Symbolic logic

Symbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical logic. Second, the rules for manipulating symbols found in symbolic logic can be implemented on a computing machine.

Symbolic logic is usually divided into two subfields, propositional logic and predicate logic. Other logics of interest include temporal logic, modal logic and fuzzy logic. See also model theory.

Modern mathematical areas arising out of formal logic are grouped under the heading mathematical logic.

Propositional logic

The area of symbolic logic called propositional logic, originally called propositional calculus but not to be confused with the school subject calculus, studies the properties of sentences formed from constants, usually designated A, B, C, ... and five logical operators, AND, OR, IMPLIES, EQUALS and NOT. The corresponding logical operations are known, respectively, as conjunction, disjunction, material conditional, biconditional, and negation. These five operators are sometimes denoted as keywords, especially in computer languages, and sometimes by special symbols (see Table of logic symbols). All except NOT are binary operators; NOT is a unary operator which precedes its operand. The values of these operators are given by truth tables.

Predicate logic

Predicate logic, originally called predicate calculus, expands on propositional logic by the introduction of variables, usually denoted by x, y, z, or other lowercase letters, and also by the introduction of sentences containing variables, called predicates, usually denoted by an uppercase letter followed by a list of variables, such as P(x) or Q(y,z). In addition, predicate logic allows so-called quantifiers, representing ALL and EXISTS.

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