Constant (logic)

In general, a constant (from Latin constans "fixed") is a sign or a language expression with a "precisely defined meaning that remains unchanged in the course of the considerations". The constant is thus a counter-term to the variable .

Logical constants

Logical constants or logical particles are signs or expressions that determine the logical structure of statements . The two statements “It is not the case that it is raining” and “It is not the case that the earth is a cube” have the same syntactic and semantic structure - they are negations. The language expression "It is not the case that ..." is the logical constant in these two structurally identical statements.

As logical constants, expressions for the negation (for example the formulation "It is not the case that ..."), the logical conjunction ("... and ..."), the logical disjunction (".. .or ... "), the logical conditional and other statement links and expressions for the quantifiers (" all "," any / r "...) of the first predicate logic stage . While it is also undisputed that expressions like “earth” or “it's raining” are not constants, there is a very wide range between these extremes that is the subject of investigation and leaves room for many different opinions. For example, the status of expressions such as “true” or “... is an element of ...” and of higher-level quantifiers (“there is a predicate for which ...”) is controversial.

The distinction between logical constants and logical variables within artificial languages is less problematic when these are interpreted , that is, when formal semantics are given for them. A commonly used definition was suggested by Christopher Peacocke in 1976 :

“A is a logical constant if it is not composed and if, for each argument sequence to which a is applied, knowledge of the fulfillment conditions of the individual elements of this argument sequence (as well as knowledge of the fulfillment conditions of the formal composition of expressions of the syntactic category of Sequence of arguments using a) is sufficient to be able to know a priori which sequences fulfill the overall expression of the corresponding syntactic category formed using a, or which extension assigns each given sequence to this expression, without having to identify the properties and relationships of the corresponding elements of the incoming individual sequences themselves knows."

- Particles, logical, in: Historical Dictionary of Philosophy , Volume 7, page 152

In this sense, the logical constants are the propositional logic , the connectives ; those of the first-level predicate logic, the first-level quantifiers and the connectors; those of modal logic use modal expressions such as "it is necessary that ..." and "it is possible that ...".

Non-logical constants

In predicate logic , in addition to the above-mentioned logical constants, one also considers other non-logical symbols that are necessary for the formulation of mathematical facts, and thus comes to a language supplemented by these symbols. Constant symbols, function symbols and relation symbols come into question as non-logical symbols . The constant symbols are distinguished from the other non-logical symbols in that they can appear in the same places as the variables in the term structure.

A typical example is the set of symbols that can be used to formulate ring theory . We have two constant symbols 0 and 1, whose intended interpretation is the zero and one element of a ring, and two function symbols that stand for addition and multiplication. These constants can be used to build terms and equations. So means about ${\ displaystyle \ {0,1, +, \ cdot \}}$

{\ displaystyle 1+ \ ldots + 1 = 0}

, whereby ones should be added here, prime number ,

{\ displaystyle p}

{\ displaystyle p}

that the ring has the characteristic . If one adds this statement made up of constants about the set of ring axioms, one arrives at the theory of rings with characteristics . ${\ displaystyle p}$ ${\ displaystyle p}$

An important proof procedure is the so-called constant expansion . In doing so, a considered language is expanded by a set of new constants in order to have a sufficient number of them available in the expanded language for purposes of proof.

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^ Tarski, Introduction to Mathematical Logic, 5th ed. (1977), p. 17
↑ This paragraph follows particularly closely: John MacFarlane: Logical Constants. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
↑ Christopher Peacocke, “What Is a Logical Constant?” Journal of Philosophy 73 (1976), pages 221-240
↑ Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to Mathematical Logic , Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1996, ISBN 3-8274-0130-5 , Chapter II, Definition 2.1
↑ Wolfgang Rautenberg : Introduction to Mathematical Logic. A textbook. 3rd, revised edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0578-2 , section 3.2, p. 76 , doi : 10.1007 / 978-3-8348-9530-1 ( springer.com ).

literature

Christopher Peacocke, “What Is a Logical Constant?” Journal of Philosophy 73 (1976), 221-240

Web links

John MacFarlane: Logical Constants. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

[1] Tarski, Introduction to Mathematical Logic, 5th ed. (1977), p. 17

[2] This paragraph follows particularly closely: John MacFarlane: Logical Constants. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

[3] Christopher Peacocke, “What Is a Logical Constant?” Journal of Philosophy 73 (1976), pages 221-240

[4] Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to Mathematical Logic , Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1996, ISBN 3-8274-0130-5 , Chapter II, Definition 2.1

[5] Wolfgang Rautenberg : Introduction to Mathematical Logic. A textbook. 3rd, revised edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0578-2 , section 3.2, p. 76 , doi : 10.1007 / 978-3-8348-9530-1 ( springer.com ).