propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

, 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 Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

(''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 free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...

), 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 to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...

, ''A''(''x'') or ''A''(''x'', ''x'', ..., ''x''), the propositional function is abstracted from

predicates Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...

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 moon onto its surface. Lava may be erupted at a volcano or through a fracture in the crust, on land or un ...

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

ice Ice is water frozen into a solid state, typically forming at or below temperatures of 0 degrees Celsius or Depending on the presence of impurities such as particles of soil or bubbles of air, it can appear transparent or a more or less opaqu ...

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

set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...

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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...

wrote in ''

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

. 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 An open formula is a formula that contains at least one free variable. An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ' ...

''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 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 ...

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

relation Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...

s, for example, a propositional function of ''n'' variables is a relation of

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. In ...

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

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...

s, of which there are

homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''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 elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

References

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

Overview

See also

References