Non-logical symbol

https://en.wikipedia.org/wiki/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 also called logical and non-logical constants).

symbols:

• logical variables

• logical constants

  • logical symbols/constants

  • non-logical symbols/constants

    • predicates and individual constants (FOL)

Language of first-order logic

• logical symbols

  • logical connectives

  • quantifiers

  • logical variables

• non-logical symbols

  • logical constants

    • predicates

    • individual constants

    • variables

    • functions

The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates.

A language of first-order logic is a formal language over the alphabet consisting of its non-logical symbols and its logical symbols. The latter include logical connectives, quantifiers, and variables that stand for statements.

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.

The logical constants, by definition, have the same meaning in all interpretations.

They include the symbols for truth-functional connectives (such as "and", "or", "not", "implies", and logical equivalence) and the symbols for the quantifiers.

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.