Logical constant
General is a constant (from Latin constans - fixed) a sign or a language expression with a " well-defined [n ] meaning, which remains unchanged in the course of the considerations ." The constant is thus an antonym of variables.
Logical constants
Logical constants or logical particles are signs or expressions that define the logical structure of statements. Thus the two statements "It is not the case that it is raining " and " It is not the case, that the earth is a cube " has the same syntactic and semantic structure - are negations. The language expression "It is not the case, dass .. " is in these two structurally identical statements, the logical constants.
As a logical constants are undisputed expressions for the negation (for example, the phrase " It is not the case, dass .. " ), the logical conjunction ( " ... and ..." ), the logical disjunction ( " ... , or ... "), the logical conditional statement and other links as well as expressions for the quantifiers ( " all "," every / r ", ...) of the first-order predicate logic. While it is also undisputed that terms like "earth " or " it's raining " are not constants, there are between these extremes, a very broad range, which is the subject of investigations and provides room for many different opinions. There is disagreement as to the status of expressions such as " true" or " ... is an element of ..." and quantifiers of a higher level ( "there is a predicate that is true of the ...").
Less of a problem to distinguish between logical and logical variables is constant within artificial languages if they are interpreted, i.e., if they for formal semantics are specified. A frequently used definition was proposed by Christopher Peacocke 1976:
"A is a logical constant, if it is not assembled and when it is applied to each argument sequence to which a, the knowledge about the fulfillment of conditions of the individual elements of this argument sequence (as well as the knowledge about the fulfillment of conditions of the formal composition of expressions of the syntactic category of argument sequences by means of a) is sufficient to know a priori to what consequences meet the formed by a total expression of the corresponding syntactic category or which extension any given sequence assigns this expression without having the properties and relations of the corresponding elements of the incoming individual episodes themselves knows. "
In this sense, the logical constants of propositional logic, the connectives; those of the first stage of predicate logic quantifiers of first order and the connectives; those of the modal logic Modalausdrücke such as " it is necessary, dass .. " and " it is possible, dass .. ".
Non- logical constants
In QL, we consider in addition to the above logical constants further non- logical symbols that are required to formulate mathematical problems, and thus comes to added to these symbols language. As a non - logical symbols come here constant symbols, function symbols and relation symbols in question. The constant symbols are distinguished from the other non-logical symbols by the fact that they can occur as the variables in the term structure to the same places.
A typical example is the set of symbols that can be used to formulate the ring theory. We have here two constant symbols 0 and 1, the intended interpretation that are zero and the unity element of a ring, and two function symbols that stand for addition and multiplication. By means of these constants terms and equations can be constructed. So does about
That the ring has the characteristic. Assuming this statement is constructed from constants to the amount of ring axioms added, one comes to the theory of rings with characteristic.
An important method of proof is the so-called constant expansion. It expanded to a considered language to a lot of new constants to many of them to have sufficient for evidentiary purposes in the so- extended language.