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A **formal interpretation**^{[1]} or **model** is the assignment of meanings to the symbols and truth-values to the sentences of a formal language. ^{[citation needed]} 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*). ^{[2]} An interpretation is a factual interpretation if it is not a logical interpretation. ^{[3]} *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. ^{[citation needed]}

Main article: Formal language |

Formal interpretations are expressed in some *formal language*. ^{[4]}^{[5]} 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.^{[citation needed]}

If the class α consists of all of the logical and non-logical signs of a formal language and the class consists of all sentences of then a formal language can be expressed as the ordered pair <α, >

- The class υ of the expressions of 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 , if <α, > is a formal language then every element of is a finite sequence of elements of α, and every element of α occurs as a member of some element of .
^{[6]}

An *interpreted formal language* can defined as the ordered triple
<α,,>. The first domain of the relation is identical with the class .

- If an extensional metalanguage is used for semantics, then is the relation of value assignment for the sentences of the language.

- For example, "(
_{1},grass is green)" means the same as "The sentence_{1}is true if and only if 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*.

- For example, "(

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

- If on the other hand, an intensional metalanguage, containing a modal operator, such as "it is necessary that", then is taken as the relation of

- 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*.

- For example, "(

- In either of these two metalanguages extensional, or intensional, truth with respect to any given interpreted language (α, ,) 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 (α,,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 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 <α,ds,,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.

^{[7]}

Main article: Formal grammar |

The expressions used in a formal interpretation must be consistent with some *formal grammar*. A *formal grammar* (also called *formation rules*) is a precise description of a the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the formal language which constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean).

Main article: Formal system |

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,,d>, where 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 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 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 *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,,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 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 _{i}_{1},...,_{i}_{n},
_{j}, *p*_{1},...,*p*_{n},*q* if d(_{j},{_{i}_{1},...,_{i}_{n}}), (_{i}_{1},*p*_{1}) and ... and (_{i}_{n},*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 , analogous to those given for interpreted formal languages.
^{[8]}

An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of 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 of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables.^{[9]}

For *n* distinct propositional symbols there are 2^{n} distinct possible interpretations. For any particular symbol *a*, for example, there are 2^{1}=2 possible interpretations: 1) *a* is assigned **T**, or 2) *a* is assigned **F**. For the pair *a*, *b* there are 2^{2}=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**.^{[9]}

Since has , that is, denumerably many propositional symbols, there are 2^{$\aleph _{0))$}=, and therefore uncountably many distinct possible interpretations of .^{[9]}

If *A* and *B* are formulas of and is an interpretation of then:

- If
*A*is a propositional symbol, then*A*is true under iff assigns the truth value true (**T**) to*A*. *A*is true under iff*A*is not true under- (
*A**B*) is true under iff either*A*is not true under or*B*is true under .^{[9]}

Some consequences of these definitions:

*A*is*false under an interpretation*iff*A*is not true under .^{[9]}- For any given interpretation a given formula is either true or false.
^{[9]} - No formula is both true and false under the same interpretation.
^{[9]} *A*is false for a given interpretation iff*A*is true for that interpretation; and*A*is true under an interpretation iff*A*is false under that interpretation.^{[9]}- If
*A*and (*A**B*) are both true under a given interpretation, then*B*is true under that interpretation.^{[9]} - If
*A*and (*A**B*), then*B*.^{[9]} *B*is a semantic consequence of*A*iff (*A**B*) is logically valid, that is,*A**B*iff (*A**B*).^{[9]}

For the purposes of a first-order formal system (we shall refer to it as so as to distinguish it from ), 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.^{[9]}

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.
^{[9]}

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.^{[9]}

The key notion in a complete account of a definition 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 . Also, instead talking about properties and relations we speak of sets of ordered n-tuples of objects. ^{[9]}

Main article: Formal proof |

A *formal proof* is a sequences of well-formed formulas of a formal language, the last one of which is a theorem of a formal system. The theorem is a syntactic consequence of all the wffs preceding it in the proof. For a wff to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous wffs in the proof sequence.

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.

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 _{j} is directly derivable from a sentence _{i}, then _{i} _{j} turns out to be a true sentence (under the customary interpretation of ""). These requirements ensure that all provable sentences also come out to be true.^{[10]}

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. ^{[11]}

Main article: Logical interpretation |

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. ^{[12]}

Main article: Descriptive 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.

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.

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.

Main article: Model (economics) |

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.

Main article: Conceptual schema |

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.

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.