Symbolic logic
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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 branch of mathematics calculus, studies the properties of sentences formed from constants, usually designated p, q, r, ... and five logical operators, AND, OR, IF...THEN, IF AND ONLY IF 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. It is to be noted that capital letters are usually used to represent propositions groups in metalanguage. For example, A & B is a metalinguistic way of conjoining either individual or grouped propositions. A & B could represent p & q, but it could also very well represent (p & Q & [NOT]r) & (p & r). Syntactic rules will always be formulated in metalanguage.
The Sheffer operator, which parallels the nand operator in computer language, is also used to mark logical incompatibility or binary inconsistency. The neither...nor operator is also commonplace, as well as the exclusive disjunction. The sheffer operator is commonly represented as |, while the exclusive Or and the neither...nor operators are represented by the w sign and a downwards arrow respectively.
All complex connectors (conditional, biconditional, sheffer, neither...nor) can be reduced to the three fundamental connectors which are the conjunction, disjunction and negation (Not). For example, the exclusive disjunction can be symbolized as an inclusive disjunction conjoined with the negation of the conjoint elements. Formally, this gives:
( p [exclusive OR] q ) is equivalent to ( [ p [inclusive OR] q] & [NOT (p & q)] )
It would seem that this aspect of symbolic logic is what makes it more enlightening concerning the basic operations of the mind than the old Aristotelian logical axioms; modus ponens and syllogism can in fact be represented more fundamentally by the properties of the three basic connectors of symbolic logic. The material conditional, which operates the link between premisses and conclusions in syllogisms, is reducible to more basic operations and is in fact a composite of the mind's basic operations, a synthesis appearing intuitively elementary and fundamental.
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. qp s
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