# Intensional logic

Last updated

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). [1] 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 [2] (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. [3] Functors belong to the most important categories in logical grammar (along with basic categories like sentence and individual name): [4] 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. [5] Thus, a functor acts like a function sign, [6] taking on input expressions, resulting in a new, output expression. [5]

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. [7]

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. [8]

Natural languages abound with intensional functors, [9] 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. [10] Gottlob Frege developed a kind of two dimensional semantics: for resolving questions like those of intensional statements, he has introduced a distinction between two semantic values: sentences (and individual terms) have both an extension and an intension. [6] 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. [11]

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

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). [12]

Modal logic can be regarded also as the most simple appearance of such studies: it extends extensional logic just with a few sentential functors: [13] these are intensional, and they are interpreted (in the metarules of semantics) as quantifying over possible worlds. For example, the Necessity operator (the 'square') when applied to a sentence A says 'The sentence "('square')A" is true in world i 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 iff 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 covers 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 quantifiers do [14] (for example by the analogous correspondents of De Morgan's laws). I.e., Something is necessary iff its negation is not possible, i.e. inconsistent. Syntactically, the operators are not quantifiers, they do not bind variables, [15] 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, [14] they are called modal operators. [15]

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. [10]

Modern modal logic began with the Clarence Irving Lewis, his work was motivated by establishing the notion of strict implication. [16] Possible worlds approach enabled more exact study of semantical questions. Exact formalization resulted in Kripke semantics (developed by Saul Kripke, Jaakko Hintikka, Stig Kanger). [13]

## 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 know in establishing semantics for modal logic in a formal way, because they had not been invented then: [17] Church has not provided formal semantic definitions. [18]

Later, possible world 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. [17]

## Notes

1. Ruzsa 2000 , p. 10
2. Ruzsa 2000 , p. 13
3. Ruzsa 2000 , p. 12
4. Ruzsa 2000 , p. 21
5. Ruzsa 2000 , p. 22
6. Ruzsa 2000 , p. 24
7. Ruzsa 2000 , pp. 22–23
8. Ruzsa 2000 , pp. 25–26
9. Ruzsa 1987 , p. 724
10. Ruzsa 2000 , pp. 246–247
11. Ruzsa 2000 , p. 128
12. Ruzsa 2000 , p. 252
13. Ruzsa 2000 , p. 247
14. Ruzsa 2000 , p. 245
15. Ruzsa 2000 , p. 269
16. Ruzsa 2000 , p. 256
17. Ruzsa 2000 , p. 297
18. Ruzsa 1989 , p. 492

## Related Research Articles

In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase, or another symbol. In the case of a word, the word's definition often implies an intension. For instance, the intensions of the word plant include properties such as "being composed of cellulose", "alive", and "organism", among others. A comprehension is the collection of all such intensions.

Saul Aaron Kripke is an American philosopher and logician in the analytic tradition. He is a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. Since the 1960s, Kripke has been a central figure in a number of fields related to mathematical logic, modal logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and recursion theory. Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts.

Modal logic is a collection of formal systems originally developed and still widely used to represent statements about necessity and possibility. For instance, the modal formula can be read as "if P is necessary, then it is also possible". This formula is widely regarded as valid when necessity and possibility are understood with respect to knowledge, as in epistemic modal logic. Whether it is also valid with legal or moral necessity is a question debated since Sophocles' play Antigone.

De dicto and de re are two phrases used to mark a distinction in intensional statements, associated with the intensional operators in many such statements. The distinction is used regularly in metaphysics and in philosophy of language.

Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes use of the notions of intensional logic, via Kripke models. Montague pioneered this approach in the 1960s and early 1970s.

In logic, the semantics of logic or formal semantics is the study of the semantics, or interpretations, of formal and natural languages usually trying to capture the pre-theoretic notion of entailment.

Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke.

In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.

David Benjamin Kaplan is an American philosopher. He is the Hans Reichenbach Professor of Scientific Philosophy at the UCLA Department of Philosophy. His philosophical work focuses on the philosophy of language, logic, metaphysics, epistemology and the philosophy of Frege and Russell. He is best known for his work on demonstratives, propositions, and reference in intensional contexts. He was elected a Fellow of the American Academy of Arts & Sciences in 1983 and a Corresponding Fellow of the British Academy in 2007.

In the philosophy of language, the descriptivist theory of proper names is the view that the meaning or semantic content of a proper name is identical to the descriptions associated with it by speakers, while their referents are determined to be the objects that satisfy these descriptions. Bertrand Russell and Gottlob Frege have both been associated with the descriptivist theory, which is sometimes called the Frege–Russell view.

Transparent intensional logic is a logical system created by Pavel Tichý. Due to its rich procedural semantics TIL is in particular apt for the logical analysis of natural language. From the formal point of view, TIL is a hyperintensional, partial, typed lambda calculus.

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.

In linguistics and grammar, a quantifier is a type of determiner, such as all, some, many, few, a lot, and no, that indicates quantity.

Donkey sentences are sentences that contain a pronoun with clear meaning but whose syntactical role in the sentence poses challenges to grammarians. Such sentences defy straightforward attempts to generate their formal language equivalents. The difficulty is with understanding how English speakers parse such sentences.

In logic, the term statement is variously understood to mean either:

Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic.

This is an index of articles in philosophy of language

Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.

Japaridze's polymodal logic (GLP), is a system of provability logic with infinitely many modal (provability) operators. This system has played an important role in some applications of provability algebras in proof theory, and has been extensively studied since the late 1980s. It is named after Giorgi Japaridze.

## References

• Melvin Fitting (2004). First-order intensional logic. Annals of Pure and Applied Logic 127:171–193. The 2003 preprint is used in this article.
• — (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), 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, 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”.