In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, a logical constant of a
language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
is a
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
that has the same
semantic value under every
interpretation of
. Two important types of logical constants are
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
s and
quantifiers. The
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
predicate (usually written '=') is also treated as a logical constant in many
systems of logic.
One of the fundamental questions in the
philosophy of logic
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application ...
is "What is a logical constant?"; that is, what special feature of certain constants makes them ''logical'' in nature?
Some symbols that are commonly treated as logical constants are:
Many of these logical constants are sometimes denoted by alternate symbols (''e.g.'', the use of the symbol "&" rather than "∧" to denote the
logical and
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...
).
Defining logical constants is a major part of the work of
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
and
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, ...
. Russell returned to the subject of logical constants in the preface to the second edition (1937) of ''
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 ...
'' noting that logic becomes linguistic: "If we are to say anything definite about them,
hey
Hey or Hey! may refer to:
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* Hey (band), a Polish rock band
Albums
* ''Hey'' (Andreas Bourani album) or the title song (see below), 2014
* ''Hey!'' (Julio Iglesias album) or the title song, 1980
* ''Hey!'' (Jullie album) or the title s ...
must be treated as part of the language, not as part of what the language speaks about."
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, ...
(1937) Preface to ''The Principles of Mathematics'', pages ix to xi The text of this book uses
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 R, their
converses
Chuck Taylor All-Stars or Converse All Stars (also referred to as "Converse", "Chuck Taylors", "Chucks", "Cons", "All Stars", and "Chucky Ts") is a model of casual shoe manufactured by Converse (a subsidiary of Nike, Inc. since 2003) that was i ...
and
complements as
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 ...
s, also taken as logical constants in the form aRb.
See also
*
Non-logical symbol
In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes a ...
*
Logical value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false'').
Computing
In some progra ...
*
Logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
References
External links
Stanford Encyclopedia of Philosophy entry on logical constants
Philosophical logic
Concepts in logic
Syntax (logic)
Logical truth
Logic symbols
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