Concept
Non-logical symbol
Overview
In mathematical logic, especially model theory, non-logical symbols are elements of a formal language whose interpretation may change depending on the model. In first-order logic, these usually consist of constant symbols, function symbols, and predicates. This is in contrast to logical constants which are required to have the same interpretation under every model, such as logical connectives and quantifiers.
A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be true or false under an interpretation. These concepts are defined and discussed in the article on first-order logic, and in particular the section on syntax.
The equality symbol is sometimes treated as a non-logical symbol and sometimes treated as a symbol of logic. If it is treated as a logical symbol, then any interpretation will be required to interpret the equality sign using true equality; if interpreted as a non-logical symbol, it may be interpreted by an arbitrary equivalence relation.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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The chief business of the logician is to classify arguments; for all testing clearly depends on classification. The classes of the logicians are defined by certain typical forms called syllogisms. For example, the syllogism called _Barbara_ is as follows: S is M; M is P: Hence, S is P. Or, to put words for letters— Enoch and Elijah were men; all men die: Hence, Enoch and Elijah must have died. The “is P” of the…
Charles S. Peirce, Chance, Love, and Logic
States what Non-logical symbol is.
[To say, that the rose is not red, implies that it is still coloured--in the first place with another colour; which however would be only one more positive judgment.] The individual however is not a universal. Hence (3) the judgment suffers disruption into one of two forms. It is either (a) the Identical judgment, an empty identical relation stating that the individual is the individual; or it is (b) what is called…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Non-logical symbol is.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Non-logical symbol is.
Related ideas
- Formal language
- First-order logic
- Interpretation (logic)
- Integer
- Mathematical logic
- Model theory
- Sentence (mathematical logic)
- Domain of discourse
- Predicate (logic)
- Formal system
- Arity
- Quantifier (logic)
- Rudolf Carnap
- Equivalence relation
- Function symbol
Place in the tradition
Part of First-order logic.