Non-logical symbol
Encyclopedia
In logic
, the formal language
s used to create expressions consist of symbol
s 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).
The non-logical symbols of a language of first-order logic
consist of predicates and individual constants. These include symbols which, 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 connective
s, 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. Main article: first order logic especially Syntax of first-order logic
The logical constants, by contrast, 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 "for all" and "there exists".
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 nonlogical symbol, it may be interpreted by an arbitrary equivalence relation.
n (a natural number), or a relation symbol of a specific arity. The additional information controls how the non-logical 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 <.
language over it.
A structure over a signature consists of a set D, known as the domain of discourse
, together with interpretations of the non-logical symbols: Every constant symbol is interpreted by an element of D, and the interpretation of an n-ary function symbol is an n-ary function on D, i.e. a function Dn → D from the n-fold cartesian product
of the domain to the domain itself. Every n-ary relation symbol is interpreted by an n-ary relation on the domain, i.e. by a subset of Dn.
An example of a structure over the signature mentioned above is the ordered group of integer
s. Its domain is the set
= {…, –2, –1, 0, 1, 2, …} 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.
introduced a terminology distinguishing between logical and non-logical 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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s used to create expressions consist of symbol
Symbol (formal)
For other uses see Symbol In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, tokens of which may be marks or a configuration of marks which form a particular pattern...
s which can be broadly divided into constants and variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants).
The non-logical symbols of a language of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
consist of predicates and individual constants. These include symbols which, in an interpretation, may stand for individual constants, variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
, functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, 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 connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s, quantifiers, and variables that stand for statements
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...
.
A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
. Consequently, a sentence
Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...
containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be true or false under an interpretation. Main article: first order logic especially Syntax of first-order logic
The logical constants, by contrast, have the same meaning in all interpretations. They include the symbols for truth-functional connectives (such as and, or, not, implies, and logical equivalence
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
) and the symbols for the quantifiers "for all" and "there exists".
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 nonlogical symbol, it may be interpreted by an arbitrary equivalence relation.
Signatures
A signature is a set of non-logical constants together with additional information identifying each symbol as either a constant symbol, or a function symbol of a specific arityArity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
n (a natural number), or a relation symbol of a specific arity. The additional information controls how the non-logical 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 first-orderFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
language over it.
A structure over a signature consists of a set D, known as the domain of discourse
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...
, together with interpretations of the non-logical symbols: Every constant symbol is interpreted by an element of D, and the interpretation of an n-ary function symbol is an n-ary function on D, i.e. a function Dn → D from the n-fold cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the domain to the domain itself. Every n-ary relation symbol is interpreted by an n-ary relation on the domain, i.e. by a subset of Dn.
An example of a structure over the signature mentioned above is the ordered group of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. Its domain is the set
= {…, –2, –1, 0, 1, 2, …} 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 CarnapRudolf Carnap
Rudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
introduced a terminology distinguishing between logical and non-logical symbols (which he called descriptive signs) of a formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
under a certain type of interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some 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.
External links
- Semantics section in Classical Logic (an entry of Stanford Encyclopedia of Philosophy)