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In mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
, the formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
s, formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
s, and theories
Theory (mathematical logic)

In mathematical logic, a theory is a set of sentence s in a formal language. For example, a first-order theory is a set of first-order logic sentences....
 which are studied have no meaningful content until they are given an interpretation within some other system. An interpretation is a semantic
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
 concept which consists in a correlation or assignment
Valuation (logic)

In logic and model theory, a valuation can be:*In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables....
 of meaning
Meaning

Meaning may refer to:...
s to the symbol
Symbol

A symbol is something such as an entity, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention....
s of a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
. The study of interpretations expressed in formal languages is called formal semantics. Giving an interpretation is synonymous with constructing a model
Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of an underlying Set along with a collection of finitary functions and relations which are defined on it....
.

When the empirical sciences attempt to axiomatize
Axiomatic system

In mathematics, an axiomatic system is any Set of axioms from which some or all axioms can be used in conjunction to logically derive theorems....
 the principles governing the subjects they study, they are creating a formal system for which reality
Reality

Reality, in everyday usage, means "the state of things as they actually exist". In a sense it is what is real. The term reality, in its widest sense, includes everything that being, whether or not it is observation or comprehension....
 is the only interpretation.






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In mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
, the formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
s, formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
s, and theories
Theory (mathematical logic)

In mathematical logic, a theory is a set of sentence s in a formal language. For example, a first-order theory is a set of first-order logic sentences....
 which are studied have no meaningful content until they are given an interpretation within some other system. An interpretation is a semantic
Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
 concept which consists in a correlation or assignment
Valuation (logic)

In logic and model theory, a valuation can be:*In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables....
 of meaning
Meaning

Meaning may refer to:...
s to the symbol
Symbol

A symbol is something such as an entity, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention....
s of a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
. The study of interpretations expressed in formal languages is called formal semantics. Giving an interpretation is synonymous with constructing a model
Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of an underlying Set along with a collection of finitary functions and relations which are defined on it....
.

When the empirical sciences attempt to axiomatize
Axiomatic system

In mathematics, an axiomatic system is any Set of axioms from which some or all axioms can be used in conjunction to logically derive theorems....
 the principles governing the subjects they study, they are creating a formal system for which reality
Reality

Reality, in everyday usage, means "the state of things as they actually exist". In a sense it is what is real. The term reality, in its widest sense, includes everything that being, whether or not it is observation or comprehension....
 is the only interpretation. The world is an interpretation (or model) of these sciences, only insofar as these sciences are true.

Interpretation of a formal language


Interpretations are expressed in a metalanguage
Metalanguage

In logic and linguistics, a metalanguage is a language used to make statements about statements in another language which is called the object language....
 which is talking about some object language
Object language

Object language has meaning in contexts of computer programming and operation, and in linguistics and logic....
, which is usually some formal language. A formal language is an organized set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of symbol
Symbol

A symbol is something such as an entity, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention....
s the essential feature of which is that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference
Reference

A reference is a relation between Object in which one object designates by linking to another object. Such relations as these may occur in a variety of domains, including logic, computer science, time, art and scholarship....
 to any meaning
Meaning (linguistics)

Linguistic strings can be made up of phenomena such as words, phrases, and sentences, each of which has a different kind of meaning. Individual words, such as the word "bachelor", refer to some abstract concept....
s of any of its expressions; it can exist before any formal interpretation is assigned to it—that is, before it has any meaning.

A formal language can be defined formally as a set A of strings (finite sequences) composed exclusively from a fixed alphabet a. It may be defned in terms of an ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
 A>. Carnap also requires that each element of a must occur in at least one string in A.

The class
Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of Set which can be unambiguously defined by a property that all its members share....
 A of the expressions of is defined as the class of all finite sequences whose members
Element (mathematics)

In mathematics, an element or member of a Set is any one of the distinct objects that make up that set....
 are the elements of the class a.


An n-place sequence
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
 can be defined as a function
Function

Function may refer to* Function , explaining why a feature survived selection* Function , an abstract entity that associates an input to a corresponding output according to some rule...
 between the n first natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s and the members of the sequence. A syntactic
Syntax

In linguistics, syntax is the study of the principles and rules for constructing Sentence s in natural languages. In addition to referring to the discipline, the term syntax is also used to refer directly to the rules and principles that govern the sentence structure of any individual language, as in "the Irish syntax"....
 axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
 may be adopted that states: For any class a and any class , if is a formal language then every element of is a finite sequence of elements of a, and every element of a occurs as a member of some element of .


An interpretation of a formal language can defined as the ordered triple . The first domain
Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
 of the relation is identical with the class .

If an extension
Extension

Extension may refer to:* A List of cheerleading stunts* The building of community capacity by outsiders, for instance agricultural extension* Extension , relating to the pulling apart of the Earth's crust and lithosphere...
al metalanguage
Metalanguage

In logic and linguistics, a metalanguage is a language used to make statements about statements in another language which is called the object language....
 is used for semantics, then is the relation of value assignment for the sentences
Sentence (mathematical logic)

In mathematical logic, a sentence of a predicate logic is a well formed formula with no free variables. A sentence is viewed by some as expressing a proposition....
 of the language.

For example, "(1,grass is green)" means the same as "The sentence 1 is true if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 grass is green." For any p and q and any element 1 of the class , if (1,p) and (1,q) then p if and only if q.

If on the other hand, an intension
Intension

Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
al metalanguage, containing a modal operator
Modal operator

In modal logic, a modal operator is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth function, and is "intuitively" characterised by expressing a modal attitude about the proposition to which the operator is applied....
, such as "it is necessary that", then is taken as the relation of designation, That is, the relation between an expression and its intension.

For example, "(1,grass is green)" means the same as "The sentence 1 designates the proposition that grass is green." For any p and q and any element 1 of the class , if (1,p) and (1,q) then p and q are identical, i.e it is logically necessary that p if and only if q.

In either of these two metalanguages extensional, or intensional, truth
Truth

semantic fields for the word truth extend from honesty, good faith, and sincerity in general, to agreement with fact or reality in particular....
 with respect to any given interpreted language can be defined as follows: A sentence 1 is true if and only if for some p, (1,p), and p.


There is another method applicable to either of these two metalanguages which takes the relation as applying not only to sentences but to a more comprehensive class d of designators. By this method, an interpreted formal language is an ordered quadruple .
In these metalanguages, d is the class of finite sequences of elements of the class a, the class of the first place members of is the class d, and that is a subclass of d.


There is also a third method, which is more explicit, which demands that in order to specify an interpreted formal language a class ds of descriptive signs of the language must be indicated. In this method, an interpreted formal language can be defined as the ordered quintuple
Using this method, ds is a subclass of a. This most explicit method is convenient as a basis for definitions of concepts such as "model
Mathematical model

A mathematical model uses mathematics language to describe a system. Mathematical models are used not only in the natural sciences and engineering disciplines but also in the social sciences ; physicists, engineers, computer sciences, and economists use mathematical models most extensively....
", "value assignment", "range
Range (mathematics)

In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
 of a sentence", "logical truth", and other logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
al concepts.


A simple example


The formal language is defined as follows:
Alphabet α :
Formal grammar : Any finite string of symbols from the alphabet of that begins with a is a formula.


A possible interpretation of would be to take as meaning the same as the decimal digit '1', as meaning the same as the digit '0', and each formula as meaning the same as a decimal numeral composed exclusively of '1's and '0's. Therefore ' ' means '101' under this interpretation of .

Interpretation of logical connectives


The symbols of a language which represent the logical connective
Logical connective

In logic, two sentences may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences....
s can be interpreted as operations on truth values of sentences. The interpretation of the logical connectives as logical operations can be expressed in terms of truth tables.

 Interpretation of logical connectives
F ? F (F ?) (F ?) (F ?) (F ?)
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T


Interpretation of a theory


An interpretation of a theory is the relationship between a theory and some contensive subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain contensive statements related to the subject matter. If every elementary statement in the theory has a contensive correspondent it is called a full interpretation, otherwise it is called a partial interpretation.

Interpretation of a formal system


A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s, or have both. A formal system is used to derive
Proof theory

Proof theory is a branch of mathematical logic that represents Mathematical proofs as formal mathematical objects, facilitating their analysis by mathematical techniques....
 one expression from one or more other expressions.

A formal system can be defined as an ordered triple , where d is the relation of direct derivability. This relation is understood in a comprehensive sense
Sense and reference

The distinction between Sinn and Bedeutung was an innovation of the German philosopher and mathematician Gottlob Frege in his 1892 paper ?ber Sinn und Bedeutung , which is still widely read today....
 such that the primitive sentences of the formal system are taken as directly derivable
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
 from the empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
 of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are laid down in such a way that every first place member of d is a member of and every second place member is a finite subset of .

It is also possible to define a formal system using only the relation d. In this way we can omit , and a in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more difficult to understand and work with.

An interpretation of a formal system is a formal language for which both syntactical rules for deduction
Deductive system

A deductive system consists of the axioms and rules of inference that can be used to formal proof the theorems of the system.Such a deductive system is intended to preserve deduction qualities in the formula s that are expressed in the system....
, and semantical rules of interpretation are given. An interpreted formal system can be defined as the ordered quadruple . Here axioms are stated, some similar to those stated for a formal system, and some like those for an interpreted formal language. Usually, we wish for d to be truth-preserving (that is, any sentence which is directly derivable from true sentences is itself true), however other modalities
Modal logic

A modal logic is any system of mathematical logic#Formal logic that attempts to deal with notions of possibility and necessity. Traditionally, there are three "modes" or "moods" or "modalities" of the Copula to be, namely, Logical possibility, probability, and Necessary_and_sufficient_conditions#Necessary_conditions....
 can also preserved in such a system. We can formulate an axiom for these purposes without use of the term "true". For any i1,...,in, j, p1,...,pn,q if d(j,), (i1,p1) and ... and (in,pn) and p1 and ... and pn, q.

For interpreted formal systems there are also alternative, more explicit definitions which include ds, or both ds and , analogous to those given for interpreted formal languages.

Interpretation of a truth-functional propositional calculus


An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol
Propositional variable

In mathematical logic, a propositional variable is a variable which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics....
 of of one or the other (but not both) of the truth values truth
Truth

semantic fields for the word truth extend from honesty, good faith, and sincerity in general, to agreement with fact or reality in particular....
 (T) and falsity
False

False is the antonym of the adjective true.False is the 2nd album of Gorefest, False .False may also refer to:* FALSE, an esoteric stack-oriented programming language...
 (F), and an assignment to the connective symbols
Logical connective

In logic, two sentences may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences....
 of of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.

For n distinct propositional symbols there are 2n distinct possible interpretations. For any particular symbol a, for example, there are 21=2 possible interpretations: 1) a is assigned T, or 2) a is assigned F. For the pair a, b there are 22=4 possible interpretations: 1) both are assigned T, 2) both are assigned F, 3) a is assigned T and b is assigned F, or 4) a is assigned F and b is assigned T.

Since has , that is, denumerably many propositional symbols, there are 2=, and therefore uncountably many
Cardinality of the continuum

In mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size of the Set of real numbers ....
 distinct possible interpretations of .

Interpretation of a sentence of truth-functional propositional logic

If F and ? are formulas of and is an interpretation of then:

  • A sentence of propositional logic is true under an interpretation iff assigns the truth value T to that sentence.
  • F is false under an interpretation iff F is not true under .
  • A sentence of propositional logic is valid iff it is true under every interpretation
F means that F is valid
  • A sentence ? of propositional logic is a semantic consequence
    Logical consequence

    Logical consequence is a fundamental concept in logic. It is the Relation that holds between a Set of Sentence and a sentence when the former Entailment the latter....
     of a sentence F iff there is no interpretation under which F is true and ? is false.
  • A sentence of propositional logic is consistent
    Consistency

    Consistency can refer to:* Consistency * Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
     iff it is true under at least one interpretation. It is inconsistent if it is not consistent.


Some consequences of these definitions:

  • For any given interpretation a given formula is either true or false.
  • No formula is both true and false under the same interpretation.
  • F is false for a given interpretation iff F is true for that interpretation; and F is true under an interpretation iff F is false under that interpretation.
  • If F and (F ?) are both true under a given interpretation, then ? is true under that interpretation.
  • If F and (F ?), then ?.
  • F is true under iff F is not true under .
  • (F ?) is true under iff either F is not true under or ? is true under .
  • A sentence ? of propositional logic is a semantic consequence of a sentence F iff (F ?) is logically valid, that is, F ? iff (F ?).


Interpretation of a first-order formal system


For the purposes of a first-order formal system
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
 (we shall refer to it as so as to distinguish it from ), we cannot simply adopt the notion of tautology
Tautology (logic)

In propositional logic, a tautology is a propositional formula that is true under any possible Valuation of its propositional variables. For example, the propositional formula is a tautology, because the statement is true for any valuation of A....
 as it is used within a truth-functional propositional calculus. There are logically valid formulas of a first-order formal system, which are not necessarily instances of any tautological schema
Axiom schema

In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a well-formed formula in the language of an axiomatic system, in which one or more schematic variables appear....
 of that system. In order to deal with well-formed formula
Well-formed formula

In computer science and mathematical logic, a well-formed formula or simply formula is a symbol or string of symbols that is generated by the formal grammar of a formal language....
s in which free variables
Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where First-order_logic#Substitution may take place....
 occur, the complete definition of an interpretation of a first-order formal system has to be rather complicated.

Preliminary account

A preliminary account of an interpretation of a first-order formal system consists in the specification of some non-empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
 (called the domain of the interpretation
Domain of discourse

The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in deductive logic, especially predicate logic....
) and four other sets of designations.

Domain of discourse

The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 which indicates entities
Object (philosophy)

In philosophy, an object is a thing, an entity, or a being. This may be taken in several senses.In its weakest sense, the word object is the most all-purpose of nouns, and can replace a noun in any sentence at all....
 that are being talked about in an interpretation. The definition of an interpretation prohibits the empty domain
Empty domain

In first-order logic the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid....
 because the validity of certain theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
s or rules of the interpreted systems depends on it. A more fundamental reason the domain cannot be empty is that an interpretation has to have some thing which it is an interpretation of. Sometimes the domain of discourse is designated in notation as . We may, for instance, designate a domain as follows.



In addition to the domain of discourse, an interpretation consists of the following designations:

A sentence is either true or false under an interpretation which assigns values to the logical variables. We might for example make the following assignments:

Individual constants

The individual constants of are non-logical symbols which are assigned the names of objects from the domain of the interpretation. For instance, we can make the following designation.

  • a: Socrates
  • b: Plato
  • c: Aristotle


Sentential variables

The sentential variables in are each assigned a truth value, true (T) or false (F).

  • P: 'T'
  • Q: 'F'


Predicates symbols

The predicate symbols of are each assigned some property or relation defined for objects in the domain. For instance, we can designate the following relations.

  • F1:
  • G1:
  • H1:


  • R2:
  • R3:
  • S2:


In general, each n-ary predicate symbol is assigned an n-ary relation.

The connectives
Logical connective

In logic, two sentences may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences....
 are given their usual truth-functional meanings, however, they may stand between formulas that for a given interpretation are neither true nor false. Quantifiers are understood to refer exclusively to members of the domain of the interpretation.

Interpretation of a sentence of first order logic

An interpretation of a sentence F of a first order formal language consists of a non-empty domain
Empty domain

In first-order logic the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid....
  together with an assignment that associates with each individual constant of F some element of , with each sentential symbol of F one of the truth-values T or F, with each n-ary operation or function symbol in F an n-ary operation whose operands are exclusively from , with each n-ary predicate in F an n-ary relation among elements of , and, optionally, with some binary predicate the identity relation among elements of .

Truth under an interpretation of a first-order formal system

The key notion in a complete account of a definition of an interpretation of a first order formal system is the satisfaction of a formula by a denumerable sequence
Countable set

In mathematics, a countable set is a Set with the same cardinality as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers....
 of objects. We must account for all of the various forms that a formula may take within . Also, instead of talking about properties
Property (philosophy)

In modern philosophy, mathematics, and logic, a property is an attribute of an Object ; thus a red object is said to have the property of redness....
 and relations
Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
 we speak of set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
s of ordered n-tuples of objects.

A sentence which is a propositional variable
Propositional variable

In mathematical logic, a propositional variable is a variable which can either be true or false. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher logics....
 standing alone by itself is true under an interpretation if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
  assigns the truth value 'T' to that variable.

Any other atomic sentence
Atomic sentence

Molecular sentence which is a sentence comprised of atomic sentences also redirects here.In Logic, Sentence are those strings of words or symbols which are either true or false....
 is true under if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 in the case of predicates of degree 1, the member of the domain that assigns to the individual constant is a member of the set which assigns to the predicate; and

in the case of predicates of degree 2, the members of the domain that assigns to the two individual constants are members, in the order in which their representations occur in the sentence, or of an ordered pair
Ordered pair

In mathematics, an ordered pair is a collection of two distinguishable objects, one being the first coordinate system , and the other being the second coordinate ....
 in the binary relation
Binary relation

In mathematics, a binary relation is an arbitrary association of elements within a set or with elements of another set.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a divisibility of p, and no othe...
 that assigns to the predicate; and

in the case of predicates of degree n, the members of the domain that assigns to the n individual constants are members, in the order in which their representations occur in the sentence, or of an ordered n-tuple
Tuple

In mathematics, a tuple is a sequence of a specific number of values, called the components of the tuple. These components can be any kind of mathematical objects, where each component of a tuple is a value of a specified type....
 in the relation that assigns to the predicate.

A molecular sentence is true under if and only if

  • it is of the form F and F is not true under ; or
  • it is of the form (F ?) and both F and ? are true under ; or
  • it is of the form (F ?) and either F or ? or both are true under ; or
  • it is of the form (F ?) and either ? is true under or F is not true under ; or
  • it is of the form (F ?) and either both F and ? are true under or neither F nor ? are true under .


A general sentence is true under if and only if it is of the form (a)F and Fa/ß is true under every ß-variant of (where Fa/ß is the result of replacing all free occurrences of the variable a in F by occurrences of an individual constant ß); or

it is of the form (a)F and Fa/ß is true under at least one ß-variant of ;

Any sentence that is not true under is false under .

Further semantic concepts for first-order formal systems
A sentence of is valid (or logically true, written ' F ') iff it is true under every interpretation.

A sentence F is a logical consequence
Logical consequence

Logical consequence is a fundamental concept in logic. It is the Relation that holds between a Set of Sentence and a sentence when the former Entailment the latter....
 of a set of sentences G (or ' G F ') iff there is no interpretation under which all the members of G are true and F is false.

A set of sentences G is consistent
Consistency

Consistency can refer to:* Consistency * Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
 (or satisfiable) iff there is an interpretation under which all the sentences of G are true.

An interpretation is a model for a sentence F (or set of sentences G) iff F is (or all the members of G are) true under .

Valid interpretations


An interpretation is a true or valid interpretation if whenever a particular sentence P implies another Q within the formal system, in its interpretation, whenever P is true, Q must necessarily be true; and whenever a sentence is refutable within the formal system, it is false in the interpretation.

A true interpretation is called a logically true interpretation if the sentences that become true in the interpretation become logically true.

Intended interpretation


One who constructs a formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
 usually has in mind from the outset some interpretation of this system. While this intended interpretation can have no explicit indication in the syntactical rules
Deductive system

A deductive system consists of the axioms and rules of inference that can be used to formal proof the theorems of the system.Such a deductive system is intended to preserve deduction qualities in the formula s that are expressed in the system....
 --since these rules must be strictly formal --the author's intention respecting interpretation naturally affects his choice of the formation
Formal grammar

In formal language theory, grammars, also called formal grammars or generative grammars, are a formalism used to describe formal languages – i.e....
 and transformation rules of the syntactical system. For example, he chooses primitive signs in such a way that certain concepts can be expressed: He chooses sentential formulas in such a way that their counterparts in the intended interpretation can appear as meaningful declarative sentences; his choice of primitive sentences must meet the requirement that these primitive sentences come out as true sentences in the interpretation; his rules of inference must be such that if by one of these rules the sentence j is directly derivable
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
 from a sentence i, then i j turns out to be a true sentence (under the customary interpretation of "" as meaning implication
Material conditional

The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic....
). These requirements ensure that all provable sentences also come out to be true.

Most formal systems have many more models than they were intended to have (the existence of non-standard models is an example). When we speak about 'models' in empirical sciences, we mean, if we want reality to be a model of our science, to speak about an intended model. A model in the empirical sciences is an intended factually-true descriptive interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants.

Standard and non-standard models of arithmetic


A distinction is made between standard and non-standard models of Peano arithmetic, which is intended to describe the addition and multiplication operations on the natural numbers. The canonical standard model is obtained by taking the set of natural numbers as the domain of discourse, and interpreting "0" as zero, "1" as one, "+" as addition, and "x" as multiplication, etcetera. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms
Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind?Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian people mathematician Giuseppe Peano....
. There also exist non-standard models of the Peano axioms
Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind?Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian people mathematician Giuseppe Peano....
, which contain elements not correlated with any natural number. All standard models are logico-mathematical interpretations, but only some non-standard models are descriptive interpretations.

Logical and descriptive interpretations


Rudolf Carnap
Rudolf Carnap

Rudolf Carnap was an influential Germany-born philosophy who was active in Europe before 1935 and in the United States thereafter. He was a leading member of the Vienna Circle and a prominent advocate of logical positivism....
, in his Introduction to Semantics makes a distinction between formal interpretations which are logical interpretations (also called mathematical interpretation or logico-mathematical interpretation) and descriptive interpretation
Descriptive interpretation

According to Rudolf Carnap, in logic, a formal interpretation is a descriptive interpretation if at least one of the undefined symbols of its formal system becomes, in the interpretation, a descriptive sign ....
s (also called a factual interpretation).

An interpretation is a descriptive interpretation if at least one of the undefined symbols of the formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
 becomes, in the interpretation, a descriptive sign (i.e., the name of single objects, or observable properties).

An interpretation is a descriptive interpretation if it is not a logical interpretation.

Method of proof by interpretation


The method of proof by interpretation is given by showing that some sentence A is not a consequence of a certain system T of axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s or other statements of a given deductive theory.

If a sentence A can be derived from the statements of the system T it remains valid for any interpretation of this system. The existence of an interpretation of T for which A is not valid is proof that the sentence cannot be derived from the system T.

If we have an arbitrary deductive theory
Theory (mathematical logic)

In mathematical logic, a theory is a set of sentence s in a formal language. For example, a first-order theory is a set of first-order logic sentences....
  which we presume to be consistent
Consistency

Consistency can refer to:* Consistency * Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
 (which may be the same theory as the set of statements of system T) the method of proof by interpretation tries to find an interpretation of the system T within theory such that not the sentence A itself, but its negation becomes a theorem of the theory .

The method of proof by interpretation can be used to establish the independence
Independence (mathematical logic)

In mathematical logic, a sentence σ is called independent of a given theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false....
 of a given axiom system by applying the method as any times as there are axioms in the system. As each axiom, in turn, is taken as the sentence A, while T consists of the remaining axioms of the system.

Mathematical models


In universal algebra
Universal algebra

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures.For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study....
 and in model theory
Model theory

In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
, a structure is a type of formal interpretation which consists of an underlying set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 along with a collection of finitary
Finitary

In mathematics or logic, a finitary operation is one, like those of arithmetic, that takes a finite number of input values to produce an output....
 functions and relations which are defined on it.

Informally, a valuation is an assignment of particular values to the variables in a mathematical statement or equation.

In model theory
Model theory

In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
, interpretation of a structure
Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of an underlying Set along with a collection of finitary functions and relations which are defined on it....
 M in another structure N (typically of a different signature
Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure....
) is a technical notion that approximates the idea of representing M inside N.

A mathematical model is a type of formal interpretation that uses mathematical
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 language to describe a system
System

System is a set of interacting or interdependent entities, real or abstract, forming an integrated whole.The concept of an "integrated whole" can also be stated in terms of a system embodying a set of relationships which are differentiated from relationships of the set to other elements, and from relationships between an element of the se...
.

Scientific models


Attempts to axiomatize
Axiomatic system

In mathematics, an axiomatic system is any Set of axioms from which some or all axioms can be used in conjunction to logically derive theorems....
 the empirical sciences use a descriptive interpretation to model reality. The aim of these attempts is to construct a formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
 for which reality
Reality

Reality, in everyday usage, means "the state of things as they actually exist". In a sense it is what is real. The term reality, in its widest sense, includes everything that being, whether or not it is observation or comprehension....
 is the only interpretation. The world is an interpretation (or model) of these sciences, only insofar as these sciences are true.

Scientific modeling is the process of generating a formal interpretation for the empirical sciences. Science offers a growing collection of methods
Scientific method

Scientific method refers to techniques for investigating phenomenon, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and Measure evidence subject to specific principles of reasoning....
, techniques and theory
Theory

For a more detailed account of theories as expressed in formal language as they are studied in mathematical logic see Theory A theory, in the general sense of the word, is an analytic structure designed to explain a set of observations....
 about different types of specialized scientific modeling.

See also

  • Causal model
    Causal model

    A causal model is an model that uses Causality logic to describe the behaviour of a system....
  • EconMult
    EconMult

    EconMult is a general fishing fleet model to be used in fishery modelling. EconMult has been developed since 1991 as a part of the Multispecies management programme by the Norwegian Research Council at the Norwegian College of Fishery Science ....
    : an economic model for fisheries
  • Formal semantics
  • Herbrand interpretation
    Herbrand interpretation

    In mathematical logic, a Herbrand interpretation is an Interpretation_ in which all constants and function symbols are assigned very simple meanings....
  • Interpretation (logic)
    Interpretation (logic)

    In logic an interpretation gives meaning to an artificial or formal language or to a Sentence of such a language by assigning a denotation to each non-logical symbol in that language or in that sentence....
  • Mathematical models
  • Meta-modeling
  • Meta-Object Facility
    Meta-Object Facility

    The Meta-Object Facility is an Object Management Group standard for model-driven engineering. The official reference page may be found at OMG's website....
    : the OMG standard for defining metamodels
  • Model Driven Engineering
  • Model theory
    Model theory

    In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
  • Model Transformation Language
    Model Transformation Language

    Presentation...
  • Morphological modelling
    Morphological analysis

    Morphological analysis or General Morphological Analysis is a method developed by Fritz Zwicky for exploring all the possible solutions to a multi-dimensional, non-quantified problem complex....
  • SEQUAL framework