Formal interpretation

Formal interpretation

A formal interpretation [,M1 Cann Ronnie, Formal Semantics: An Introduction] ] or model is the assignment of meanings to the symbols and truth-values to the sentences of a formal language. Fact|date=May 2008 The study of formal interpretations is called formal semantics. 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").Rudolf Carnap, "Introduction to Symbolic Logic and its Applications"] An interpretation is a factual interpretation if it is not a logical interpretation. "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. Fact|date=May 2008

Formal language

Formal interpretations are expressed in a metalanguage which is talking about some object language, which is usually a "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.Fact|date=May 2008

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 υ 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 many-one relation 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}.

Interpreted formal languages

An "interpreted 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 : { riangle, square }: Formal grammar : Any finite string of symbols from the alphabet of mathcal{W} that begins with a ' riangle' is a formula.

A possible interpretation of mathcal{W} would be to take ' riangle' 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 ' riangle square riangle' means '101' under this interpretation of mathcal{W}.

Formal systems

A formal interpretation is an interpretation of some "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.

Interpreted formal systems

An "interpreted 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, "p"1,...,"p"n,"q" if mathcal{D}d(mathcal{I}j,{mathcal{I}i1,...,mathcal{I}in}), mathcal{D}(mathcal{I}i1,"p"1) and ... and mathcal{D}(mathcal{I}in,"p"n) and "p"1 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.Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971]

For "n" distinct propositional symbols there are 2"n" 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 2aleph_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:

# If "A" is a propositional symbol, then "A" is true under mathcal{I} iff mathcal{I} assigns the truth value true (T) to "A".
#"A" is true under mathcal{I} iff "A" is not true under mathcal{I}
# ("A"imp"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 not"A" is true for that interpretation; and "A" is true under an interpretation iff not"A" is false under that interpretation.
*If "A" and ("A"imp"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" imp "B"), then models_{mathrm P}"B".
*"B" is a semantic consequence of "A" iff ("A" imp "B") is logically valid, that is, "A" models_{mathrm P} "B" iff models_{mathrm P}("A" imp "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 the following designations:

# To each propositional symbol, one or the other (but not both) of the truth values truth (T) and falsity (F).
# To each individual constant, some member of the domain of the interpretation.
# To each function symbol, a function with arguments and values in the domain.
# 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.

True interpretations

A formal interpretation is a "true 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 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 imp mathcal{I}j turns out to be a true sentence (under the customary interpretation of "imp"). 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. [The Concept and the Role of the Model in Mathematics and Natural and Social Sciences]

Logical interpretations

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 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. [Cambridge Dictionary of Philosophy]

Descriptive interpretations

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.

Mathematical models

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.

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

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".

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

Scientific models

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, and theory about different types of specialized scientific modeling.

Economic models

In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and quantitative relationships between them.

Structure of models

A 'conceptual model is a representation of some phenomenon, data or theory by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference etc. A conceptual model has an ontology, that is the set of expressions in the model which are "intended" to denote some aspect of the modeled object. Here we are deliberately vague as to how expressions are constructed in a model and particularly what the logical structure of formulas in a model actually is. In fact, we have made no assumption that models are encoded in any logical system at all, although we briefly address this issue below. Moreover, the definition given here is oblivious about whether two expressions really should denote the same thing. Note that this notion of ontology is different from (and weaker than) ontology as is sometimes understood in philosophy; in our sense there is no claim that the expressions actually denote anything which exists "physically" or "spatio-temporally" (to use W. Quine's formulation).

For example, a stochastic model of stock prices includes in its ontology a sample space, random variables, the mean and variance of stock prices, various regression coefficients etc. Models of quantum mechanics in which pure states are represented as unit vectors in a Hilbert space include in their ontologies observables, dynamics, measurement operators etc. It is possible that observables and states of quantum mechanics are as physically real as the electrons they model, but by adopting this purely formal notion of ontology we avoid altogether this question.

Use of models

The purpose of a model is to provide an "argumentative framework" for applying logic and mathematics that can be independently evaluated (for example by testing) and that can be applied for reasoning in a range of situations. Models are used throughout the natural and social sciences, psychology and the philosophy of science. Some models are predominantly statistical (for example portfolio models used in finance); others use calculus, linear algebra or convexity, see mathematical model. Of particular political significance are models used in economics, since they are used to justify decisions regarding taxation and government spending. This often leads to hotly contested debates in the academic world as well as in the political arena; see for instance supply side economics.

Abstract models are used primarily as a reusable tool for discovering new facts, for providing systematic logical arguments as explicatory or pedagogical aids, for evaluating hypotheses theoretically, and for devising experimental procedures to test them. Reasoning within models is determined by a set of logical principles, although rarely is the reasoning used completely mathematical.

In some cases, abstract models can be used to implement computer simulations that illustrate the behavior of a system over time. Simulations are used everywhere in science, especially in economics, engineering, biology, ecology etc., to discover the effects of changing a variable. The validity of different simulation methodologies is a subject of debate in the philosophy and methodology of science.

The "automated" use of modeling has been identified as a significant issue in the creation of artificial intelligence. Some researchers argue a system without a model cannot achieve understanding, while others argue that running full, consistent models is too computationally costly for either machines or animals, and that much intelligent behavior is reactive or instinctive.

See also

* Causal model
* EconMult: an economic model for fisheries
* Ecosystem model
* 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
* Quality of models


* 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.

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