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Chapter 9 Overview

Critical Thinking

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Chapter 9 Overview Despite the power and rigor of categorical logic, it lacks flexibility. Its 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 truth values of the individual claims determine the truth values of their compounds, and whether or not a given conclusion follows from a given set of premises. 1. Truth-functional logic is a precise and useful method for testing the validity of arguments. a. Also called propositional or sentential logic, truth-functional logic is the logic of sentences. b. It 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 truth-functional logic makes it a good introduction to nonmathematical symbolic systems. 2. The vocabulary of truth-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 truth value. i. We use T and F to represent the two possible truth values. ii. When the truth value of a claim is not known, we use a truth table to indicate all possibilities.
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iii. Thus, for a single variable P, we write: P T F c. Whatever truth 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 truth table: P ~P T F F T ii. This truth table is a definition of negation. iii. ~P is read "not-P." This is our first truth-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. ii. Accordingly, each symbol receives a precise definition with a truth table, and never deviates from that definition.
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