I’ve occasionally run across the term “intension” but never understood it; I still don’t (OED: “5. Logic. The internal quantity or content of a notion or concept, the sum of the attributes contained in it; the number of qualities connoted by a term”), but I found the discussion of its history linked by Sarah Lobar in this OUPblog post interesting enough to pass on:
“Why the “S” in “intension”?” by Mary Spencer
The peculiar spelling of the logical term “intension” has always given pause to laymen readers of English logical prose. From time to time, in fact, even the initiate tend to become confused. […] It turns out that (1) Sir William Hamilton did not introduce the word “intension ” into the logical vocabulary, and (2) neither he nor the man who did introduce it were in any sort of muddle about its meaning.
Click the link for the exciting facts!
I think the easiest way into the concept is through the intensional and extensional definitions of sets. Consider these definitions of a set S:
D1 = {Delaware, Pennsylvania, New Jersey, Georgia, Connecticut, Massachusetts, Maryland, South Carolina, New Hampshire, Virginia, New York, North Carolina, Rhode Island, Vermont, Kentucky, Tennessee, Ohio, Louisiana, Indiana, Mississippi, Illinois, Alabama, Maine, Missouri, Arkansas, Michigan, Florida, Texas, Iowa, Wisconsin, California, Minnesota, Oregon, Kansas, West Virginia, Nevada, Nebraska, Colorado, North Dakota, South Dakota, Montana, Washington, Idaho, Wyoming, Utah, Oklahoma, New Mexico, Arizona, Alaska, Hawaii}.
D2 = {s|s is one of the United States}, where | means “such that”.
D1 enumerates the members of S and we call it an extensional definition. D2, on the other hand, gives a rule for membership in S and we call it an intensional definition. Extensionally, D1 and D2 define the same set and so are extensionally equal, but they are not in the same form and so are not intensionally equal.
There is only one extensional definition of any set: writing out the elements in a different order has no effect on it. However, there may be many intensional definitions; what is more, whether two intensional definitions are distinct is a question:
D3 = {s|s is a member of the federal state called United States}.
D4 = {ss|ss est l’un des États-Unis}
Are D2, D3, and D4 intensionally equal or not? Probably they are.
Now when I say that D1 and D2 are extensionally equal, that’s true in 2020. It was not true in 1950, when Alaska and Hawaii were not states; it may not be true in 2050 for a variety of reasons.
Now we can talk about intensional and extensional sentences. The sentence “Everyone knows that Mark Twain is Mark Twain” is true, and the sentence “Mark Twain is Mark Twain” is extensional because it simply says that the extensionally defined sets {Mark Twain} and {Mark Twain} are obviously equal. However, the sentence “Everyone knows that Mark Twain is the author of ‘Corn-Pone Opinions'” is false, and so the sentence “Mark Twain is the author of ‘Corn-Pone Opinions’, even though it is true, is intensional, because it says that the intensionally distinct set definitions {Mark Twain} and {s|s wrote “Corn-Pone Opinions”} are extensionally the same. A similar situation arises with the extensionally equal sets {the planet Venus}, {the planet Phosphorus}, {the planet Hesperus}, {s|s is the morning star}, and {s|s is the evening star}: whether you know that they are equal depends on your knowledge about Venus.
And so on. (I hope that helps.)
I’m sorry, but I’ll unapologetically veer off…
This reminds me of connexion. In H.G. Wells’s short story The Purple Pileus, Mr. Coombes, a henpecked husband, tries to raise a fuss about his wife’s friends coming over and playing music on a Sunday.
What on earth does it mean to “study your connexion”?
Something like ‘consider the feelings of your customers’, as far as I can tell – the problem being the Sunday more than the music.
Here’s the OED on studying, and a Dickens character studying a connection:
10. transitive. To pay practical regard to, show consideration for (a person’s wishes, feelings, or interests). Hence (colloquial): to accommodate the feelings or convenience of, to humour (a person). Now chiefly Caribbean.
1853 C. Dickens Bleak House ii. 9 I [sc. a tradesman] have been accustomed to study the leaders of my high connexion.
And on connections:
7. A body, or circle of persons connected together, or with whom one is connected, by political or religious ties, or by commercial relations; a body of fellow-worshippers, of political sympathizers, a circle of clients, customers, etc.
And from later in the story:
‘Comic songs a’ Sunday, it was getting to, and driving trade away’.
And now I know that there are two Sir William Hamiltons – although no doubt it would require philosophy to have Nelson always hanging round.
Much obliged!
(I just noticed “bright drab”, too. It wasn’t in the color discussion.)
Thanks @JC. The clearest definition between extension vs. intension is when the set in question is empty:
* my prize collection of unicorns; compare
* my prize collection of rocking-horse poop.
Their extensions are equal (empty); but they do not ‘mean’ the same thing. That difference is captured as ‘intension’. We could imagine a possible world in which those collections were non-empty.
And yeah there’s a whole nother rabbit-hole about identity of extension (Morning Star/Evening Star, Cicero/Tully, …) vs intension of those descriptive phrases or names. Cue Kripke Semantics.
I’m surprised someone like Geach struggled with it: the ‘s’ is there because there’s an ‘s’ in ‘extension’; and because it’s a quite different sense to ‘intention’ in Philosophy of Mind.
the sentence “Mark Twain is the author of ‘Corn-Pone Opinions’, even though it is true, is intensional, because it says that the intensionally distinct set definitions {Mark Twain} and {s|s wrote “Corn-Pone Opinions”} are extensionally the same.
Why are they intensionally distinct? They refer to the same thing, just as your first two sets do. What does the fact that someone might not know that Mark Twain wrote “Corn-Pone Opinions” have to do with it? Someone might not know the states; someone might not know anything at all.
Edit: Never mind, I misunderstood; see my later comment.
It seems like a variant of de re/de dicto distinction. Though it probably is simpler.
In grammar, you can define (with some precision) what noun is, but it is between impractical and impossible to list all nouns. But personal pronouns can and in practice are just listed. The interesting situation is when the set is not known. Like a set of people who wrote the Bible. And in math, like with Riemann hypothesis (and Fermat’s last until recently) the main question is whether the set is empty at all.
Intensionally different because although they describe the same thing in different ways, they could be describing different things – even in this universe it would be possible to discover that someone else had written (part of) Corn-Pone Opinions.
Whereas ‘the author of Corn-Pone Opinions’ and ‘the person who wrote Corn-Pone Opinions’ are (probably) intensionally the same – different words for the same definition.
I’m writing this out mostly so someone who understands can tell me if I’m right
I don’t feel like Mark Twain is a very good example, though, because someone who said that everyone knows that Mark Twain is Mark Twain might well actually mean that everyone knows that Samuel Clemens is Mark Twain…
I’m coming to a hazy conclusion that an extensional definition is one which describes the extent of a set – listing the members of it – while an intensional definition describes the intent of a set – the things that it would contain, whether you actually know what those things are or not.
In which case the names do make a kind of sense, although the spelling is still not very useful, as earlier meanings of ‘intension’ have to do with intensity rather than intent.
Intensionally different because although they describe the same thing in different ways, they could be describing different things
But the same is true of the list of states… Oh, wait, looking back I see that I misunderstood through not reading carefully enough (not the first time this has happened): JC writes “Extensionally, D1 and D2 define the same set and so are extensionally equal, but they are not in the same form and so are not intensionally equal.” So it all makes sense, though I certainly won’t remember any of it tomorrow.
So how did Latin even get intentio, L&S has both that and lautgesetzliche(?) intensio with essentially the same meaning (but more quotations for the T form). (And both from intendo, that was the first thing I checked).
I’m coming to a hazy conclusion that an extensional definition is one which describes the extent of a set – listing the members of it – while an intensional definition describes the intent of a set – the things that it would contain, whether you actually know what those things are or not.
@Jen, while that’s not wrong, it does start down a slippery slope: ‘intension’ is not saying anything about what’s in the mind (the ‘intent’) or knowledge of the speaker/hearer. ‘Intension’ is the form or meaning of the words/formula used, abstracted away from a particular utterance by a particular speaker. (Yeah, yeah, this presupposes all sorts of dangerous ideologies about meaning being distinct from usage.)
Logicians have far fewer hangups about abstract meaning; they’re used to writing ‘let f(x) denote the heavenly body that appears brightly lit around sunrise …’; the tree in the quad and all that.
In mathematical type theory, there is a very precise (and very technical) difference between extensional and intensional variants. I feel like I should understand it, it’s using the same sort of notation as denotational semantics and I’m pretty sure I had that under my belt in 1984 (3rd year CS). What’s 36 years between old friends?
(I don’t think I threw away the text book either, but I don’t know what box it’s in).
And then there are ostensional definitions: point at one member and mention a rule from which the other members then logically follow.
@lars
Re sio vs tio
The former is used with some 3rd conjugation verbs with infinitive in tere, dere, rgere, etc. There are some doublets and I find perditio but dispersio strange, maybe some of these are formed with reference to or in analogy with other verbal forms e.g. responsum from respondere vs factum from facere.
“Intensional” when used in linguistics:
Intensional Relative Clauses and the Semantics of Variable Objects
I couldn’t understand a word.
Yes, that was my point, there was a -TT- > -s- sound change at one point so tendo tetendi tensum tendere is expected — but L&S show side forms tenno and tentum in this particular case, I don’t know why, and thus intentio beside intensio. No Classical †extentio, though it could just be lack of attestation.
(de Vaan thinks the ppp with -t- was conflated with that of stative teneo teteni/tenui tentum tenere but is silent as regards tenno).
@BR, too Montagovian? (Don’t answer, I just wanted to use the word). But in that paper, an intensional relative clause is simply one with a verb of intension. The fun comes when trying to model the semantics of objects that only enter the discourse because people have intensions about them, and evidently it’s even harder when the intension is in a relative clause attached to a definite noun phrase. And then I stopped reading.
Probably because at some point, likely as soon as /tː/ appeared in the language, it was no longer obvious that tend-tum should give tensum and not ten(t)tum. Conversely, ss was generalized to a few words that wouldn’t regularly have it.
When the Indic branch got to that point, its inherited /st zd zdʰ/ were replaced wholesale by tt, dd, ddʰ except (for st at least) in a few “derivationally isolated” words that weren’t transparent anymore.
Tenno might be backformed from tensum.
I do like the idea of understanding extension and intension in terms of extent and intent. I do think AntC brings up a good point, which I was thinking about it, but I still think “intent” is helpful in thinking about it. At least it is for me.
In John Cowen’s example in the first comment, with the states, I would imagine the intent of the creator of D1 and D2, as far as what they describe, is the same. So, yeah, in some sense “intent” isn’t the right word. But we might imagine the words themselves as having an intent, with a literalistic bent. And with other examples, like the empty set example, seeing intent/intension as a pair, and extent/extension as a pair more clearly helps with understanding the terms extension and intension.
As for the S in the spelling, I’ve no clue about the Latin, but I do think it makes sense to both match the S in “extension”, and to differentiate it from “intention”.
@lars,
Looking at loads of 2nd conjugation verbs,
Perfect-supine
stem in [not in]ui-itum
stem in inui-entum
stem in avi-autum
stem in evi-etum
stem in ovi-otum
stem in gi-ctum
stem in idi-isum
stem in udi-visum (therefore ausum ctum, *admiscitum > admistum and *mixui but *admiscui? ) and seriously weird verbs like spondeo, spondere, spopondi.
So what I am saying is the nouns in sio corresponding to 2nd conjugation verbs are those where the verb has s in the supine, example: visio, responsio, and these verbs can be identified from the form of the perfect (not the infinitive) I do not know
if 3rd conjugation verbs work like this
@lars
Something ate the middle of my comment, I have replaced .LT by “from” and .GT by “to”. Here is the missing bit:
stem in udi-visum (therefore ausum from *avisum?)
stem in edi-essum
stem in ndi-nsum
stem in rdi-rsum
stem in xi-ctum
stem in si-sum
Outliers
censui, censum
arcui, arctum
arsi, arsurum
docui, doctum
ferbui, fervitum
imminui, imminitum
admiscui, admistum
miscui, mixtum
pavi, pavitum
retinui, retinitum
spopondi, sponsum
stridi, striditum
I think the outliers are mainly either -itum replacements by analogy (together with *citum to ctum, *admiscitum to admistum and *mixui but *admiscui? ) and seriously weird verbs like spondeo, spondere, spopondi.
If I recall correctly, Vox Latina has a section about these patterns as well, I think they are based on old regular sound changes but smoothed a little by analogy. And then analogical things like teteni > tenui happen in the perfect so you get tenui vs tentum, I happened to see that in de Vaal.
…and eventually tenui vs *tenūtum, if French is any guide.
AntC’s example using the many descriptions of the empty set is an excellent one, and I would have added it if I had thought of it at the time.
I myself belong to the set {s|(s is a follower of Quine) and (s is a follower of Kripke)}, which perhaps many people would suppose to be yet another intensional description of the empty set. But not so!
(On the other hand, I am definitely not a follower of Kripkenstein.
I wrote reflexion for two years, because our Physics master insisted on it. He had written our textbook and must have had a struggle over the spelling, perhaps with the publisher (or perhaps just with his wife).