From Latin intensiōn-em "stretching, straining" (Oxford English Dictionary).
This term has undergone a fairly severe shift in how it is used in semantic theory. Prior to the mid-20th century, the "intension" of a term was understood to be the conditions which an object must meet in order for the term to apply to it. The more conditions a term imposes, the more "intensive" its meaning — the more intension it has. For example, square imposes all the same requirements as rectangle, plus the additional requirement of having sides of equal length, so square has more intension than rectangle. Conversely, the fewer conditions a term imposes, the more objects it will apply to — that is, the more "extensive" its meaning, and the more extension it has.
The earliest attestation of intension in English is from an 1836 lecture by William Hamilton, later published in Hamilton (1860): "The Internal Quantity of a notion,— its Intension or Comprehension, is made up of those different attributes of which the concept is the conceived sum; that is, the various characters connected by the concept itself into a single whole in thought." As Hamilton implies, this is essentially the same as what other authors call "comprehension," following the Port-Royal Logic of Arnauld and Nicole (1662).
As Spencer (1971) pointed out, Hamilton (1852) attributes the term to Leibniz "and his followers," and it seems clear that Leibniz was the first to use this term in semantics (though not, of course, in English). The best-known passage is in French, in the posthumously published Leibniz (1765); I am unsure whether this is Leibniz' earliest use of the term:
Car disant ‘tout homme est animal’, je veux dire que tous les hommes sont compris dans tous les animaux; mais j’entends en même temps que l’idée de l’animal est comprise dans l’idée de l’homme. L’animal comprend plus d’individus que l’homme, mais l’homme comprend plus d’idées ou plus de formalités; l’un a plus d’exemples, l’autre plus de degrés de realité; l’un a plus d’extension, l’autre plus d’intension.
(For in saying 'Every man is an animal', I mean that all men are included in all animals; but I understand at the same time that the idea of animal is included in the idea of man. Animal includes more individuals than man, but man includes more ideas or more formalities; the one has more examples, the other more degrees of reality; the one has more extension, the other more intension.)
It should perhaps be noted that many editions of this work use the word intensité in this passage, not intension.
Lourié (2012) argues that Leibniz viewed himself as building on the medieval scholastic notion of intensio, and especially the work of Richard Swineshead, in adopting this term, but also notes that medieval uses were exclusively in physics (referring, for example, to the intensity of physical force), not semantics.
Carnap (1947) initiated, perhaps unintentionally, the shift in meaning which has resulted in people now understanding the intension of an expression to be a function mapping each possible world (or analogous index) onto the expression's extension at that index. However, one has to look pretty hard in Carnap to find the idea of intensions as functions.
He does prominently define the notion of "having the same intension" as being L-equivalent, where two expressions A and B are L-equivalent iff the sentence 'A ≡ B ' holds in every state description. If A and B are 1-place predicates, this amounts to saying that A and B are true of the same individuals in all state descriptions. A state description is a set containing, for every atomic formula, either that formula or its negation, understood as representing a possible state of affairs. Thus, A and B will have the same intension under this definition iff they hold of the same individuals as each other, under all possible circumstances. It is easy to see how this agrees with the traditional Leibnizian notion, where A and B have the same intension iff they impose the same conditions on the things they apply to.
As Carnap notes, defining "having the same intension" does not tell us what intensions are. He proceeds to "look around for entities which might be taken ... as intensions," and identifies propositions as the intensions of sentences, properties as the intensions of predicates, and "individual concepts" as the intensions of individual expressions such as proper names. For the bulk of Carnap's discussion, the notions of proposition, property, and individual concept are not defined or formalized. But on p. 181, he writes:
...we might say that an individual concept...is an assignment of exactly one individual to every state... However, we shall actually take not these states but the state-descriptions; and not the individuals but the individual constants.... Thus we shall take any assignment of exactly one individual constant to each state-description...(in other words, any function from state-descriptions to individual constants) as representing an individual concept...
Here, the idea of an intension as a function from indices to extensions is clearly expressed — though not for intensions in general, but only for individual concepts, and the actual formalization replaces extralinguistic entities (states and individuals) with expressions representing them.
Perhaps someone will correct me, but as far as I am aware, the first fully general definition of intensions as functions from indices to extensions is not until Montague (1968): "Suppose that A is a possible interpretation for a pragmatic language L, A = ⟨I, U, F⟩ and ζ is a term of L. Then IntA(ζ), or the intension of ζ, is that function H with domain I such that, for each i in I, H(i) = Exti,A(ζ)."
- Arnauld, Antoine and Pierre Nicole (1662) La logique, ou L'Art de penser. Jean Guignart, Charles Savreux, & Jean de Lavnay.
- Carnap, Rudolf (1947) Meaning and Necessity. University of Chicago Press.
- Hamilton, William (1852) Discussions on Philosophy and Literature, Education and University Reform. Longman, Brown, Green and Longmans.
- Hamilton, William (1860) Lectures on Metaphysics and Logic. William Blackwood and Sons.
- Leibniz, Gottfried (1765) Nouveaux essais sur l’entendement humain. Œuvres philosophiques, latines et françoises, 1–496. Rudolf Erich Raspe.
- Lourié, Basil (2012) 'Intensio: Leibniz in Creating a New Term for the Modal Logic', Studia Humana 1.3/4.59–65.
- Montague, Richard (1968). 'Pragmatics', in R. Klibansky (ed.) Contemporary philosophy: A survey, 102–122. La Nuova Italia Editrice.
- Spencer, Mary (1971) 'Why the "S" in "Intension"?', Mind 80.317.114–115.