notes_12_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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Unformatted text preview: Branden Fitelson Philosophy 12A Notes 1 ' & $ % Announcements and Such Todays Music: Big Star I have posted my solutions to HW #4 (with the shortest proofs I know). Ive posted HW #5, which is due on Thursday @ 4pm. A few LMPL symbolization problems (chapter 5). Mostly, working with LMPL Interpretations (chapter 6). + Ive 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. LMPL-validity 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 + +- + - --- + + +--- + So, I 1 is such that: D = { ,, } , Ext (F) = { , } , Ext (G) = { } , Ext (H) = ( is the null set ), Ext (I) = { , } , and Ext (J) = { , } . What are the I-truth-values 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 HWs 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 , truth-table procedures for proving both negative ( ) and affirmative ( ) semantical claims....
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notes_12_2x2 - Branden Fitelson Philosophy 12A Notes 1 '...

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