Extensional logic

TheInfoList

OR:

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 quanti ...
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 quantifie ...
, 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
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 Temporal may refer to: Entertainment * Temporal (band), an Australian metal band * ''Temporal'' (Radio Tarifa album), 1997 * ''Temporal'' (Love Spirals Downwards album), 2000 * ''Temporal'' (Isis album), 2012 * ''Temporal'' (video game), a 20 ...
, 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 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 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 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''). Computing In some program ...
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 statement 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 ...
s. 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 period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...
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 phil ...
developed a kind of two dimensional semantics: for resolving questions like those of
intensional statement 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 ...
s, 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 is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend othe ...

# Modal logic

Modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend othe ...
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 In 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 valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
). 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 period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...
. 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, 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 m ...
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 philosopher and logician in the analytic tradition. He was a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and eme ...
, Jaakko Hintikka, Stig Kanger).

# Type-theoretical intensional logic

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scienc ...
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 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 m ...
s 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.

*
Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal d ...
* 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 The Temperature Paradox or Partee's Paradox is a classic puzzle in formal semantics and philosophical logic. Formulated by Barbara Partee in the 1970s, it consists of the following argument, which speakers of English judge as wildly invalid. # Th ...
* Relevance

# References

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