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 nonlogical symbols (sometimes also called logical and nonlogical constants).
The nonlogical symbols of a language of firstorder 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 firstorder logic is a formal language over the alphabet consisting of its nonlogical symbols and its logical symbols. The latter include logical connectives, quantifiers, and variables that stand for statements.
A nonlogical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence containing a nonlogical 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 firstorder logic, and in particular the section on syntax.
The logical constants, by contrast, have the same meaning in all interpretations. They include the symbols for truthfunctional connectives (such as "and", "or", "not", "implies", and logical equivalence) and the symbols for the quantifiers "for all" and "there exists".
The equality symbol is sometimes treated as a nonlogical 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 nonlogical symbol, it may be interpreted by an arbitrary equivalence relation.
YouTube Encyclopedic

1/3Views:11 76131 160845

The OTHER HALF of Japanese Structure  nonlogical topic/comment structure  Lesson 60

Sirius Andrew Wilkow vs Non logical liberal on health insurance

ModelTheoretic Semantics
Transcription
Signatures
A signature is a set of nonlogical constants together with additional information identifying each symbol as either a constant symbol, or a function symbol of a specific arity n (a natural number), or a relation symbol of a specific arity. The additional information controls how the nonlogical symbols can be used to form terms and formulas. For instance if f is a binary function symbol and c is a constant symbol, then f(x, c) is a term, but c(x, f) is not a term. Relation symbols cannot be used in terms, but they can be used to combine one or more (depending on the arity) terms into an atomic formula.
For example a signature could consist of a binary function symbol +, a constant symbol 0, and a binary relation symbol <.
Models
Structures over a signature, also known as models, provide formal semantics to a signature and the firstorder language over it.
A structure over a signature consists of a set (known as the domain of discourse) together with interpretations of the nonlogical symbols: Every constant symbol is interpreted by an element of and the interpretation of an ary function symbol is an ary function on that is, a function from the fold cartesian product of the domain to the domain itself. Every ary relation symbol is interpreted by an ary relation on the domain; that is, by a subset of
An example of a structure over the signature mentioned above is the ordered group of integers. Its domain is the set of integers. The binary function symbol is interpreted by addition, the constant symbol 0 by the additive identity, and the binary relation symbol < by the relation less than.
Informal semantics
Outside a mathematical context, it is often more appropriate to work with more informal interpretations.
Descriptive signs
Rudolf Carnap introduced a terminology distinguishing between logical and nonlogical symbols (which he called descriptive signs) of a formal system under a certain type of interpretation, defined by what they describe in the world.
A descriptive sign is defined as any symbol of a formal language which designates things or processes in the world, or properties or relations of things. This is in contrast to logical signs which do not designate any thing in the world of objects. The use of logical signs is determined by the logical rules of the language, whereas meaning is arbitrarily attached to descriptive signs when they are applied to a given domain of individuals.^{[1]}
See also
References
 ^ Carnap, Rudolf (1958). Introduction to symbolic logic and its applications. New York: Dover.
 Notes
 Hinman, P. (2005), Fundamentals of Mathematical Logic, A K Peters, ISBN 9781568812625
External links
 Semantics section in Classical Logic (an entry of Stanford Encyclopedia of Philosophy)