Formal interpretation

For other uses, see Model, and Interpretation (disambiguation).

In mathematical logic, the formal languages, formal systems, and theories which are studied have no meaningful content until they are given an interpretation within some other system. A formal interpretation is a semantic concept which consists in the assignment of meanings to the symbols of a formal language. The study of formal interpretations is called formal semantics. Giving an interpretation is synonymous with constructing a model. Models are constructed to enable reasoning within an idealized logical framework about these processes and are an important component of scientific theories.

Interpretation of a formal language

Main article: Formal language

Formal interpretations are expressed in a metalanguage which is talking about some object language, which is usually some formal language. A formal language is an organized set of symbols 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 to any meanings 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 ${\mathcal {L}}$ can be defined formally as a set A of strings (finite sequences) on a fixed alphabet α. Some authors, including Carnap, define the language as the ordered pair <α, A>. Carnap also requires that each element of α must occur in at least one string in A.

The class A of the expressions of ${\mathcal {L}}$ is defined as the class of all finite sequences whose members are the elements of the class α.

An n-place sequence can be defined as a function between the n first natural numbers and the members of the sequence. A syntactic axiom may be adopted that states: For any class α and any class ${\mathcal {I}}$, if <α, ${\mathcal {I}}$> is a formal language then every element of ${\mathcal {I}}$ is a finite sequence of elements of α, and every element of α occurs as a member of some element of ${\mathcal {I}}$.

An interpretation of a formal language can defined as the ordered triple <α,${\mathcal {I}}$,${\mathcal {D}}$>. The first domain of the relation ${\mathcal {D}}$ is identical with the class ${\mathcal {I}}$.

If an extensional metalanguage is used for semantics, then ${\mathcal {D}}$ is the relation of value assignment for the sentences of the language.

For example, "${\mathcal {D}}$(${\mathcal {I}}$₁,grass is green)" means the same as "The sentence ${\mathcal {I}}$₁ is true if and only if grass is green." For any p and q and any element ${\mathcal {I}}$₁ of the class ${\mathcal {I}}$, if ${\mathcal {D}}$(${\mathcal {I}}$₁,p) and ${\mathcal {D}}$(${\mathcal {I}}$₁,q) then p if and only if q.

If on the other hand, an intensional metalanguage, containing a modal operator, such as "it is necessary that", then ${\mathcal {D}}$ is taken as the relation of designation, That is, the relation between an expression and its intension.

For example, "${\mathcal {D}}$(${\mathcal {I}}$₁,grass is green)" means the same as "The sentence ${\mathcal {I}}$₁ designates the proposition that grass is green." For any p and q and any element ${\mathcal {I}}$₁ of the class ${\mathcal {I}}$, if ${\mathcal {D}}$(${\mathcal {I}}$₁,p) and ${\mathcal {D}}$(${\mathcal {I}}$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 with respect to any given interpreted language <α, ${\mathcal {I}}$,${\mathcal {D}}$> can be defined as follows: A sentence ${\mathcal {I}}$₁ is true if and only if for some p, ${\mathcal {D}}$(${\mathcal {I}}$₁,p), and p.

There is another method applicable to either of these two metalanguages which takes the relation ${\mathcal {D}}$ 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 <α,${\mathcal {I}}$,d,${\mathcal {D}}$ >.

In these metalanguages, d is the class of finite sequences of elements of the class α, the class of the first place members of ${\mathcal {D}}$ is the class d, and that ${\mathcal {I}}$ 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 <α,ds,${\mathcal {I}}$,d,${\mathcal {D}}$ >

Using this method, ds is a subclass of α. This most explicit method is convenient as a basis for definitions of concepts such as "model", "value assignment", "range of a sentence", "logical truth", and other logical concepts.

A simple example

The formal language ${\mathcal {W}}$ is defined as follows:

Alphabet α : { $\triangle$, $\square$ }

Formal grammar : Any finite string of symbols from the alphabet of ${\mathcal {W}}$ that begins with a '$\triangle$' is a formula.

A possible interpretation of ${\mathcal {W}}$ would be to take '$\triangle$' as meaning the same as the decimal digit '1', '$\square$' 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 '$\triangle$ $\square$ $\triangle$' means '101' under this interpretation of ${\mathcal {W}}$.

Interpretation of logical connectives

Main article: Logical connective

The symbols of a language which represent the logical connectives can be interpreted as operations. The interpretation of the logical connectives as logical operations can be expressed in terms of truth tables.

Interpretation of a theory

Main article: Theory (mathematical logic)

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

Main article: 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 axioms, or have both. A formal system is used to derive one expression from one or more other expressions.

A formal system can be defined as an ordered triple <α,${\mathcal {I}}$,${\mathcal {D}}$d>, where ${\mathcal {D}}$d is the relation of direct derivability. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set 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 ${\mathcal {D}}$d is a member of ${\mathcal {I}}$ and every second place member is a finite subset of ${\mathcal {I}}$.

It is also possible to define a formal system using only the relation ${\mathcal {D}}$d. In this way we can omit ${\mathcal {I}}$, and α 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, and semantical rules of interpretation are given. An interpreted formal system can be defined as the ordered quadruple <α,${\mathcal {I}}$,${\mathcal {D}}$d,${\mathcal {D}}$>. 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 ${\mathcal {D}}$d to be truth-preserving (that is, any sentence which is directly derivable from true sentences is itself true), however other modalities can also preserved in such a system. We can formulate an axiom for these purposes without use of the term "true". For any ${\mathcal {I}}$_i1,...,${\mathcal {I}}$_in, ${\mathcal {I}}$_j, p₁,...,p_n,q if ${\mathcal {D}}$d(${\mathcal {I}}$_j,{${\mathcal {I}}$_i1,...,${\mathcal {I}}$_in}), ${\mathcal {D}}$(${\mathcal {I}}$_i1,p₁) and ... and ${\mathcal {D}}$(${\mathcal {I}}$_in,p_n) and p₁ and ... and p_n, q.

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

Interpretation of a truth-functional propositional calculus

An interpretation of a truth-functional propositional calculus ${\mathcal {P}}$ is an assignment to each propositional symbol of ${\mathcal {P}}$ of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of ${\mathcal {P}}$ 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 ${\mathcal {P}}$ has $\aleph _{0}$, that is, denumerably many propositional symbols, there are 2^{$\aleph _{0}$}=${\mathfrak {c}}$, and therefore uncountably many distinct possible interpretations of ${\mathcal {P}}$.

Truth under an interpretation of a truth-functional propositional calculus

If A and B are formulas of ${\mathcal {P}}$ and ${\mathcal {I}}$ is an interpretation of ${\mathcal {P}}$ then:

1. If A is a propositional symbol, then A is true under ${\mathcal {I}}$ iff ${\mathcal {I}}$ assigns the truth value true (T) to A.
2. $\neg$A is true under ${\mathcal {I}}$ iff A is not true under ${\mathcal {I}}$
3. (A$\to$B) is true under ${\mathcal {I}}$ iff either A is not true under ${\mathcal {I}}$ or B is true under ${\mathcal {I}}$.

Some consequences of these definitions:

• A is false under an interpretation ${\mathcal {I}}$ iff A is not true under ${\mathcal {I}}$.
• For any given interpretation a given formula is either true or false.
• No formula is both true and false under the same interpretation.
• A is false for a given interpretation iff $\neg$A is true for that interpretation; and A is true under an interpretation iff $\neg$A is false under that interpretation.
• If A and (A$\to$B) are both true under a given interpretation, then B is true under that interpretation.
• If $\models _{\mathrm {P} }$A and $\models _{\mathrm {P} }$(A $\to$ B), then $\models _{\mathrm {P} }$B.
• B is a semantic consequence of A iff (A $\to$ B) is logically valid, that is, A $\models _{\mathrm {P} }$ B iff $\models _{\mathrm {P} }$(A $\to$ B).

Interpretation of a first-order formal system

For the purposes of a first-order formal system (we shall refer to it as ${\mathcal {Q}}$ so as to distinguish it from ${\mathcal {P}}$), we cannot simply adopt the notion of tautology 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 of that system. In order to deal with well-formed formulas in which free variables 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 (called the domain of the interpretation) and four other sets of designations.

Domain of discourse

Main article: domain of discourse

The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is a set which indicates entities that are being talked about in an interpretation. The definition of an interpretation prohibits the empty domain because the validity of certain theorems or rules of the interpreted systems depends on it. Sometimes the domain of discourse is designated in notation as ${\mathcal {D}}$

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

1. To each propositional symbol, one or the other (but not both) of the truth values truth (T) and falsity (F).
2. To each individual constant, some member of the domain of the interpretation.
3. To each function symbol, a function with arguments and values in the domain.
4. To each predicate symbol, some property or relation defined for objects in the domain.

The connectives 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.

Satisfiability of formulas of first-order formal systems

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 of objects. We must account for all of the various forms that a formula may take within ${\mathcal {Q}}$. Also, instead of talking about properties and relations we speak of sets of ordered n-tuples of objects.

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

Main article: Intended interpretation

One who constructs a syntactical system 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 --since these rules must be strictly formal --the author's intention respecting interpretation naturally affects his choice of the formation 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 ${\mathcal {I}}$_j is directly derivable from a sentence ${\mathcal {I}}$_i, then ${\mathcal {I}}$_i $\to$ ${\mathcal {I}}$_j turns out to be a true sentence (under the customary interpretation of "$\to$" as meaning implication). 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.

Logical and descriptive interpretations

Main articles: Logical interpretation and Descriptive interpretation

Rudolf Carnap, 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 interpretations (also called a factual interpretation).

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

Standard and non-standard models of arithmetic

Main article: Non-standard model

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 0, "1" as 1, "+" as the addition, and "x" as the multiplication. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. There also exist non-standard models of the Peano axioms, 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.

Mathematical models

Main article: Structure (mathematical logic)

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

Main article: Valuation (mathematics)

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

Main article: Interpretation (model theory)

In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N.

Main article: Mathematical model

A mathematical model is a type of formal interpretation that uses mathematical language to describe a system.

Scientific models

Main article: Scientific model

Attempts to axiomatize the empirical sciences use a descriptive interpretation to model reality. The aim of these attempts is to construct a formal system for which reality 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, techniques and theory about different types of specialized scientific modeling.

See also

• Causal model
• EconMult: an economic model for fisheries
• Formal semantics
• Herbrand interpretation
• Interpretation (logic)
• Mathematical models
• Meta-modeling
• Meta-Object Facility: the OMG standard for defining metamodels
• Model Driven Engineering
• Model theory
• Model Transformation Language
• Morphological modelling
• SEQUAL framework

References

• Carnap, Rudolf, Introduction to Semantics
• Carnap, Rudolf, Introduction to Symbolic Logic and its Applications
• R. Frigg and S. Hartmann, Models in Science. Entry in the Stanford Encyclopedia of Philosophy.
• W. Quine, From a Logical Point of View, Harper Torchbooks, 1961.