propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

, a propositional function or a predicate is a sentence expressed in a way that would assume the value of

true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...

or false, except that within the sentence there is a variable (''x'') that is not defined or specified (thus being a

free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...

), which leaves the statement undetermined. The sentence may contain several such variables (e.g. ''n'' variables, in which case the function takes ''n'' arguments).

Overview

As a

mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ...

, ''A''(''x'') or ''A''(''x'', ''x'', ..., ''x''), the propositional function is abstracted from predicates or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for ''x'' will produce a specific proposition that can be described as either true or false, even though "''x'' is hot" on its own has no value as either a true or false statement. However, when a value is assigned to ''x'', such as

lava Lava is molten or partially molten rock (magma) that has been expelled from the interior of a terrestrial planet (such as Earth) or a Natural satellite, moon onto its surface. Lava may be erupted at a volcano or through a Fissure vent, fractu ...

, the function then has the value ''true''; while one assigns to ''x'' a value like

ice Ice is water that is frozen into a solid state, typically forming at or below temperatures of 0 ° C, 32 ° F, or 273.15 K. It occurs naturally on Earth, on other planets, in Oort cloud objects, and as interstellar ice. As a naturally oc ...

, the function then has the value ''false''. Propositional functions are useful in

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

for the formation of sets. For example, in 1903

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...

wrote in ''

The Principles of Mathematics ''The Principles of Mathematics'' (''PoM'') is a 1903 book by Bertrand Russell, in which the author presented Russell's paradox, his famous paradox and argued his thesis that mathematics and logic are identical. The book presents a view of ...

'' (page 106): :"...it has become necessary to take ''propositional function'' as a

primitive notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to Intuition (knowledge), intuition or taken ...

. Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types. A Propositional Function, or a predicate, in a variable ''x'' is an open formula ''p''(''x'') involving ''x'' that becomes a proposition when one gives ''x'' a definite value from the set of values it can take. According to Clarence Lewis, "A

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

is any expression which is either true or false; a propositional function is an expression, containing one or more variables, which becomes a proposition when each of the variables is replaced by some one of its values from a discourse domain of individuals." Clarence Lewis (1918) ''A Survey of Symbolic Logic'', page 232,

University of California Press The University of California Press, otherwise known as UC Press, is a publishing house associated with the University of California that engages in academic publishing. It was founded in 1893 to publish scholarly and scientific works by faculty ...

, second edition 1932, Dover edition 1960 Lewis used the notion of propositional functions to introduce relations, for example, a propositional function of ''n'' variables is a relation of

arity In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...

''n''. The case of ''n'' = 2 corresponds to

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

s, of which there are

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...

s (both variables from the same set) and

heterogeneous relation In mathematics, a binary relation associates some elements of one set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x, y), where x i ...

References

{{reflist Functions and mappings Mathematical relations Concepts in logic Predicate logic Logical expressions

Overview

See also

References