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Unformatted text preview: Chapter 9 Overview Despite the power and rigor of categorical logic, it lacks flexibility. I ts methods really only apply to syllogisms in which each of the two premises can be translated into a standard- form categorical claim. For this reason, even an introductory treatment of logic calls for some discussion of modern symbolic logic. Truth-functional or propositional logic is the simplest part of symbolic logic, though you will find it both rigorous enough to let you carry out systematic proofs and broad enough to handle a wide range of ordinary arguments. This chapter shows how to work with complex arrangements of individual sentences. You will use letters to represent sentences, and a few special symbols to represent the standard relations among sentences: roughly speaking, the relations that correspond to the English words "not," "and," "or," and "if-then." Truth tables and rules of proof show how the t ruth values of the individual claims determine the t ruth values of their compounds, and whether or not a given conclusion follows from a given set of premises. 1. T ruth-functional logic is a precise and useful method for testing the validity of arguments. a. Also called propositional or sentential logic, t ruth-functional logic is the logic of sentences. b. I t has applications as wide-ranging as set theory and the fundamental principles of computer science, as well as being useful for the examination of ordinary arguments. c. Finally, the precision of t ruth-functional logic makes it a good introduction to nonmathematical symbolic systems. 2. The vocabulary of t ruth-functional logic consists of claim variables and truth-functional symbols . a. Claim variables are capital letters that stand for claims. i. In categorical logic, we sometimes used capital letters to represent terms (nouns and noun phrases). Keep those distinct from the same capital letters that now represent whole sentences. ii. Each claim variable stands for a complete sentence. b. Every claim variable has a t ruth value. i. We use T and F to represent the two possible t ruth values. ii. When the truth value of a claim is not known, we use a t ruth table to indicate all possibilities. iii. Thus, for a single variable P, we write: P T F c. Whatever t ruth value a claim has, its negation (contradictory claim) has the opposite value. i. Using ~P to mean the negation of P, we produce the following t ruth table: P ~P T F F T ii. This t ruth table is a definition of negation. iii. ~P is read "not-P." This is our first t ruth-functional symbol. d. The remaining truth-functional symbols cover relations between two claims. i. Each symbol corresponds, more or less, to an ordinary English word; but you will find the symbols clearer and more rigid than their ordinary-language counterparts....
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This note was uploaded on 06/15/2011 for the course HUM 115 taught by Professor Miller during the Spring '11 term at Craven CC.
- Spring '11