symbolic logic: Truth-functional Analysis

The first part of symbolic logic is known as truth-functional analysis, the propositional calculus, or the sentential calculus; it deals with statements that can be assigned truth values (true or false). Combinations of these statements are called truth functions, and their truth values can be determined from the truth values of their components.

The basic connectives in truth-functional analysis are usually negation, conjunction, and alternation. The negation of a statement is false if the original statement is true and true if the original statement is false; negation corresponds to “it is not the case that,” or simply “not” in ordinary language. The conjunction of two statements is true only if both are true; it is false in all other instances. Conjunction corresponds to “and” in ordinary language. The alternation, or disjunction, of two statements is false only if both are false and is true in all other instances; alternation corresponds to the nonexclusive sense of “or” in ordinary language (Lat. vel), as opposed to the exclusive “either … or … but not both” (Lat. aut).

Other connectives commonly used in truth-functional analysis are the conditional and the biconditional. The conditional, or implication, corresponds to “if … then” or “implies” in ordinary language, but only in a weak sense. The conditional is false only if the antecedent is true and the consequent is false; it is true in all other instances. This kind of implication, in which the connection between the antecedent and the consequent is merely formal, is known as material implication. The biconditional, or double implication, is the equivalence relation and is true only if the two statements have the same truth value, either true or false. In any truth function one may substitute an equivalent expression for all or any part of the function. The validity of arguments may be analyzed by assigning all possible combinations of truth values to the component statements; such an array of truth values is called a truth table.

Sections in this article:

• Introduction
• Analysis of the Foundations of Mathematics
• The Predicate Calculus
• Truth-functional Analysis
• Bibliography

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