Intensional logic is an approach to predicate logic

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

 that extends first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 extensional and intensional entities is parallel to the distinction between sense and reference

Sense and reference

Sinn and bedeutung are usually translated, respectively, as sense and reference. Two different aspects of some terms' meanings, a term's reference is the object that the term refers to, while the term's sense is the way that the term refers to that object.Sinn and bedeutung were introduced by...

.

Its place inside logic

Logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 is the study of proof and deduction

Deductive reasoning

Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

 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

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

, temporal

Temporal logic

In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...

, dynamic

Dynamic logic

Dynamic logic may mean:* In theoretical computer science, dynamic logic is a modal logic for reasoning about dynamic behaviour* In digital electronics, dynamic logic is a technique used for combinatorial circuit design* A different concept proposed by Leonid Perlovsky...

, epistemic

Epistemic logic

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy,...

 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 belong to the most important categories in logical grammar (alongside 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 in them 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

Reference is derived from Middle English referren, from Middle French rèférer, from Latin referre, "to carry back", formed from the prefix re- and ferre, "to bear"...

 of an individual name (the "designated" object named by that) is called its extension

Extension (semantics)

In any of several studies that treat the use of signs - for example, in linguistics, logic, mathematics, semantics, and semiotics - the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of...

; and as for sentences, their truth value is called also 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

Intensional statement

In logic, an intensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive...

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

Aristotle

Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

 had already studied modal syllogism

Syllogism

A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

s. Gottlob Frege

Gottlob Frege

Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

 developed a kind of two dimensional semantics: for resolving questions like those of intensional statement

Intensional statement

In logic, an intensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preserve logical value. An intensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive...

s, he has introduced a distinction between two semantic values

Sense and reference

Sinn and bedeutung are usually translated, respectively, as sense and reference. Two different aspects of some terms' meanings, a term's reference is the object that the term refers to, while the term's sense is the way that the term refers to that object.Sinn and bedeutung were introduced by...

: sentences (and individual terms) have both an extension and an intension

Intension

In linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase or other symbol. In the case of a word, it is often implied by the word's definition...

. 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

Proof calculus

In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules...

 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
  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...
  
  
• Modal logic
  Modal logic
  Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
  
  

Modal logic

Modal logic

Modal logic

Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

 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. Moreover, they are related to one another by similar dualities

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

 like quantifiers do (for example by the analogous correspondents of De Morgan's laws). Syntactically, they are not quantifiers, they do not bind variables, they appear in the grammar as sentential functors, they are called modal operator

Modal operator

In modal logic, a modal operator is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional, and is "intuitively" characterised by expressing a modal attitude about the proposition to which the operator is applied...

s.

As mentioned, precursors of modal logic includes Aristotle

Aristotle

Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

. Medieval scholastic discussions accompanied its development, for example about de re versus de dicto

De dicto and de re

De dicto and de re are two phrases used to mark important distinctions in intensional statements, associated with the intensional operators in many such statements. The distinctions are most recognized in philosophy of language and metaphysics....

 modalities: said in recent terms, in the de re modality the modal functor is applied to an open sentence

Open sentence

In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined...

, the variable is bound

Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place...

 by a quantifier whose scope includes the whole intensional subterm.

Modern modal logic began with the Clarence Irving Lewis

Clarence Irving Lewis

Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in...

, his work was motivated by establishing the notion of strict implication. Possible world

Possible world

In philosophy and logic, the concept of a possible world is used to express modal claims. The concept of possible worlds is common in contemporary philosophical discourse and has also been disputed.- Possibility, necessity, and contingency :...

s approach enabled more exact study of semantical questions. Exact formalization resulted in Kripke semantics

Kripke semantics

Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...

 (developed by Saul Kripke

Saul Kripke

Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...

, Jaakko Hintikka

Jaakko Hintikka

Kaarlo Jaakko Juhani Hintikka is a Finnish philosopher and logician.Hintikka was born in Vantaa. After teaching for a number of years at Florida State University, Stanford, University of Helsinki, and the Academy of Finland, he is currently Professor of Philosophy at Boston University...

, Stig Kanger).

Type theoretical intensional logic

Already in 1951, Alonzo Church

Alonzo Church

Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

 had developed an intensional calculus

Proof calculus

In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules...

. 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 yet that time: Church has not provided formal semantic definitions.

Later, possible world

Possible world

In philosophy and logic, the concept of a possible world is used to express modal claims. The concept of possible worlds is common in contemporary philosophical discourse and has also been disputed.- Possibility, necessity, and contingency :...

 approach to semantics provided tools for a comprehensive study in intensional semantics. Richard Montague

Richard Montague

Richard Merett Montague was an American mathematician and philosopher.-Career:At the University of California, Berkeley, Montague earned an B.A. in Philosophy in 1950, an M.A. in Mathematics in 1953, and a Ph.D. in Philosophy 1957, the latter under the direction of the mathematician and logician...

 could preserve the most important advantages of Church's intensional calculus in his system. Unlike its forerunner, Montague grammar

Montague grammar

Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on formal logic, especially higher order predicate logic and lambda calculus, and makes use of the notions of intensional logic, via Kripke models...

was built in a purely semantical way: a simpler treatment became possible, thank to the new formal tools invented since Church's work.