Extensional logic
   HOME

TheInfoList



OR:

Intensional logic is an approach to
predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...
that extends
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
, which has quantifiers that range over the individuals of a universe (''
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values t ...
''), by additional quantifiers that range over terms that may have such individuals as their value (''
intension In any of several fields of study that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language—an intension is any property or quality connoted by a word, phrase, or another s ...
s''). The distinction between intensional and extensional entities is parallel to the distinction between
sense and reference In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the ...
.


Overview

Logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
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 Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowledg ...
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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
(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 Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
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 A reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''nam ...
of an individual name (the "designated" object named by that) is called its extension; and as for sentences, their
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
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 Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
had already studied modal
syllogism A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defin ...
s.
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
developed a kind of
two-dimensional semantics Two-dimensionalism is an approach to semantics in analytic philosophy. It is a theory of how to determine the sense and reference of a word and the truth-value of a sentence. It is intended to resolve the puzzle: How is it possible to discover emp ...
: 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 In any of several fields of study that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language—an intension is any property or quality connoted by a word, phrase, or another s ...
. 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 Transparent intensional logic (frequently abbreviated as TIL) 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 vie ...
*
Modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...


Modal logic

Modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
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 Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistem ...
, 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 In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are nam ...
). 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 operator A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non- truth-functional in the following se ...
s. As mentioned, precursors of modal logic include
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
. 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 An open formula is a formula that contains at least one free variable. An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ...
, the variable is bound by a quantifier whose scope includes the whole intensional subterm. Modern modal logic began with the
Clarence Irving Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964) was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logician, he later branched into epis ...
. His work was motivated by establishing the notion of strict implication. The
possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...
s approach enabled more exact study of semantical questions. Exact formalization resulted in
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
(developed by
Saul Kripke Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...
,
Jaakko Hintikka Kaarlo Jaakko Juhani Hintikka (; ; 12 January 1929 – 12 August 2015) was a Finnish philosopher and logician. Hintikka is regarded as the founder of formal epistemic logic and of game semantics for logic. Life and career Hintikka was born in ...
, Stig Kanger).


Type-theoretical intensional logic

Already in 1951,
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
had developed an intensional
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
. 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 world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...
s approach to semantics provided tools for a comprehensive study in intensional semantics.
Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...
could preserve the most important advantages of Church's intensional calculus in his system. Unlike its forerunner,
Montague grammar 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 th ...
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 In logic, extensionality, or extensional equality, refers to principles that judge objects to be equality (mathematics), equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned wi ...
* Frege–Church ontology *
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
* Temperature paradox *
Relevance Relevance is the connection between topics that makes one useful for dealing with the other. Relevance is studied in many different fields, including cognitive science, logic, and library and information science. Epistemology studies it in gener ...


Notes


References

* Melvin Fitting (2004). First-order intensional logic. ''
Annals of Pure and Applied Logic The ''Annals of Pure and Applied Logic'' is a peer-reviewed scientific journal published by Elsevier that publishes papers on applications of mathematical logic in mathematics, in computer science, and in other related disciplines.
'' 127:171–193. Th
2003 preprint
is used in this article. *Melvin Fitting (2007)
Intensional Logic
In the ''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
''. * . Translation of the title: “Classical, modal and intensional logic”. * . Original: “The Development of Logic”. Translation of the title of the Appendix by Ruzsa, present only in Hungarian publication: “The last two decades”. * . Translation of the title: “Syntax and semantics of logic”. * . * Translation of the title: “Introduction to modern logic”.


External links

* {{Formal semantics Non-classical logic Philosophical logic Predicate logic