Predicate variable

TheInfoList

OR:

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capital roman letters such as $P$, $Q$ and $R$, or lower case roman letters, e.g., $x$. In
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, they can be more properly called metalinguistic variables. In
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...
, predicate variables correspond to
propositional variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propos ...
s which can stand for
well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...
s of the same logic, and such variables can be quantified by means of (at least) second-order quantifiers.

# Notation

Predicate variables should be distinguished from predicate constants, which could be represented either with a different (exclusive) set of predicate letters, or by their own symbols which really do have their own specific meaning in their
domain of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...
: e.g. $=, \ \in , \ \le,\ <, \ \sub,...$. If letters are used for both predicate constants and predicate variables, then there must be a way of distinguishing between them. One possibility is to use letters ''W'', ''X'', ''Y'', ''Z'' to represent predicate variables and letters ''A'', ''B'', ''C'',..., ''U'', ''V'' to represent predicate constants. If these letters are not enough, then numerical subscripts can be appended after the letter in question (as in ''X''1, ''X''2, ''X''3). Another option is to use Greek lower-case letters to represent such metavariable predicates. Then, such letters could be used to represent entire well-formed formulae (wff) of the predicate calculus: any free variable terms of the wff could be incorporated as terms of the Greek-letter predicate. This is the first step towards creating a higher-order logic.

# Usage

If the predicate variables are not defined as belonging to the vocabulary of the predicate calculus, then they are predicate metavariables, whereas the rest of the predicates are just called "predicate letters". The metavariables are thus understood to be used to code for axiom schema and theorem schemata (derived from the axiom schemata). Whether the "predicate letters" are constants or variables is a subtle point: they are not constants in the same sense that $=, \ \in , \ \le,\ <, \ \sub,$ are predicate constants, or that $1,\ 2,\ 3,\ \sqrt,\ \pi,\ e\$ are numerical constants. If "predicate variables" are only allowed to be bound to predicate letters of zero
arity Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics ...
(which have no arguments), where such letters represent
propositions In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
, then such variables are '' propositional variables'', and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate calculus, or
second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
. If predicate variables are also allowed to be bound to predicate letters which are unary or have higher arity, and when such letters represent '' propositional functions'', such that the domain of the arguments is mapped to a range of different propositions, and when such variables can be bound by quantifiers to such sets of propositions, then the result is a higher-order predicate calculus, or
higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...
.