Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition.

So, for example, an intensional definition of ' Prime Minister ' might be the most senior minister of a cabinet in the executive branch of government in a parliamentary system.

A partitio is simply an intensional definition.

A genus – differentia definition is a type of intensional definition, and it is composed by two parts:

A genus – differentia definition is a type of intensional definition which defines a species ( that is, a type — not necessarily a biological category ) as a subtype of a genus satisfying certain conditions ( the differentia ).

One way is by intensional definition, using a rule or semantic description:

In either case, we might want to know the properties of that object, which we might then list in an intensional definition.

The opposite approach is the intensional definition, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.

In logic and mathematics, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined.

For example, an intensional definition of bachelor is ' unmarried man '.

As becomes clear, intensional definitions are best used when something has a clearly defined set of properties, and it works well for sets that are too large to list in an extensional definition.

It is impossible to give an extensional definition for an infinite set, but an intensional one can often be stated concisely — there is an infinite number of even numbers, impossible to list, but they can be defined by saying that even numbers are integer multiples of two.

Definition by genus and difference, in which something is defined by first stating the broad category it belongs to and then distinguished by specific properties, is a type of intensional definition.

For example, an intensional definition of " square number " can be " any number that can be expressed as some integer multiplied by itself.

Similarly, an intensional definition of a game, such as chess, would be the rules of the game ; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

See also intensional definition versus extensional definition.

See also extensionality, and also intensional definition versus extensional definition

Note that it also creates a class from the extension of the intensional set.

Predicates may also be defined by facts and rules and therefore neither be purely extensional nor intensional, but any datalog program can be rewritten into an equivalent program without such predicate symbols with duplicate roles.

These semantic values can be interpreted, transferred also for functors ( except for intensional functors, they have only intension ).

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.

* The inclusion of a single point of interaction between syntax and the interfaces ( conceptual / intensional and sensori-motor ), commonly called the point of Spell-Out.

Such expressions always, or nearly always, produce intensional statements when added ( in some intelligible manner ) to an extensional statement, and thus they ( or more complex expressions like " It is possible that ") are sometimes called intensional operators.

Functors for which this assumption does not hold are called intensional.

In the history of western thought, essence has often served as a vehicle for doctrines that tend to individuate different forms of existence as well as different identity conditions for objects and properties ; in this eminently logical meaning, the concept has given a strong theoretical and common-sense basis to the whole family of logical theories based on the " possible worlds " analogy set up by Leibniz and developed in the intensional logic from Carnap to Kripke, which was later challenged by " extensionalist " philosophers such as Quine.

More recently a number of theorists have sought to account for relevance in terms of " possible world logics " in intensional logic.

* Economy of representation is the principle that grammatical structures must exist for a purpose, i. e. the structure of a sentence should be no larger or more complex than required to satisfy constraints on grammaticality, which are equivalent to constraints on the mapping between the conceptual / intensional and sensori-motor interfaces in the optimal system that minimalism seeks to explore.

That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co-extensive expressions ( in the relevant language ) such that one of them occurs in the statement, and if the other one is put in its place ( uniformly, so that it replaces the former expression wherever it occurs in the statement ), the result is a ( different ) statement with a different logical value.

Typically ( although it depends on the type theory used ), the axiom of choice will hold for functions between types ( intensional functions ), but not for functions between setoids ( extensional functions ).

James Garson has given some results in the areas of adequacy for intensional logics outfitted with such a semantics.

He is best known for his work on demonstratives, on propositions, and on reference in intensional contexts.

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.

As mentioned, motivations for settling problems that belong today to intensional logic have a long past.

Thus sometimes similar patterns repeated themselves for the history of development of intensional logic like earlier for that of extensional logic.

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.

Later, possible world approach to semantics provided tools for a comprehensive study in intensional semantics.

Significantly, modal logics can be developed to accommodate most of these idioms ; it is the fact of their common logical structure ( the use of " intensional " sentential operators ) that make them all varieties of the same thing.

In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional equality ( the equality on the original set ) and extensional equality ( the equivalence relation, or the equality on the quotient set ).

The needs of thoroughly intensional theories such as untyped lambda calculus have been met in denotational semantics.

But the definitions of the functions are not equal, and in that intensional sense the functions are not the same.

He proposed both intensional and extensional variants of the theory.

A fundamental distinction is extensional vs intensional Type Theory.

In contrast in intensional Type Theory type checking is decidable, but the representation of standard mathematical concepts is somewhat complex, since extensional reasoning requires using setoids or similar constructions.

It is a subject of current discussion whether this tradeoff is unavoidable and whether the lack of extensional principles in intensional Type Theory is a feature or a bug.

Here Lewis recognizes Barcan Marcus as the first logician to extend propositional logic as a higher order intensional logic.

This strict implication was not primitive, but defined in terms of negation, conjunction, and a prefixed unary intensional modal operator,.

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 distinction is best understood by examples of intensional contexts of which we will consider three: a context of thought, a context of desire, and a context of modality.

Willard Van Orman Quine thought that the failure of substitutivity in intensional contexts ( e. g., " Sally believes that p ," " It is necessarily the case that q ") shows that modal logic is an impossible project.