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Intensional logic embraces the logical study of intensional languages. While in extensional languages all of their functors are extensional (and that suffices in many formal languages developed for formalizing special fields in mathematics or science),^[1] intensional languages have at least one intensional functor.^[2]

Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes).^[3] Logic is not a closed, accomplished science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language^[4] (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 use directly of the underlying natural language.^[5] Functors belong to the most important categories in logical grammar (alongside with sentence and individual name): a functor can be regarded as an "unaccomplished" expression with argument places to fill in. It acts like a function, taking on input expressions, resulting in a new, output expression.^[6]

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 logical categories: the truth of a sentence and the "designated" object named by an individual name is called its extension.^[7]

Some functors are simpler than other ones. 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 function turning the extension of its input(s) into the extendsion 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.^[7]

Natural languages abound with intensional functors,^[8] this can be illustrated by intensional statements. Extensional logic cannot reach inside such fine logical structures of the language, it 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.^[9] Gottlob Frege developed a two-dimensional approach to semantics, introducing a distinction between two semantic values: sentences (and individual terms) have both an extension and an intension.^[10] These semantic values can be interpreted, transferred also for functors (except for intensional functors, they have only intension).

There are some intensional logic systems that claim to fully analyze the common language:

Modal logic

Modal logic is historically the earliest are in intensional logic. It can be regarded also as the most simple appearance of such studies: it extends extensional logic just with a few sentential functors:^[11] these are intensional, and they are interpreted (in the metarules of semantics) as quantifying over possible worlds. Moreoever, they are related to one another by similar dualities like quantifiers do.^[12] Syntactically, they are not quantifiers, they do not bind variables,^[13] they appear in the grammar as sentential functors,^[12] they are called modal operators.^[13]

As mentioned, precursors of modal logic includes Aristotle. Medieval scholastic 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.^[9]

Modern modal logic began with the Clarence Irving Lewis, his work was motivated by establishing the notion of strict implication.^[14] Possible worlds approach enabled more exact study of semantical questions. Exact formalization resulted in Kripke semantics (developed by Saul Kripke, Jaakko Hintikka, Stig Kanger).^[15]

Notes

^ Ruzsa 2000, p. 26
^ Ruzsa 1988, p. 182
^ Ruzsa 2000, p. 10
^ Ruzsa 2000, p. 13
^ Ruzsa 2000, p. 12
^ Ruzsa 2000, p. 22
^ ^a ^b Ruzsa 2000, pp. 22–23 Cite error: The named reference "logical_semantics" was defined multiple times with different content (see the help page).
^ Ruzsa 1987, p. 724
^ ^a ^b Ruzsa 2000, pp. 246–247
^ Ruzsa 2000, p. 24
^ Ruzsa 2000, p. 247
^ ^a ^b Ruzsa 2000, p. 245
^ ^a ^b Ruzsa 2000, p. 269
^ Ruzsa 2000, p. 256
^ Ruzsa 2000, p. 247

References

Ruzsa, Imre (1987), "Függelék. Az utolsó két évtized", in Kneale, William; Kneale, Martha (eds.), A logika fejlődése, Budapest: Gondolat, pp. 695–734, ISBN 963 281 780 X
Ruzsa, Imre (1988), Logikai szintaxis és szemantika, vol. I, Budapest: Akadémiai Kiadó, ISBN 963 05 4720 1
Ruzsa, Imre (2000), Bevezetés a modern logikába, Osiris tankönyvek, Budapest: Osiris, ISBN 963 379 978 3

External links

Intensional Logic - Stanford Encyclopedia of Philosophy

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[1] Ruzsa 2000, p. 26

[2] Ruzsa 1988, p. 182

[3] Ruzsa 2000, p. 10

[4] Ruzsa 2000, p. 13

[logical_grammar-5] Ruzsa 2000, p. 12

[functor-6] Ruzsa 2000, p. 22

[logical_semantics-7] Ruzsa 2000, pp. 22–23 Cite error: The named reference "logical_semantics" was defined multiple times with different content (see the help page).

[8] Ruzsa 1987, p. 724

[Aristotle-9] Ruzsa 2000, pp. 246–247

[10] Ruzsa 2000, p. 24

[11] Ruzsa 2000, p. 247

[sf-12] Ruzsa 2000, p. 245

[qpw-13] Ruzsa 2000, p. 269

[14] Ruzsa 2000, p. 256

[15] Ruzsa 2000, p. 247

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 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 logical categories: the truth of a sentence and the "designated" object named by an individual name is called its [[extension]].<ref name=logical_semantics>{{harvnb|Ruzsa|2000|pp=22–23}}</ref>
+Some functors are simpler than other ones. 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 function turning the extension of its input(s) into the extendsion 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''.<ref name=logical_semantics>{{harvnb|Ruzsa|2000|pp=25–26}}</ref>
 Natural languages abound with intensional functors,<ref>{{harvnb|Ruzsa|1987|p=724}}</ref> this can be illustrated by [[intensional statement]]s. [[Extensional logic]] cannot reach inside such fine logical structures of the language, it 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 [[syllogism]]s.<ref name=Aristotle>{{harvnb|Ruzsa|2000|pp=246–247}}</ref> [[Gottlob Frege]] developed a [[Two dimensionalism|two-dimensional]] approach to semantics, introducing a distinction between two semantic values: sentences (and individual terms) have both an [[extension]] and an [[intension]].<ref>{{harvnb|Ruzsa|2000|p=24}}</ref> These semantic values can be interpreted, transferred also for functors (except for intensional functors, they have only intension).