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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 3 Feb 2009 13:03:05 -0600
Message-id: <D0DC6921-694C-46B9-9430-F3B9B6F79C23@xxxxxxxx>
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 
>>>> 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 
>> "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.    (01)

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.    (02)

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".    (03)

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.    (04)

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.    (05)

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.    (06)

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.    (07)

> 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.    (08)

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.    (09)

-chris    (010)

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