Logical constant
Encyclopedia
In logic
, a logical constant of a language
is a symbol
that has the same semantic value under every interpretation
of
. Two important types of logical constants are logical connective
s and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic
.
Some symbols that are commonly treated as logical constants are:
For many logical constants also other symbols are commonly used to denote them, such as the symbol "&" for logical and.
One of the fundamental questions in the philosophy of logic
is "What is a logical constant?"; that is, what special feature of certain constants that makes them logical in nature?
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, a logical constant of a language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
is a symbolSymbol (formal)
For other uses see Symbol In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, tokens of which may be marks or a configuration of marks which form a particular pattern...
that has the same semantic value under every interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
of
. Two important types of logical constants are logical connectiveLogical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
.
Some symbols that are commonly treated as logical constants are:
| Symbol | Meaning in English |
|---|---|
| T | "true" |
| F | "false" |
| ¬ | "not" |
| ∧ | "and" |
| ∨ | "or" |
| → | "implies", "if...then" |
| ∀ | "for all" |
| ∃ | "there exists", "for some" |
| = | "equals" |
 | "necessary" |
 | "possible" |
For many logical constants also other symbols are commonly used to denote them, such as the symbol "&" for logical and.
One of the fundamental questions in the philosophy of logic
Philosophy of logic
Following the developments in Formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply...
is "What is a logical constant?"; that is, what special feature of certain constants that makes them logical in nature?