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| {{Mergeto|Interpretation (logic)|Talk:Interpretation_(logic)#Merge_proposal|date=March 2009}} | |||
| In ], the ]s, ]s, and ] which are studied have no meaningful content until they are given an '''interpretation''' within some other system.<ref>Exner & Rosskopf, ''Logic in Elementary Mathematics''</ref> An interpretation is a ] concept which consists in a correlation or ] of ]s to the ]s of a ]. The study of interpretations expressed in formal languages is called ]. ''Giving an interpretation'' is synonymous with ''constructing a ]''. The term interpretation is also synonymous with the term ]. | |||
| In ] and in ], a ] 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. Model theory studies the models of various ]. A theory is a set of sentences in a formal language with a particular (]), while a model is a structure whose interpretation of the symbols of the signature causes the sentences of the theory to be true. Model theory is closely related to universal algebra and algebraic geometry, although the methods of model theory focus more on logical considerations than those fields. | |||
| An ] can be regarded as an auxiliary notion, an important step in a specific way for defining the concept of truth formally (e.g. for first-order theories). It enables us to give meanings to terms (truth to sentences) of a language which deals with (free) variables. | |||
| When the ]s attempt to ] the principles governing the subjects they study, they are creating a formal system for which ] 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 == | |||
| {{Main|Formal language}} | |||
| Interpretations are expressed in a ] which is talking about some ], which is usually some ''formal language''.<ref name="fsai"></ref> A ''formal language'' is an organized ] of ]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 is purely ] in nature and can be defined, then, without any ] to any ]s of any of its expressions; it can exist before any interpretation is assigned to it—that is, before it has any meaning or ]. | |||
| A formal language <math>\mathcal{L}</math> can be defined formally as a set ''A'' of strings (finite sequences) composed exclusively from a fixed alphabet α. It may be defined in terms of an ] <α, ''A''>.<ref name="itslaia"/> A ] is a list and description of the ]s of a formal language. It may be required in some languages that each element of α must occur in at least one string in ''A''.<ref>Rudolf Carnap, ''Introduction to Symbolic Logic and its Applications''</ref> | |||
| :The ] A of the expressions of <math>\mathcal{L}</math> is defined as the class of all ]s whose ] are the elements of the class α. | |||
| :An n-place ] can be defined as a ] between the n first ]s and the members of the sequence. A ] ] may be adopted that states: For any class α and any class <math>\mathcal{I}</math>, if <α, <math>\mathcal{I}</math>> is a formal language then every element of <math>\mathcal{I}</math> is a finite sequence of elements of α, and every element of α occurs as a member of some element of <math>\mathcal{I}</math>.<ref name="itslaia"/> | |||
| An '''interpretation of a formal language''' can defined as the ordered triple | |||
| <α,<math>\mathcal{I}</math>,<math>\mathcal{D}</math>>. The ] of the relation <math>\mathcal{D}</math> is identical with the class <math>\mathcal{I}</math>. | |||
| ::If an ]al ] is used for ], then <math>\mathcal{D}</math> is the relation of ] for the ] of the language. | |||
| :::For example, "<math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,grass is green)" means the same as "The sentence <math>\mathcal{I}</math><sub>1</sub> is true ] grass is green." For any ''p'' and ''q'' and any element <math>\mathcal{I}</math><sub>1</sub> of the class <math>\mathcal{I}</math>, if <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,''p'') and <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,''q'') then ''p'' if and only if ''q''. | |||
| ::If on the other hand, an ]al metalanguage, containing a ], such as "it is necessary that", then <math>\mathcal{D}</math> is taken as the relation of ''designation'', That is, the relation between an expression and its intension. | |||
| :::For example, "<math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,grass is green)" means the same as "The sentence <math>\mathcal{I}</math><sub>1</sub> designates the proposition that grass is green." For any ''p'' and ''q'' and any element <math>\mathcal{I}</math><sub>1</sub> of the class <math>\mathcal{I}</math>, if <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,''p'') and <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,''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, ] with respect to any given interpreted language <α, <math>\mathcal{I}</math>,<math>\mathcal{D}</math>> can be defined as follows: A sentence <math>\mathcal{I}</math><sub>1</sub> is true if and only if for some ''p'', <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>1</sub>,''p''), and ''p''. | |||
| :There is another method applicable to either of these two metalanguages which takes the relation <math>\mathcal{D}</math> 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,<math>\mathcal{I}</math>,d,<math>\mathcal{D}</math> >. | |||
| :In these metalanguages, d is the class of finite sequences of elements of the class α, the class of the first place members of <math>\mathcal{D}</math> is the class d, and that <math>\mathcal{I}</math> 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 ]s of the language must be indicated. In this method, an ''interpreted formal language'' can be defined as the ordered quintuple <α,ds,<math>\mathcal{I}</math>,d,<math>\mathcal{D}</math> > | |||
| :Using this method, ds is a subclass of α. This most explicit method is convenient as a basis for definitions of concepts such as "]", "value assignment", "] of a sentence", "]", and other '']al'' concepts.<ref name="itslaia"/> | |||
| === A simple example === | |||
| The formal language <math>\mathcal{W}</math> is defined as follows: | |||
| : Alphabet α : { <math>\triangle</math>, <math>\square</math> } | |||
| : Formal grammar : Any finite string of symbols from the alphabet of <math>\mathcal{W}</math> that begins with a '<math>\triangle</math>' is a formula. | |||
| A possible interpretation of <math>\mathcal{W}</math> would be to take '<math>\triangle</math>' as meaning the same as the decimal digit '1', '<math>\square</math>' 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 '<math>\triangle</math> <math>\square</math> <math>\triangle</math>' means '101' under this interpretation of <math>\mathcal{W}</math>.<ref name="metalogic"/> | |||
| === logical symbols === | |||
| {{Main|Logical connective}} | |||
| The logical symbols of a language (other than quantifiers) represent truth-functions (]): functions that take truth-values as arguments and return truth-values (operations on truth values of sentences). The assignments of truth-functions (logical operations) to the logical symbols of a language can be exactly defined by means of ]. | |||
| {| class-"wikitable" style="text-align:center" border="1" align="center" | |||
| |+ Interpretation of logical connectives | |||
| ! Φ !! Ψ !! {{not}}Φ !! (Φ {{and}} Ψ) !! (Φ {{or-}} Ψ) !! (Φ {{imp}} Ψ) !! (Φ {{eqv}} Ψ) | |||
| |- | |||
| | 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 | |||
| |} | |||
| Thus if in a langauge the symbol "{{not}}" represents the truth-function negation, and "Φ" is a sentence in that langauge then if Φ is True then {{not}}Φ is False, and if Φ is false then {{not}}Φ is true. | |||
| == Interpretation of a theory == | |||
| {{main|Theory (mathematical logic)}} | |||
| An '''interpretation of a theory''' is the relationship between a theory and some contensive subject matter when there is a ] 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'''.<ref>Curry, Haskell, ''Foundations of Mathematical Logic'' p.48</ref> | |||
| == Interpretation of a formal system == | |||
| {{main|Formal system}} | |||
| A ''formal system'' (also called a ''logical calculus'', or a ''logical system'') consists of a formal language together with a ] (also called a ''deductive system''). The deductive apparatus may consist of a set of ]s (also called ''inference rules'') or a set of ]s, or have both. A formal system is used to ] one expression from one or more other expressions. | |||
| A ''formal system'' can be defined as an ordered triple <α,<math>\mathcal{I}</math>,<math>\mathcal{D}</math>d>, where <math>\mathcal{D}</math>d is the relation of direct derivability. This relation is understood in a comprehensive ] such that the primitive sentences of the formal system are taken as directly ] from the ] 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 <math>\mathcal{D}</math>d is a member of <math>\mathcal{I}</math> and every second place member is a finite subset of <math>\mathcal{I}</math>. | |||
| It is also possible to define a ''formal system'' using only the relation <math>\mathcal{D}</math>d. In this way we can omit <math>\mathcal{I}</math>, 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 ], and ] are given. An ''interpreted formal system'' can be defined as the ordered quadruple <α,<math>\mathcal{I}</math>,<math>\mathcal{D}</math>d,<math>\mathcal{D}</math>>. 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 <math>\mathcal{D}</math>d to be ] (that is, any sentence which is directly derivable from true sentences is itself true), however other ] can also preserved in such a system. We can formulate an axiom for these purposes without use of the term "true". For any <math>\mathcal{I}</math><sub>i</sub><sub><sub>1</sub></sub>,...,<math>\mathcal{I}</math><sub>i</sub><sub><sub>n</sub></sub>, | |||
| <math>\mathcal{I}</math><sub>j</sub>, ''p''<sub>1</sub>,...,''p''<sub>n</sub>,''q'' if <math>\mathcal{D}</math>d(<math>\mathcal{I}</math><sub>j</sub>,{<math>\mathcal{I}</math><sub>i</sub><sub><sub>1</sub></sub>,...,<math>\mathcal{I}</math><sub>i</sub><sub><sub>n</sub></sub>}), <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>i</sub><sub><sub>1</sub></sub>,''p''<sub>1</sub>) and ... and <math>\mathcal{D}</math>(<math>\mathcal{I}</math><sub>i</sub><sub><sub>n</sub></sub>,''p''<sub>n</sub>) and ''p''<sub>1</sub> and ... and ''p''<sub>n</sub>, ''q''. | |||
| For ''interpreted formal systems'' there are also alternative, more explicit definitions which include ds, or both ds and <math>\mathcal{D}</math>, analogous to those given for interpreted formal languages. | |||
| <ref name="itslaia"/> | |||
| === Interpretation of a truth-functional propositional calculus === | |||
| An '''interpretation of a ]''' <math>\mathcal{P}</math> is an ] to each ] of <math>\mathcal{P}</math> of one or the other (but not both) of the ]s ] ('''T''') and ] ('''F'''), and an assignment to the ] of <math>\mathcal{P}</math> of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of ].<ref name="metalogic">Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971</ref> | |||
| For ''n'' distinct propositional symbols there are 2<sup>''n''</sup> distinct possible interpretations. For any particular symbol ''a'', for example, there are 2<sup>1</sup>=2 possible interpretations: 1) ''a'' is assigned '''T''', or 2) ''a'' is assigned '''F'''. For the pair ''a'', ''b'' there are 2<sup>2</sup>=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'''.<ref name="metalogic"/> | |||
| Since <math>\mathcal{P}</math> has <math>\aleph_0</math>, that is, ] many propositional symbols, there are 2<sup><math>\aleph_0</math></sup>=<math>\mathfrak c</math>, and therefore ] distinct possible interpretations of <math>\mathcal{P}</math>.<ref name="metalogic"/> | |||
| ==== Interpretation of a sentence of truth-functional propositional logic ==== | |||
| If '''Φ''' and '''Ψ''' are ]s of <math>\mathcal{P}</math> and <math>\mathcal{I}</math> is an interpretation of <math>\mathcal{P}</math> then: | |||
| * A sentence of propositional logic is ''true under an interpretation'' <math>\mathcal{I}</math> iff <math>\mathcal{I}</math> assigns the truth value '''T''' to that sentence. If a sentence is ] under an interpretation, then that interpretation is called a ''model'' of that sentence. | |||
| * '''Φ''' is ''false under an interpretation'' <math>\mathcal{I}</math> iff '''Φ''' is not true under <math>\mathcal{I}</math>.<ref name="metalogic"/> | |||
| * A sentence of propositional logic is ''logically valid'' iff it is true under every interpretation | |||
| :{{models}}'''Φ''' means that '''Φ''' is logically valid | |||
| * A sentence '''Ψ''' of propositional logic is a '']'' of a sentence '''Φ''' iff there is no interpretation under which '''Φ''' is true and '''Ψ''' is false. | |||
| * A sentence of propositional logic is '']'' 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.<ref name="metalogic"/> | |||
| *No formula is both true and false under the same interpretation.<ref name="metalogic"/> | |||
| *'''Φ''' is false for a given interpretation iff {{not}}'''Φ''' is true for that interpretation; and '''Φ''' is true under an interpretation iff {{not}}'''Φ''' is false under that interpretation.<ref name="metalogic"/> | |||
| *If '''Φ''' and ('''Φ''' {{imp}}'''Ψ''') are both true under a given interpretation, then '''Ψ''' is true under that interpretation.<ref name="metalogic"/> | |||
| *If <math> \models_{\mathrm P}</math>'''Φ''' and <math> \models_{\mathrm P}</math>('''Φ''' {{imp}} '''Ψ'''), then <math> \models_{\mathrm P}</math>'''Ψ'''.<ref name="metalogic"/> | |||
| *{{not}}'''Φ''' is true under <math>\mathcal{I}</math> iff '''Φ''' is not true under <math>\mathcal{I}</math>. | |||
| * ('''Φ''' {{imp}}'''Ψ''') is true under <math>\mathcal{I}</math> iff either '''Φ''' is not true under <math>\mathcal{I}</math> or '''Ψ''' is true under <math>\mathcal{I}</math>.<ref name="metalogic"/> | |||
| * A sentence '''Ψ''' of propositional logic is a semantic consequence of a sentence '''Φ''' iff ('''Φ''' {{imp}} '''Ψ''') is ], that is, '''Φ''' <math> \models_{\mathrm P}</math> '''Ψ''' iff <math> \models_{\mathrm P}</math>('''Φ''' {{imp}} '''Ψ''').<ref name="metalogic"/> | |||
| === Interpretation of a first-order formal system === | |||
| For the purposes of a ] (we shall refer to it as <math> \mathcal{Q}</math> so as to distinguish it from <math> \mathcal{P} </math>), we cannot simply adopt the notion of ] as it is used within a truth-functional propositional calculus. There are ] formulas of a first-order formal system, which are not necessarily instances of any ] of that system. In order to deal with ]s in which ] occur, the complete definition of an interpretation of a first-order formal system has to be rather complicated.<ref name="metalogic"/> | |||
| The ] establishes that any satisfiable formula of first-order logic is satisfiable in a ] domain of interpretation. Hence, countable domains (i.e. domains whose cardinality is countable) are sufficient for interpretation of first-order logic if one is only interested in a single sentence at a time.<ref name=sakharov /> | |||
| ==== Preliminary account ==== | |||
| A preliminary account of an ''interpretation of a first-order formal system'' consists in the specification of some non-] (called the ]) and four other sets of designations. | |||
| ===== Domain of discourse ===== | |||
| {{Main|domain of discourse}} | |||
| The ''domain of discourse'', sometimes called the ''universe of discourse'', ''logical discourse'', or simply ''discourse'', is a ] which indicates ] that are being talked about in an interpretation. The definition of an interpretation prohibits the ] because the validity of certain ]s or ] 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.'' | |||
| The domain of discourse forms the ] of any ] that occur in any statements in the language. As for structures, the ] of an interpretation is defined as the cardinality of the domain.<ref>http://www.earlham.edu/~peters/courses/logsys/glossary.htm Glossary of First-Order Logic</ref> The truth-value of a formula under a given interpretation is intuitively clear; mathematically it is defined recursively by the ], also known as "Tarski's definition of truth". | |||
| Sometimes the domain of discourse is designated in notation as <math> \mathcal{D} </math>. We may, for instance, designate a domain as follows. | |||
| *<math> \mathcal{D} </math>: {Socrates, Plato, Aristotle} | |||
| 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: | |||
| ======Non-empty domain requirement====== | |||
| It is stated above that an interpretation must specify a non-empty domain as the universe of disourse. An important reason for this is so that equivalences like: | |||
| ::<math>(\phi \lor \exists x \psi) \leftrightarrow \exists x (\phi \lor \psi)</math>, | |||
| where ''x'' is not free in φ, are logically valid. This equivalence is not logically valid when empty structures are permitted (e.g. let φ be <math>\forall y ( y = y)</math> and ψ be <math> x = x</math>). So the proof theory of first-order logic becomes much more complicated when empty structures are permitted, but the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study have non-empty domains.<ref>{{Citation | last1=Hailperin | first1=Theodore | title=Quantification theory and empty individual-domains | id={{MathSciNet | id = 0057820}} | year=1953 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=18 | pages=197–200}}</ref><ref>{{Citation | last1=Quine | first1=W. V. | title=Quantification and the empty domain | id={{MathSciNet | id = 0064715}} | year=1954 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=19 | pages=177–179}}</ref> The difficulty with empty domains is certain inference rules that permit quantifiers to be passed across logical connectives. For concreteness, look at | |||
| ::<math>\forall y ( y = y) \lor \exists x ( x = x) </math> | |||
| This is satisfied by an empty domain. To put this in ], we want to move the existential quantifier to obtain | |||
| ::<math>\exists x ( \forall y ( y = y) \lor x = x) </math> | |||
| But this new formula is not satisfied by an empty domain, as there is no element with which the existential quantifier can be instantiated. The underlying issue is that the scope of the existential quantifier has changed to include the left disjunct. | |||
| Empty relations, however, don't cause this problem since there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. | |||
| ===== Individual constants ===== | |||
| The ''individual constants'' of <math> \mathcal{Q}</math> are ] 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 ===== | |||
| {{Main|Propositional variable}} | |||
| The sentential variables in <math> \mathcal{Q}</math> are each assigned a ], true ('''T''') or false ('''F'''). The interpretation of a propositional variable is one of the two truth-values ''true'' and ''false.''<ref>{{cite book | |||
| | last = Mates | |||
| | first = Benson | |||
| | title = Elementary Logic, Second Edition | |||
| | publisher = ] | |||
| | date = 1972 | |||
| | location = New York | |||
| | pages = p. 56 | |||
| | isbn = 019501491X }}</ref> | |||
| * ''P'': 'T' | |||
| * ''Q'': 'F' | |||
| ===== Predicates symbols ===== | |||
| {{Main|Predicate variable}} | |||
| The predicate symbols of <math> \mathcal{Q}</math> are each assigned some property or relation defined for objects in the domain. For instance, we can designate the following relations. | |||
| * F<sup>1</sup>: {Socrates, Plato} | |||
| * G<sup>1</sup>: {Plato, Aristotle} | |||
| * H<sup>1</sup>: {Socrates, Plato, Aristotle} | |||
| * R<sup>2</sup>: {<Socrates, Plato>,<Socrates, Aristotle>,<Socrates, Socrates>} | |||
| * R<sup>3</sup>: {<Socrates, Plato, Aristotle>,<Aristotle, Socrates, Plato>} | |||
| * S<sup>2</sup>: {<Socrates, Socrates>,<Plato, Plato>,<Aristotle, Aristotle>} | |||
| In general, each n-ary predicate symbol is assigned an n-ary relation. | |||
| The ] are given their usual truth-functional meanings, however, they may stand between formulas that for a given interpretation are neither true nor false. ]s are understood to refer exclusively to members of the domain of the interpretation.<ref name="metalogic"/> | |||
| ==== Interpretation of a sentence of first order logic ==== | |||
| An interpretation of a sentence Φ of a first order formal language <math>\mathcal{Q}</math> consists of a non-] <math>\mathcal{D}</math> together with an assignment that associates with each individual constant of Φ some element of <math>\mathcal{D}</math>, with each sentential symbol of Φ one of the truth-values '''T''' or '''F''', with each n-ary operation or function symbol in Φ an n-ary operation whose operands are exclusively from <math>\mathcal{D}</math>, with each n-ary predicate in Φ an n-ary relation among elements of <math>\mathcal{D}</math>, and, optionally, with some binary predicate <math>\mathcal{K}</math> the ] among elements of <math>\mathcal{D}</math>. | |||
| ==== 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 ] of objects. We must account for all of the various forms that a formula may take within <math> \mathcal{Q} </math>. Also, instead of talking about ] and ] we speak of ]s of ]s of objects. <ref name="metalogic"/> | |||
| A sentence which is a ] standing alone by itself is true under an interpretation <math>\mathcal{I}</math> ] <math>\mathcal{I}</math> assigns the truth value 'T' to that variable. | |||
| Any other ] is true under <math>\mathcal{I}</math> ] in the case of predicates of degree 1, the member of the domain that <math>\mathcal{I}</math> assigns to the individual constant is a member of the set which <math>\mathcal{I}</math> assigns to the predicate; and | |||
| in the case of predicates of degree 2, the members of the domain that <math>\mathcal{I}</math> assigns to the two individual constants are members, in the order in which their representations occur in the sentence, or of an ] in the ] that <math>\mathcal{I}</math> assigns to the predicate; and | |||
| in the case of predicates of degree ''n'', the members of the domain that <math>\mathcal{I}</math> assigns to the ''n'' individual constants are members, in the order in which their representations occur in the sentence, or of an ordered ] in the relation that <math>\mathcal{I}</math> assigns to the predicate. | |||
| A ] is true under <math>\mathcal{I}</math> if and only if | |||
| *it is of the form {{not}}Φ and Φ is not true under <math>\mathcal{I}</math>; or | |||
| *it is of the form (Φ {{and}} Ψ) and both Φ and Ψ are true under <math>\mathcal{I}</math>; or | |||
| *it is of the form (Φ {{or-}} Ψ) and either Φ or Ψ or both are true under <math>\mathcal{I}</math>; or | |||
| *it is of the form (Φ {{imp}} Ψ) and either Ψ is true under <math>\mathcal{I}</math> or Φ is not true under <math>\mathcal{I}</math>; or | |||
| *it is of the form (Φ {{eqv}} Ψ) and either both Φ and Ψ are true under <math>\mathcal{I}</math> or neither Φ nor Ψ are true under <math>\mathcal{I}</math>. | |||
| A general sentence is true under <math>\mathcal{I}</math> if and only if it is of the form (α)Φ and Φα/β is true under every β-variant of <math>\mathcal{I}</math> (where Φα/β is the result of replacing all free occurrences of the variable α in Φ by occurrences of an individual constant β); or | |||
| it is of the form ({{exist}}α)Φ and Φα/β is true under at least one β-variant of <math>\mathcal{I}</math>; | |||
| Any sentence that is not true under <math>\mathcal{I}</math> is false under <math>\mathcal{I}</math>. | |||
| ==== Further semantic concepts for first-order formal systems ==== | |||
| A sentence of <math>\mathcal{Q}</math> is valid (or ], written ' {{models}}Φ ') iff it is true under every interpretation. | |||
| A sentence Φ is a ] of a set of sentences Γ (or ' Γ {{models}} Φ ') iff there is no interpretation under which all the members of Γ are true and Φ is false. | |||
| A set of sentences Γ is ] (or satisfiable) iff there is an interpretation under which all the sentences of Γ are true. | |||
| A sentence which is false under every interpretation is called unsatisfiable.<ref name=sakharov>Alex Sakharov From MathWorld--A Wolfram Web Resource.</ref> | |||
| An interpretation <math>\mathcal{I}</math> is a '''model''' for a sentence Φ (or set of sentences Γ) iff Φ is (or all the members of Γ are) true under <math>\mathcal{I}</math>. | |||
| == 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 ]. | |||
| == Intended interpretation == | |||
| {{main|Intended interpretation}} | |||
| One who constructs a ] usually has in mind from the outset some interpretation of this system. While this ''intended interpretation'' can have no explicit indication in the ] --since these rules must be strictly formal --the author's intention respecting interpretation naturally affects his choice of the ] and ] 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 ]s; 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 <math>\mathcal{I}</math><sub>j</sub> is directly ] from a sentence <math>\mathcal{I}</math><sub>i</sub>, then <math>\mathcal{I}</math><sub>i</sub> {{imp}} <math>\mathcal{I}</math><sub>j</sub> turns out to be a true sentence (under the customary interpretation of "{{imp}}" as meaning ]). These requirements ensure that all provable sentences also come out to be true.<ref name="itslaia"/> | |||
| 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. <ref>The Concept and the Role of the Model in Mathematics and Natural and Social Sciences</ref> | |||
| === Standard and non-standard models of arithmetic === | |||
| {{main|Non-standard model}} | |||
| A distinction is made between ''standard'' and ''non-standard'' models of ], 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 ] as the domain of discourse, and interpreting "0" as zero, "1" as one, "+" as addition, and "x" as multiplication, etcetera. All models that are ] to the one just given are also called standard; these models all satisfy the ]. There also exist ], 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. <ref>Cambridge Dictionary of Philosophy</ref> | |||
| == Logical and descriptive interpretations == | |||
| {{main|Logical interpretation|Descriptive interpretation}} | |||
| ], in his ''Introduction to Semantics'' makes a distinction between formal interpretations which are '']s'' (also called ''mathematical interpretation'' or ''logico-mathematical interpretation'') and '']s'' (also called a ''factual interpretation'').<ref name="itslaia">], ''Introduction to Symbolic Logic and its Applications''</ref> | |||
| An interpretation is a ''descriptive interpretation'' if at least one of the undefined symbols of the ] becomes, in the interpretation, a '']'' (i.e., the name of single objects, or observable properties). | |||
| An interpretation is a ''descriptive interpretation'' if it is not a logical interpretation.<ref name="itslaia"/> | |||
| == 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 Θ of ]s or other statements of a given deductive theory. | |||
| If a sentence A can be derived from the statements of the system Θ it remains valid for any interpretation of this system. The existence of an interpretation of Θ for which A is not valid is proof that the sentence cannot be derived from the system Θ. | |||
| If we have an arbitrary ] <math>\mathcal{J}</math> which we presume to be ] (which may be the same theory as the set of statements of system Θ) the method of proof by interpretation tries to find an interpretation of the system Θ within theory <math>\mathcal{J}</math> such that not the sentence A itself, but its negation becomes a theorem of the theory <math>\mathcal{J}</math>. | |||
| The method of proof by interpretation can be used to establish the ] 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 Θ consists of the remaining axioms of the system. | |||
| == Mathematical models == | |||
| {{main|Structure (mathematical logic)}} | |||
| In ] and in ], a ''structure'' is a type of formal interpretation which consists of an underlying ] along with a collection of ] functions and relations which are defined on it. | |||
| {{main|Valuation (mathematics)}} | |||
| Informally, a ''valuation'' is an assignment of particular values to the variables in a mathematical statement or equation. | |||
| {{main|Interpretation (model theory)}} | |||
| In ], '''interpretation''' of a ] ''M'' in another structure ''N'' (typically of a different ]) is a technical notion that approximates the idea of representing ''M'' inside ''N''. | |||
| {{main|Mathematical model}} | |||
| A ''mathematical model'' is a type of formal interpretation that uses ] language to describe a ]. | |||
| == Scientific models == | |||
| {{main|Scientific model}} | |||
| Attempts to ] the empirical sciences use a ''descriptive interpretation'' to model reality. The aim of these attempts is to construct a ] for which ] 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 ], ] and ] about different types of specialized scientific modeling. | |||
| == See also == | |||
| * ] | |||
| * ]: an economic model for fisheries | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ]: the OMG standard for defining metamodels | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| == References == | |||
| * ], ''Introduction to Semantics'' | |||
| * ], ''Introduction to Symbolic Logic and its Applications'' | |||
| * R. Frigg and S. Hartmann, . Entry in the ''Stanford Encyclopedia of Philosophy''. | |||
| * W. Quine, ''From a Logical Point of View'', Harper Torchbooks, 1961. | |||
| {{reflist}} | |||
| {{logic}} | |||
| ] | |||
| ] | |||
Latest revision as of 23:40, 23 April 2009
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