# predicate calculus

predicate calculus
See functional calculus. Also called predicate logic.
[1945-50]

* * *

Part of modern symbolic logic which systematically exhibits the logical relations between propositions involving quantifiers such as "all" and "some.

" The predicate calculus usually builds on some form of the propositional calculus and introduces quantifiers, individual variables, and predicate letters. A sentence of the form "All F's are either G's or H's" is symbolically rendered as (∀x)[Fx ⊃ (Gx ∨ Hx)], and "Some F's are both G's and H's" is symbolically rendered as (∃x)[Fx ∧ (Gx ∧ Hx)]. Once conditions of truth and falsity for the basic types of propositions have been determined, the propositions formulable within the calculus are grouped into three mutually exclusive classes: (1) those that are true on every possible specification of the meaning of their predicate signs, such as "Everything is F or is not F"; (2) those false on every such specification, such as "Something is F and not F"; and (3) those true on some specifications and false on others, such as "Something is F and is G." These are called, respectively, the valid, inconsistent, and contingent propositions. Certain valid proposition types may be selected as axioms or as the basis for rules of inference. There exist multiple complete axiomatizations of first-order (or lower) predicate calculus ("first-order" meaning that quantifiers bind individual variables but not variables ranging over predicates of individuals). See also logic.

* * *

logic
also called  Logic Of Quantifiers,

that part of modern formal or symbolic logic which systematically exhibits the logical relations between sentences that hold purely in virtue of the manner in which predicates or noun expressions are distributed through ranges of subjects by means of quantifiers such as “all” and “some” without regard to the meanings or conceptual contents of any predicates in particular. Such predicates can include both qualities and relations; and, in a higher-order form called the functional calculus, it also includes functions, which are “framework” expressions with one or with several variables that acquire definite truth-values only when the variables are replaced by specific terms. The predicate calculus is to be distinguished from the propositional calculus, which deals with unanalyzed whole propositions related by connectives (such as “and,” “if . . . then,” and “or”).

The traditional syllogism is the most well-known sample of predicate logic, though it does not exhaust the subject. In such arguments as “All C are B and no B are A, so no C are A,” the truth of the two premises requires the truth of the conclusion in virtue of the manner in which the predicates B and A are distributed with reference to the classes specified by C and B, respectively. If, for example, the predicate A belonged to only one of the B's, the conclusion then could possibly be false—some C could be an A.

Modern symbolic logic (formal logic), of which the predicate calculus is a part, does not restrict itself, however, to the traditional syllogistic forms or to their symbolisms, a very large number of which have been devised. The predicate calculus usually builds upon some form of the propositional calculus. It then proceeds to give a classification of the sentence types that it contains or deals with, by reference to the different manners in which predicates may be distributed within sentences. It distinguishes, for example, the following two types of sentences: “All F's are either G's or H's,” and “Some F's are both G's and H's.” The conditions of truth and falsity in the basic sentence types are determined, and then a cross-classification is made that groups the sentences formulable within the calculus into three mutually exclusive classes—(1) those sentences that are true on every possible specification of the meaning of their predicate signs, as with “Everything is F or is not F”; (2) those false on every such specification, as with “Something is F and not F”; and (3) those true on some specifications and false on others, as with “Something is F and is G.” These are, respectively, the tautologous, inconsistent, and contingent sentences of the predicate calculus. Certain tautologous sentence types may be selected as axioms or as the basis for rules for transforming the symbols of the various sentence types; and rather routine and mechanical procedures may then be laid down for deciding whether given sentences are tautologous, inconsistent, or contingent—or whether and how given sentences are logically related to each other. Such procedures can be devised to decide the logical properties and relations of every sentence in any predicate calculus that does not contain predicates (functions) that range over predicates themselves—i.e., in any first-order, or lower, predicate calculus.

Calculi that do contain predicates ranging freely over predicates, on the other hand—called higher-order calculi—do not permit the classification of all their sentences by such routine procedures. As was proved by Kurt Gödel (Gödel, Kurt), a 20th-century Moravian-born American mathematical logician, these calculi, if consistent, always contain well-formed formulas such that neither they nor their negations can be derived (shown tautologous) by the rules of the calculus. Such calculi are, in the precise sense, incomplete. Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. See also propositional calculus.

* * *

Universalium. 2010.

### Look at other dictionaries:

• predicate calculus — The logical calculus in which the expressions include predicate letters, variables, and quantifiers, names, and operation letters, as well as the expressions for truth functions and the propositional variables of the propositional calculus . The… …   Philosophy dictionary

• predicate calculus — noun Date: 1950 the branch of symbolic logic that uses symbols for quantifiers and for arguments and predicates of propositions as well as for unanalyzed propositions and logical connectives called also functional calculus compare propositional… …   New Collegiate Dictionary

• predicate calculus — noun The branch of logic that deals with quantified statements such as there exists an x such that... or for any x, it is the case that... , where x is a member of the domain of discourse. See Also: predicate logic …   Wiktionary

• predicate calculus — pred′icate cal′culus n. pho functional calculus • Etymology: 1945–50 …   From formal English to slang

• predicate calculus — noun a system of symbolic logic that represents individuals and predicates and quantification over individuals (as well as the relations between propositions) • Syn: ↑functional calculus • Hypernyms: ↑symbolic logic, ↑mathematical logic, ↑formal… …   Useful english dictionary

• predicate calculus — noun the branch of symbolic logic concerned with propositions containing predicates, names, and quantifiers …   English new terms dictionary

• Monadic predicate calculus — In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. All atomic formulae have the form P(x), where P… …   Wikipedia

• Calculus (disambiguation) — Calculus is Latin for pebble, and has a number of meanings in English: In mathematics and computer science Calculus , in its most general sense, is any method or system of calculation. To modern theoreticians the answer to the question what is a… …   Wikipedia

• Predicate variable — In first order logic, a predicate variable is a predicate letter which can stand for a relation (between terms) but which has not been specifically assigned any particular relation (or meaning). In first order logic (FOL) they can be more… …   Wikipedia

• Predicate logic — In mathematical logic, predicate logic is the generic term for symbolic formal systems like first order logic, second order logic, many sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas… …   Wikipedia