Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (extensions), by additional quantifiers that range over terms that may have such individuals as their value (intensions). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.
Overview
Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes). Logic is not a closed, completed science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language (sentences regarded as atomic, or splitting them to predicates applied to individual terms, or even revealing such fine logical structures like modal, temporal, dynamic, epistemic ones).
In order to achieve its special goal, logic was forced to develop its own formal tools, most notably its own grammar, detached from simply making direct use of the underlying natural language.Functors (also known as function words) belong to the most important categories in logical grammar (along with basic categories like sentence and individual name): a functor can be regarded as an "incomplete" expression with argument places to fill in. If we fill them in with appropriate subexpressions, then the resulting entirely completed expression can be regarded as a result, an output. Thus, a functor acts like a function sign, taking on input expressions, resulting in a new, output expression.
Semantics links expressions of language to the outside world. Also logical semantics has developed its own structure. Semantic values can be attributed to expressions in basic categories: the reference of an individual name (the "designated" object named by that) is called its extension; and as for sentences, their truth value is their extension.
As for functors, some of them are simpler than others: extension can be attributed to them in a simple way. In case of a so-called extensional functor we can in a sense abstract from the "material" part of its inputs and output, and regard the functor as a function turning directly the extension of its input(s) into the extension of its output. Of course, it is assumed that we can do so at all: the extension of input expression(s) determines the extension of the resulting expression. Functors for which this assumption does not hold are called intensional.
Natural languages abound with intensional functors; this can be illustrated by intensional statements. Extensional logic cannot reach inside such fine logical structures of the language, but stops at a coarser level. The attempts for such deep logical analysis have a long past: authors as early as Aristotle had already studied modal syllogisms.Gottlob Frege developed a kind of two-dimensional semantics: for resolving questions like those of intensional statements, Frege introduced a distinction between two semantic values: sentences (and individual terms) have both an extension and an intension. These semantic values can be interpreted, transferred also for functors (except for intensional functors, they have only intension).
As mentioned, motivations for settling problems that belong today to intensional logic have a long past. As for attempts of formalizations, the development of calculi often preceded the finding of their corresponding formal semantics. Intensional logic is not alone in that: also Gottlob Frege accompanied his (extensional) calculus with detailed explanations of the semantical motivations, but the formal foundation of its semantics appeared only in the 20th century. Thus sometimes similar patterns repeated themselves for the history of development of intensional logic like earlier for that of extensional logic.
There are some intensional logic systems that claim to fully analyze the common language:
- Transparent intensional logic
- Modal logic
Modal logic
Modal logic is historically the earliest area in the study of intensional logic, originally motivated by formalizing "necessity" and "possibility" (recently, this original motivation belongs to alethic logic, just one of the many branches of modal logic).
Modal logic can be regarded also as the most simple appearance of such studies: it extends extensional logic just with a few sentential functors: these are intensional, and they are interpreted (in the metarules of semantics) as quantifying over possible worlds. For example, the Necessity operator (the 'box') when applied to a sentence A says 'The sentence "('box')A" is true in world i if and only if it is true in all worlds accessible from world i'. The corresponding Possibility operator (the 'diamond') when applied to A asserts that "('diamond')A" is true in world i if and only if A is true in some worlds (at least one) accessible to world i. The exact semantic content of these assertions therefore depends crucially on the nature of the accessibility relation. For example, is world i accessible from itself? The answer to this question characterizes the precise nature of the system, and many exist, answering moral and temporal questions (in a temporal system, the accessibility relation relates states or 'instants' and only the future is accessible from a given moment. The Necessity operator corresponds to 'for all future moments' in this logic. The operators are related to one another by similar dualities to those relating existential and universal quantifiers (for example by the analogous correspondents of De Morgan's laws). I.e., Something is necessary if and only if its negation is not possible, i.e. inconsistent. Syntactically, the operators are not quantifiers, they do not bind variables, but govern whole sentences. This gives rise to the problem of referential opacity, i.e. the problem of quantifying over or 'into' modal contexts. The operators appear in the grammar as sentential functors, they are called modal operators.
As mentioned, precursors of modal logic include Aristotle. Medieval scholarly discussions accompanied its development, for example about de re versus de dicto modalities: said in recent terms, in the de re modality the modal functor is applied to an open sentence, the variable is bound by a quantifier whose scope includes the whole intensional subterm.
Modern modal logic began with the Clarence Irving Lewis. His work was motivated by establishing the notion of strict implication. The possible worlds approach enabled more exact study of semantical questions. Exact formalization resulted in Kripke semantics (developed by Saul Kripke, Jaakko Hintikka, Stig Kanger).
Type-theoretical intensional logic
Already in 1951, Alonzo Church had developed an intensional calculus. The semantical motivations were explained expressively, of course without those tools that we now use for establishing semantics for modal logic in a formal way, because they had not been invented then: Church did not provide formal semantic definitions.
Later, the possible worlds approach to semantics provided tools for a comprehensive study in intensional semantics. Richard Montague could preserve the most important advantages of Church's intensional calculus in his system. Unlike its forerunner, Montague grammar was built in a purely semantical way: a simpler treatment became possible, thank to the new formal tools invented since Church's work.
See also
- Extensionality
- Frege–Church ontology
- Kripke semantics
- Temperature paradox
- Relevance
Notes
- Ruzsa 2000, p. 10
- Ruzsa 2000, p. 13
- Ruzsa 2000, p. 12
- Ruzsa 2000, p. 21
- Ruzsa 2000, p. 22
- Ruzsa 2000, p. 24
- Ruzsa 2000, pp. 22–23
- Ruzsa 2000, pp. 25–26
- Ruzsa 1987, p. 724
- Ruzsa 2000, pp. 246–247
- Ruzsa 2000, p. 128
- Ruzsa 2000, p. 252
- Ruzsa 2000, p. 247
- Ruzsa 2000, p. 245
- Ruzsa 2000, p. 269
- Ruzsa 2000, p. 256
- Ruzsa 2000, p. 297
- Ruzsa 1989, p. 492
References
- Melvin Fitting (2004). First-order intensional logic. Annals of Pure and Applied Logic 127:171–193. The 2003 preprint Archived 2008-07-04 at the Wayback Machine is used in this article.
- Melvin Fitting (2007). Intensional Logic. In the Stanford Encyclopedia of Philosophy.
- Ruzsa, Imre (1984), Klasszikus, modális és intenzionális logika (in Hungarian), Budapest: Akadémiai Kiadó, ISBN 963-05-3084-8. Translation of the title: “Classical, modal and intensional logic”.
- Ruzsa, Imre (1987), "Függelék. Az utolsó két évtized", in Kneale, William; Kneale, Martha (eds.), A logika fejlődése (in Hungarian), Budapest: Gondolat, pp. 695–734, ISBN 963-281-780-X. Original: “The Development of Logic”. Translation of the title of the Appendix by Ruzsa, present only in Hungarian publication: “The last two decades”.
- Ruzsa, Imre (1988), Logikai szintaxis és szemantika (in Hungarian), vol. 1, Budapest: Akadémiai Kiadó, ISBN 963-05-4720-1. Translation of the title: “Syntax and semantics of logic”.
- Ruzsa, Imre (1989), Logikai szintaxis és szemantika, vol. 2, Budapest: Akadémiai Kiadó, ISBN 963-05-5313-9.
- Ruzsa, Imre (2000), Bevezetés a modern logikába, Osiris tankönyvek (in Hungarian), Budapest: Osiris, ISBN 963-379-978-3 Translation of the title: “Introduction to modern logic”.
External links
- Fitting, Melvin. "Intensional logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
Intensional logic is an approach to predicate logic that extends first order logic which has quantifiers that range over the individuals of a universe extensions by additional quantifiers that range over terms that may have such individuals as their value intensions The distinction between intensional and extensional entities is parallel to the distinction between sense and reference OverviewLogic is the study of proof and deduction as manifested in language abstracting from any underlying psychological or biological processes Logic is not a closed completed science and presumably it will never stop developing the logical analysis can penetrate into varying depths of the language sentences regarded as atomic or splitting them to predicates applied to individual terms or even revealing such fine logical structures like modal temporal dynamic epistemic ones In order to achieve its special goal logic was forced to develop its own formal tools most notably its own grammar detached from simply making direct use of the underlying natural language Functors also known as function words belong to the most important categories in logical grammar along with basic categories like sentence and individual name a functor can be regarded as an incomplete expression with argument places to fill in If we fill them in with appropriate subexpressions then the resulting entirely completed expression can be regarded as a result an output Thus a functor acts like a function sign taking on input expressions resulting in a new output expression Semantics links expressions of language to the outside world Also logical semantics has developed its own structure Semantic values can be attributed to expressions in basic categories the reference of an individual name the designated object named by that is called its extension and as for sentences their truth value is their extension As for functors some of them are simpler than others extension can be attributed to them in a simple way In case of a so called extensional functor we can in a sense abstract from the material part of its inputs and output and regard the functor as a function turning directly the extension of its input s into the extension of its output Of course it is assumed that we can do so at all the extension of input expression s determines the extension of the resulting expression Functors for which this assumption does not hold are called intensional Natural languages abound with intensional functors this can be illustrated by intensional statements Extensional logic cannot reach inside such fine logical structures of the language but stops at a coarser level The attempts for such deep logical analysis have a long past authors as early as Aristotle had already studied modal syllogisms Gottlob Frege developed a kind of two dimensional semantics for resolving questions like those of intensional statements Frege introduced a distinction between two semantic values sentences and individual terms have both an extension and an intension These semantic values can be interpreted transferred also for functors except for intensional functors they have only intension As mentioned motivations for settling problems that belong today to intensional logic have a long past As for attempts of formalizations the development of calculi often preceded the finding of their corresponding formal semantics Intensional logic is not alone in that also Gottlob Frege accompanied his extensional calculus with detailed explanations of the semantical motivations but the formal foundation of its semantics appeared only in the 20th century Thus sometimes similar patterns repeated themselves for the history of development of intensional logic like earlier for that of extensional logic There are some intensional logic systems that claim to fully analyze the common language Transparent intensional logic Modal logicModal logicModal logic is historically the earliest area in the study of intensional logic originally motivated by formalizing necessity and possibility recently this original motivation belongs to alethic logic just one of the many branches of modal logic Modal logic can be regarded also as the most simple appearance of such studies it extends extensional logic just with a few sentential functors these are intensional and they are interpreted in the metarules of semantics as quantifying over possible worlds For example the Necessity operator the box when applied to a sentence A says The sentence box A is true in world i if and only if it is true in all worlds accessible from world i The corresponding Possibility operator the diamond when applied to A asserts that diamond A is true in world i if and only if A is true in some worlds at least one accessible to world i The exact semantic content of these assertions therefore depends crucially on the nature of the accessibility relation For example is world i accessible from itself The answer to this question characterizes the precise nature of the system and many exist answering moral and temporal questions in a temporal system the accessibility relation relates states or instants and only the future is accessible from a given moment The Necessity operator corresponds to for all future moments in this logic The operators are related to one another by similar dualities to those relating existential and universal quantifiers for example by the analogous correspondents of De Morgan s laws I e Something is necessary if and only if its negation is not possible i e inconsistent Syntactically the operators are not quantifiers they do not bind variables but govern whole sentences This gives rise to the problem of referential opacity i e the problem of quantifying over or into modal contexts The operators appear in the grammar as sentential functors they are called modal operators As mentioned precursors of modal logic include Aristotle Medieval scholarly discussions accompanied its development for example about de re versus de dicto modalities said in recent terms in the de re modality the modal functor is applied to an open sentence the variable is bound by a quantifier whose scope includes the whole intensional subterm Modern modal logic began with the Clarence Irving Lewis His work was motivated by establishing the notion of strict implication The possible worlds approach enabled more exact study of semantical questions Exact formalization resulted in Kripke semantics developed by Saul Kripke Jaakko Hintikka Stig Kanger Type theoretical intensional logicAlready in 1951 Alonzo Church had developed an intensional calculus The semantical motivations were explained expressively of course without those tools that we now use for establishing semantics for modal logic in a formal way because they had not been invented then Church did not provide formal semantic definitions Later the possible worlds approach to semantics provided tools for a comprehensive study in intensional semantics Richard Montague could preserve the most important advantages of Church s intensional calculus in his system Unlike its forerunner Montague grammar was built in a purely semantical way a simpler treatment became possible thank to the new formal tools invented since Church s work See alsoExtensionality Frege Church ontology Kripke semantics Temperature paradox RelevanceNotesRuzsa 2000 p 10 Ruzsa 2000 p 13 Ruzsa 2000 p 12 Ruzsa 2000 p 21 Ruzsa 2000 p 22 Ruzsa 2000 p 24 Ruzsa 2000 pp 22 23 Ruzsa 2000 pp 25 26 Ruzsa 1987 p 724 Ruzsa 2000 pp 246 247 Ruzsa 2000 p 128 Ruzsa 2000 p 252 Ruzsa 2000 p 247 Ruzsa 2000 p 245 Ruzsa 2000 p 269 Ruzsa 2000 p 256 Ruzsa 2000 p 297 Ruzsa 1989 p 492ReferencesMelvin Fitting 2004 First order intensional logic Annals of Pure and Applied Logic 127 171 193 The 2003 preprint Archived 2008 07 04 at the Wayback Machine is used in this article Melvin Fitting 2007 Intensional Logic In the Stanford Encyclopedia of Philosophy Ruzsa Imre 1984 Klasszikus modalis es intenzionalis logika in Hungarian Budapest Akademiai Kiado ISBN 963 05 3084 8 Translation of the title Classical modal and intensional logic Ruzsa Imre 1987 Fuggelek Az utolso ket evtized in Kneale William Kneale Martha eds A logika fejlodese in Hungarian Budapest Gondolat pp 695 734 ISBN 963 281 780 X Original The Development of Logic Translation of the title of the Appendix by Ruzsa present only in Hungarian publication The last two decades Ruzsa Imre 1988 Logikai szintaxis es szemantika in Hungarian vol 1 Budapest Akademiai Kiado ISBN 963 05 4720 1 Translation of the title Syntax and semantics of logic Ruzsa Imre 1989 Logikai szintaxis es szemantika vol 2 Budapest Akademiai Kiado ISBN 963 05 5313 9 Ruzsa Imre 2000 Bevezetes a modern logikaba Osiris tankonyvek in Hungarian Budapest Osiris ISBN 963 379 978 3 Translation of the title Introduction to modern logic External linksFitting Melvin Intensional logic In Zalta Edward N ed Stanford Encyclopedia of Philosophy
