q , for various p ’s and q ’s. This means you must construct (or, find ) LMPL interpretations on which p 1 ,...,p n are all true, but q is false. • On page 2 of my “Working with LMPL Interpretations” handout, I have included two problems of this kind. There, I explain in detail how I arrived at my interpretations. This is a method you should emulate. • On your HW’s and exams, you will not need to explain how you arrived at your interpretations. But, you will need to demonstrate that your interpretations really are counterexamples ( i.e. , that they really are interpretations on which p 1 ,...,p n are all true, but q is false). UCB Philosophy Chapter 6 06/22/10 Branden Fitelson Philosophy 12A Notes 4 ' & Digression: How Do We Prove Claims in LMPL? • In LSL, we had systematic , truthtable procedures for proving both negative ( ) and affirmative ( ) semantical claims....
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 Fall '06
 Buechner
 Philosophy, Logic, Trigraph, Branden Fitelson, Philosophy 12A Notes

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