Intensional logic

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Intensional logic is an approach to
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...
that extends
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...
, 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 (predicate logic), the set of tuples of values that satisfy the predicate * Exte ...
''), 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 Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objectiv ...
s''). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.

# Overview

Logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
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 Semantics (from grc, wikt:σημαντικός, σημαντικός ''sēmantikós'', "significant") is the study of reference, Meaning (philosophy), meaning, or truth. The term can be used to refer to subfields of several distinct discipline ...
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 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 ''name'' ...
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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability ...
s.
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
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 In any of several fields of study that treat the use of signs — for example, in linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objectiv ...
. 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 is a collection of formal systems developed to represent statements about Modality (natural language), necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and Formal semantics (natural ...

# Modal logic

Modal logic Modal logic is a collection of formal systems developed to represent statements about Modality (natural language), necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and Formal semantics (natural ...
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
metarule Meta (from the Ancient Greek, Greek μετά, ''wikt:meta-, meta'', meaning "after" or "beyond") is a prefix meaning "more comprehensive" or "transcending". In modern nomenclature, ''meta''- can also serve as a prefix meaning self-referential, a ...
s 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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
. 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. Modern modal logic began with the
Clarence Irving Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logici ...
, 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 Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American philosopher and logician in the analytic philosophy, analytic tradition. He was a Distinguished Professor of Philosophy at the Graduate Center of the City University o ...
, Jaakko Hintikka, Stig Kanger).

# Type-theoretical intensional logic

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, list of logicians, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoreti ...
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 generalizati ...
. 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 __notoc__ 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 logic, higher-order predicate logic and lambda calc ...
was built in a purely semantical way: a simpler treatment became possible, thank to the new formal tools invented since Church's work.

*
Extensionality In logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science inves ...
* Frege–Church ontology * Kripke semantics * Temperature paradox *
Relevance Relevance is the concept of one topic being Premise, connected to another topic in a way that makes it useful to consider the second topic when considering the first. The concept of relevance is studied in many different fields, including cogn ...

# References

* Melvin Fitting (2004). First-order intensional logic. '' Annals of Pure and Applied Logic'' 127:171–193. Th
2003 preprint