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by: Jerald Stamm

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# Symbolic Logic PHIL 012

Jerald Stamm
Penn State
GPA 3.79

Staff

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## Popular in PHIL-Philosophy

This 0 page Class Notes was uploaded by Jerald Stamm on Sunday November 1, 2015. The Class Notes belongs to PHIL 012 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/233149/phil-012-pennsylvania-state-university in PHIL-Philosophy at Pennsylvania State University.

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Date Created: 11/01/15
Formal Proofs Subproofs and Assumptions PHIL 012 February 21 2001 Overview 0 Subproofs 39 Disjunction Elimination 39 Negation Introduction 0 Rules for Assumptions 0 Sample proofs Subproofs There are two circumstances so far when we will use a subproof o Disjunction Elimination Proof by cases 0 Negation Introduction reduclio ad absurdum Disjunction Elimination v Elim 0 You would use Disjunction Elimination when 1 You have a statement whose major connective is a disjunction 2 You are reasonably sure that what you are trying to prove either as a nal or an intermediate step follows from either side of the disjunction o In order to use Disjunction elimination you will introduce two subproofs 0 Each subproof begins with the assumption that one of the sides of the disjunction is true 0 Each subproof then utilizes that assumption together with any steps from the parent proof to establish the truth of the same statement 0 In order to make sure we use subproofs properly we use tch notion vertical and horizontal lines to indicate the scope of each subproof and to distinguish assumptions from other steps A V B A Assumption 0 Steps in subproof l O O P B Assumption 0 Steps in subproof 2 O O P Example Given P Q v R S prove P v R using v Elim v Intro 3 Elim 5 v Intro 6 Com v v Elim 1 24 58 Negation Introduction Intro You would use Negation Introduction when you want to prove a claim by showing that the assumption of its negation leads to a contradiction Proof by Negation Introduction involves the use of one subproof Your rst step is to introduce as an assumption the opposite of what you are trying to prove Then using steps derived from the assumption and possibly the parent proof you generate a contradiction previous steps in proof if any Assumption Steps in subproof A contradiction Intro QA Q Note that there is a more limited form of contradiction and a more general form The limited form requires a contradiction of the form Q Q which is the opposite of a tautology The more general involves a set of unsatisfiable statements such as Cubea Teta The limited form always will be false in every language as it is logically contradictory independent of meaning The more general form is a function of the meanings of the terms in the language Example Given V v W and V prove W v X 1 VVW 2 y 39 WvX 3 WvX 4 W X DeM3 5 W AElim 6 V W AIntro 25 7 1VVW DeM6 8VVW VVW AIntro 17 9 W v X Intro 38 Rules for Subproofs 0 You may not use a step of a subproof outside of its scope 0 Once a subproof is complete no steps apart from its conclusion including its assumption are usable 0 You may use steps from a parent proof at any time or the parent of a parent provided that the parent proofs are in the same scope Sample Proofs 1 G 2 H 39GHVI 3 GAH AIntrol2 4 GAHVI vIntro3 1 AABIVC 39 CvB 2AB 3B ZAElim 4BvC 3vIntro 5CvB 4Comv l7 CvB 6AIntro 8 CvB vElim12567

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