As Chris knows, the various meanings (or levels of granularity) do not stop
there. (01)
If one accepts the "*extensional* rendering of *intensional* notions" one
can argue for an extensional criterion of identity for these properties.
Arriving at the seemingly odd situation where an intensional notion has an
extensional criterion. (02)
For philosophers who take possible worlds seriously, such as David Lewis,
the phrase "*extensional* rendering of *intensional* notions" does not make
sense as (for them) the objects in other possible worlds are just as real as
those in our so-called actual world (or present actual world). Hence there
is just the extensional rendering. So they do not end up in this odd
situation. (03)
For some people, this "*extensional* rendering" is not fine grained enough.
See http://plato.stanford.edu/entries/logic-intensional/ - "The property of
being an equilateral triangle is coextensive with the property of being an
equiangular triangle, though clearly meanings differ. Then one might say,
"it is trivial that an equilateral triangle is an equilateral triangle," yet
one might deny that "it is trivial that an equilateral triangle is an
equiangular triangle"." (04)
Some people argue that where properties have a different intension/meaning
they are different properties. In this case, it is not possible to have an
extensional criterion of identity. It turns out to be difficult to devise a
sensible intensional criteria of identity. It seems to me that the ISO
Standards process (taken literally) drives one in this direction by pushing
one towards a definition which is what tells you what something is - hence
things with different definitions and different things. Try following ISO
with 'equilateral triangle' and 'equiangular triangle'. I suspect most
people do not take it literally. (05)
Ironically, extension also has a different related meaning.
http://en.wikipedia.org/wiki/Extension_(metaphysics) "In metaphysics,
extension is, roughly speaking, the property of "taking up space"."
For many 4D-ists, (4D) extension is regarded as the (extensional) criterion
of identity for individuals (i.e. things that take up space). (06)
Hence 4D-ists can say that they have an extensional criterion of identity,
extending the meaning extension over both more specific senses. (07)
Regards
Chris (08)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
> bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
> Sent: 03 February 2009 19:03
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Ontology and Category Theory
>
> On Feb 3, 2009, at 1:45 AM, Pat Hayes wrote:
> > On Feb 2, 2009, at 8:20 PM, Len Yabloko wrote:
> >> LY>> No. Earlier in this thread http://ontolog.cim3.net/forum/ontolog-
> forum/2009-01/msg00523.html
> >>>> I already proposed category in which extensions are objects and
> >>>> intensions are morphisms.
> >>>
> >> PH>Hmm. Im afraid this simply does not make sense to me. First, we
> >> have
> >>> to find out what you mean by 'extension' and 'intension'. In my
> >>> language these are usually used in the adjectival mode, to refer
> >>> to ways of understanding relations.
> >>
> >> This is the most general definition I could find
> http://en.wikipedia.org/wiki/Intension
> >> "Intension refers to the possible things a word or phrase could
> >> describe. It stands in contradistinction to extension (or
> >> denotation), which refers to the actual things the word or phrase
> >> does describe"
> >
> > I see what they are trying to say, but its not a very good
> > definition. The trouble is, the set of possible things is still a
> > kind of extension.
>
> Yes indeed, but that is *exactly* what modern possible world semantics
> for modal logic provides -- an *extensional* rendering of
> *intensional* notions. This is why the impact of possible world
> semantics among linguists, philosophers, and theoretical computer
> scientists was so dramatic -- the formerly obscure, medieval notion of
> intension, or meaning, was suddenly given (at least on the face of it)
> a rigorous and precise mathematical analysis.
>
> The problem with the Wikipedia definition above is not that intensions
> turn out to be extensional entities (that is, sets or functions), but
> simply that it picks out the wrong ones. There is in fact a variety
> of intensions in possible world semantics, but the simplest example is
> that of a property, that is, the intension of a 1-place predicate. In
> possible world semantics, the EXtension of such a predicate P is, as
> Pat indicates, the set of things it applies to *in fact*, that is, in
> the actual world. P's INtension, in possible world semantics, is a
> *function* from possible worlds to sets -- intuitively, the set of
> things in a given world to which P applies. Thus, the EXtension of
> "Red" is the set of things that are, in fact, red. Its INtension is
> the function RED that, for a given possible world w, returns the set
> of red things in w. RED, being a function, is a purely extensional
> entity -- it is defined by its domain and the values it returns on
> that domain; any function on the same domain that returns the same
> values is identical to the function RED. It is rightfully thought of
> as the *intension*, or *meaning*, of "Red", however, insofar as it
> faithfully preserves the logical properties of the predicate in
> contexts where its meaning, rather than its extension, becomes
> relevant -- so-called "intensional contexts".
>
> An intensional context is one in which traditional extensional
> substitution principles fail. In standard, extensional first-order
> logic, there are two particularly salient examples of such
> principles. The first is that names that denote the same thing can
> always be substituted one for the other; thus, for example, if Mark
> Twain = Sam Clemens, then if Twain wrote Huckleberry Finn, it follows
> that Clemens wrote Huckleberry Finn. The second is that coextensional
> predicates -- i.e., predicates that apply to the same things -- can
> always be substituted one for the other. Thus, if, as it happens, the
> honor students at Podunk High School consist of exactly the members of
> its football team, then if every honor student of PHS won a college
> scholarship it follows that every member of the PHS football time won
> a college scholarship.
>
> There are, however, contexts in which these principles appear to
> fail. Belief contexts are perhaps the most famous in regard to
> names. For example, from the fact that Pat believes that Twain wrote
> Huckleberry Finn it does not seem to follow, as a matter of logic,
> that he believes that Clemens wrote Huckleberry Finn; he might not
> know that Twain and Clemens are one and the same. Contexts involving
> the so-called alethic modalities of possibility and necessity can
> cause the second substitutivity principle to fail. Suppose again that
> the honor students at Podunk High School comprise exactly of the
> members of its football team. However, while, it is surely possible
> that a PHS honor student not be on the PHS football team, it is
> obviously not possible that a member of the PHS football team not be
> on the PHS football team; in that context, substituting "member of the
> PHS football team" for "PHS honor student" turns a true sentence into
> a false one; the former is not substitutable for the latter without
> breaking the substitution principle for coextensional predicates.
>
> An *intensional* logic is a logic that can support intensional
> contexts, that is, a logic in which at least one of the classical
> extensional substitution principles is invalid. In particular,
> possible world semantics, by unpacking possibility and necessity in
> terms of quantification over possible worlds and by assigning
> predicates intensions as above, gives us a modal logic that supports
> failures of predicate substitution like the one above: The intension
> of "PHS honor student" is the function that returns the set of PHS
> honor students at each world; the intension of "PHS football team
> member" is the function that returns the set of PHS football players
> at each world. At the actual world, according to our story above,
> these functions return the same set; but at other possible worlds,
> they do not. In particular, while there are worlds in which a PHS
> honor student is not a PHS football player, there are obviously no
> worlds in which a PHS football player is not a PHS football player.
>
> Thus, even though intensions are, in fact, extensional entities in
> possible world semantics, they support intensional contexts. The
> trick, of course, is that, by bringing possible worlds into the
> picture, we get a larger semantical universe in which we can
> characterize intensionality in terms of extensions at *all* possible
> worlds rather than only the actual world.
>
> > It would be more accurate to say that the intension of a phrase is
> > its meaning or sense (in Frege's terminology) and the extension is
> > the set of things which are described by the phrase.
>
> Yes, precisely, where "the intension of a phrase" is spelled out as
> above and "the set of things described by the phrase" is the value of
> the phrase's intension at the actual world.
>
> -chris
>
>
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