[CL] Working on a definition of "ontology"

[Pat Hayes]

>
>Unfortunately, my formal education is almost that old, 40+ years.

Mine too :-)
...

>>>And can we say: "A unary relation applied to an 
>>>instance/individual is a proposition"?
>>
>>You could, though that's not strictly correct in most logics (you 
>>get a truthvalue, not a proposition.). We just finished extending 
>>CL to IKL, and in IKL a proposition *is* a ZERO-ary relation, see
>>http://www.ihmc.us:16080/users/phayes/IKL/GUIDE/GUIDE.html.
>
>To me, a "proposition" is a sentence that is either true or false.

Wait, wait. Sentences and propositions are related but not identical. 
Sentences are strictly syntactic kinds of things, you can write them 
down using symbols and check their identities by doing a 
string-match. Propositions are murkier, but the word is usually 
understood to mean something semantic, not a mere string of 
characters. Sentences *express* propositions, or maybe (in some 
frameworks) *denote* them, but they aren't the *same as* 
propositions. To illustrate the need to distinguish them, "Pat Hayes 
is a person" and "Patrick J. Hayes is a person" are distinct 
sentences but they express the same proposition.

In IKL, like in most logics, what a sentence actually denotes is 
simply a truth-value, which is the *extension* of a proposition; to 
refer to the proposition you have to use a special form
(that (Person "George W Bush"))
which acts like a name, or a term, rather than a sentence. Then you 
can say things like
(believes Cheney (that (Person "George W. Bush")))
which expresses a relation between a person and a proposition. But if 
you were to do this to the sentence alone, it wouldn't make sense:
(believes Cheney (Person "George W. Bush"))
since that 'inner' sentence denotes a truth-value, so this is just 
like saying (believes Cheney true); while
(believes Cheney '(Person "George W. Bush")')
says that a believes relation holds between Cheney and a string of 
characters, which doesn't seem right.

>  It can be proved, disproved, observed, etc., and you get the truth 
>value when you do one of those things ("evaluate it").  And I agree 
>that it is zero-ary.  Given the "classifier" Person, there is a 
>unary relation Person(?x).

Better, (Lambda (?x) (Person ?x))  to bind the variable. But OK, I agree.

>But Person("George W. Bush") applies it to an individual, and is 
>zero-ary; it no longer has any free variables.

Actually what it doesn't have is any names bound by a lambda. But I 
see what you mean, yes.

>  In this case, the truth value is clear.

There is an opening there for a witty political remark, but I will 
avoid temptation.

>  There are other predicates that could be applied to the same 
>individual and engender much debate, but they would still be 
>propositions.
>
>>>The reason I care is that I am currently working with another 
>>>standards group
>>>that is developing "logical formulations" of "unary relations" 
>>>that don't have
>>>names, and they are very careful to distinguish those from "concepts", which
>>>are "types"/"monadic predicates" that do have names.
>>
>>Oh dear, these guys sound to be SERIOUSLY confused. (I wonder why 
>>so many people are writing logic standards who know sweet nothing 
>>about logic?)
>
>Oh, they are.  (I'm one of them.)  The database and object 
>programming worlds have been doing "logic" for years, and now that 
>"semantics" and "logic" is getting funding and attention, they want 
>to legitimize what they are doing, but not change it, of course. 
>This group is database logic people and computational linguistics 
>people, who are trying to capture the intended logic of definitions 
>and rules that are originally expressed in natural language.
>
>The goal is laudable, but the participation is voluntary and 
>variously funded.  CL and OWL were lucky in that regard; the 
>modeling world got UML.  It's how standards happen.

Well, CL is now pretty much finished and so is OWL, so if y'all want 
any voluntary logical help, let me know.

>>>  They lack a term for "unary relation".
>>
>>The textbook term, if you want one, is 'property'. Predicates 
>>denote properties. Unfortunately, RDF has used "property" to refer 
>>to *binary* relations. Sigh.
>
>I rather liked 'property' myself.  I think using that term may 
>unconfuse some of the text.  The linguists liked 'characteristic' 
>and then narrowed its use.
>
>>>  The "unary relation" is the "intension" of a set
>>
>>What?? That is meaningless. Sets don't have intensions: that is 
>>exactly WHY they are sets.
>
>Let me try this a bit more carefully.  In mathematics, it is not 
>uncommon to define a "concept" as
>   'the set of all x such that <expression involving x>'

Well, that would be a very odd way to describe a *concept*. 
Mathematics is mostly about *sets* (or, if you like category theory, 
about things like morphisms)

>Because of all the closed/open world issues that surround database 
>logic, this group is very worried about distinguishing the 
>"extension" of the set -- its actual members -- from the "intension" 
>of the set -- the rule for membership.

Fine, that is an excellent distinction to be concerned with, provided 
you say it right. It's not the set that has the intension, but the 
concept. The set IS the extension of the concept, which also has an 
intension. But it is incoherent to talk about the "INtension of a 
SET". Sets just ARE extensional: that is part of the very idea of a 
mathematical set.

>So the <expression involving x> is what they mean by the "intension" 
>of the set.

Well, again, this is muddled. There is an expression, yes. 
Expressions, like sentences, are purely syntactic. Then there are two 
semantic things: the intension, or intensional meaning, and the 
extension. So there are at least three distinct things here, and it 
is fatal to get them muddled up with one another. In CL, these would 
be a relation name (i.e. a name used to denote a relation), the thing 
the name denotes, and the relational extension of that thing.

Speaking of 'the intension of a set' is particularly harmful because 
the whole point of extensional objects like sets is that they 
obliterate important intensional distinctions, so the typical mapping 
from intensions to extensions is many-to-one and therefore not 
invertible. So as well as being philosophically muddled, the phrase 
'intension of a set' gets the mathematics screwed up, by suggesting 
that there is a mapping from sets to 'their' intensions.

BTW, CL (and IKL) already make these distinctions: in CL, what 
relation and function symbols denote is always the INTENSION, which 
has a corresponding extension mapped to it by a semantic 'ext' 
function. One of the things we managed to do quite well, I think, was 
to incorporate this extension/intension distinction right into the 
heart of the formal model theory, building on some mathematical ideas 
provided by Chris Menzel (RDF also has this distinction built in, in 
the same way). So your group might find it useful to use CL as a 
foundational logic. I think y'all might be surprised at how flexible 
and expressive it is in this regard.

>To unconfuse many of their definitions, what is needed is a term, 
>which seems to be 'property', that means "a function of an arbitrary 
>Thing that returns true or false".

Right, that is a property *understood extensionally*. Which is the 
classical way to understand all of logic. However, in CL, a property 
is simply an individual "thing" with an *associated* such function.

It might be best to define it as you have done, using strictly 
mathematical terminology, in order to be unambiguous. BTW, this 
amounts to simply a set (of Things, ie a subset of the universe), 
note: the set of all the things that evaluate to True.  So you could 
just call these *sets*; provided of course that you don't want to 
have cases where the set of Things might change according to the 
context, or some other oddity.

I can't help remarking that what y'all seem to be doing here is 
slowly re-inventing Tarskian model theory. To repeat: why not take a 
look at the CL or IKL specs and see if they might already have what 
you need? At the very least, it would be something to build on, and 
its a foundation that we can warrant is as about as strong as 
foundations can get. Think of us as the guys you hire to pour the 
concrete over the rebar, before y'all try  putting up your building; 
except that of course we do it for free :-)

>  The 'property' is monadic, whereas the <expression involving x> can 
>involve other variables that are bound within the expression and 
>sometimes in the context in which the whole structure appears.  So 
>in a given occurrence, the expression resolves to a monadic property.

Keeping track of bound and free variables is one topic that is pretty 
much finished now, and y'all should be using some off-the-shelf way 
to do it. Montague used lambda notation, its hard to beat it.

>>This is completely and utterly confused. Not just about 
>>terminology: about substance. They sound to have use/mention 
>>confusions as well.
>
>It isn't completely and utterly confused, but there is confusion, 
>and it is compounded by the terminology.  The problem isn't 
>understanding use/mention

Mixing up sentences with propositions is a use/mention kind of confusion.

>; it is separating concepts, creating anchor concepts, and writing 
>clear definitions, and in some cases, about reconciling the way 
>people say things with what they mean.
>
>>>Any recommendations?
>>
>>Tell these guys to go and read a logic textbook, or hire someone 
>>who has. Seriously: they are just totally off the wall at the 
>>moment.
>
>With all due respect, "reading" a logic textbook isn't sufficient.

Yes, sorry I was a bit too flip there. But seriously, this is not a 
topic that is entirely virgin territory. There is now about a 
century's worth of serious work on these topics, and if folk want to 
do a sensible job, they need to somehow have professional-level 
access to this stuff, either by training or by consulting. Would we 
expect someone with no knowledge of 20th-century physics to write a 
physics standard?

>  Read, understand, integrate with the views of others who have read 
>the semantics and linguistics textbooks, presumably after you have 
>read those, and then construct some "ontology" that they can all 
>understand

I don't think its necessary to go that far.

>, is what would be required, if anyone had the time and funding to do that.
>
>As I said, CL got lucky -- people who knew what to do and how to do 
>it got funded.

I have never got a cent of funding to work on CL. CL was done 
entirely by volunteer effort. After raising this issue recently at an 
NSF meeting I was told that 'standards work' is not considered to be 
research and hence is ineligible for most government funding, and 
this this amounts to a policy decision that is now a decade old.

Pat
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