Predicate logic. In informal usage, the term "predicate logic" occasionally refers to first-order logic.
Some authors consider the predicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development.[2] Predicate logics also include logics mixing modal operators and quantifiers. See Modal logic, Saul Kripke, Barcan Marcus formulae, A. N. Prior, and Nicholas Rescher. See also[edit] [edit] Jump up ^ Eric M. References[edit] A. Propositional calculus. Usually in Truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false.
[clarification needed] Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic. History[edit] Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus[1] and expanded by the Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic which was focused on terms. Propositional logic was eventually refined using symbolic logic. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic. Terminology[edit] The language of a propositional calculus consists of , and , propositional variables by Basic concepts[edit]
Axiom schema. In mathematical logic, an axiom schema (plural: axiom schemata) generalizes the notion of axiom.
An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. Two very well known instances of axiom schemata are the: induction schema that is part of Peano's axioms for the arithmetic of the natural numbers;axiom schema of replacement that is part of the standard ZFC axiomatization of set theory. See also[edit] First-order logic.
A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.
Sometimes "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective "first-order" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted.[1] In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Introduction[edit] . X in .