Propositions
sowa <sowa@turing.pacss.binghamton.edu>
Date: Tue, 10 May 94 21:18:28 EDT
From: sowa <sowa@turing.pacss.binghamton.edu>
Message-id: <9405110118.AA10365@turing.pacss.binghamton.edu>
To: cg@cs.umn.edu, interlingua@ISI.EDU
Subject: Propositions
Cc: sowa@turing.pacss.binghamton.edu
Following are some comments on Chris Menzel's comments on Pat Hayes' comments
(one > for Chris and two >> for Pat. Three >>> mark the original stimulus
but I forget who started this round.).
> Pat Hayes wrote:
>> >(1) From what I have heard on the interlingua mailing list and in private
>> >conversations, propositions CAN BE HANDLED via an ontology, just as we
>> >handle other important concepts....
>>
>> Where does your optimism come from? As far as I am aware, every attempt to
>> formalise the notion of 'proposition' has failed, and all the technical
>> results which might be relevant to the possibility are negative.
> I think that's too strong, Pat. First of all, there surely are
> formalizations of the notion of 'proposition' that have been
> successful by at least a number of measures, e.g., Montague's
> definition of a proposition as a function from possible worlds to
> truth values. For some purposes, this notion is quite adequate (cf.,
> e.g., Montague himself, Cresswell's work, or David Lewis's *On the
> Plurality of Worlds*), though its shortcomings, e.g., for the analysis
> of propositional attitudes, are well known. These shortcomings have
So far, I agree with Chris's responses to Pat. I also believe that the
shortcomings of Montague's definition make it unusable as a definition
of proposition for our purposes. The major problem is that it is so
coarse grained that all tautologies reduce to exactly the same proposition.
That means that if Bill knows "p implies p" then he knows all the theorems
of logic. That is far too coarse for practical purposes. It also means
that if Bill knows p that he also knows all the implications of p. Again
a generally bad assumption.
> spawned a number of more fine-grained (as well as type-free) analyses
> from, e.g., Ray Turner, Ed Zalta, George Bealer, and others that are
> much more successful in dealing with the challenges on which the
> possible worlds account founders. (Propositions on these accounts,
> BTW, are all a special case of n-place relation (n=0, obviously)).
These are OK. I prefer the definition I gave, which is quite simple
and allows you to have as fine-grained or coarse-grained a definition
of "proposition" as you like: namely, a proposition is defined as an
equivalence class of sentences in some formal language. Then you can
pick any axiomatization you like for your equivalence class. If you
choose identity, you get the very fine-grained definition that every
sentence states a distinct proposition; that has the drawback that
such trivial variants as p&q or q&p are considered distinct. If you
choose biconditional as your defining relation, then you get the
coarse-grained definition that puts all tautologies in the same pot.
The axiomatization that I prefer is the one that I described earlier
in terms of existential-conjunctive logic. But if you prefer a different
one, you can choose any equivalence relation you like to define your
classes of sentences.
>> There is one important difference between propositions and other kinds of
>> thing. In a logical language of the usual kind, things are denoted by
>> terms; but propositions seem to correspond to sentences. The complexity
>> comes in getting the nature of this correspondence clear. One can't
>> (usually) say that sentences denote propositions. But it is hard to see
>> what, other than sentences, should be considered to convey or describe
>> propositions. One can always enrich the term structure of (a theory in) the
>> language so as to make it have a term for every sentence, but then one is
>> skirting close to the paradoxical territory of self-reference: see
>> McCarthy's old theory for a well-worked-out example which didnt work.
> Careful not to overgeneralize; the accounts of Turner and Bealer both
> permit the construction of a term for every sentence, yet both are
> provably consistent (relative to some fragment of ZF, of course).
Again, I prefer Chris's position (one >) to Pat's (two >>).
>> Another other source of complication is that the relationship of sentence
>> to proposition is not 1:1. Several different sentences can express the same
>> proposition, everyone agrees (eg permute a few conjunctions). So the
>> natural idea would be to define a normal form which eliminates the
>> variation. If anyone is aware of a plausible candidate for such a normal
>> form, I'd love to hear why it is plausible.
> On Bealer's approach, for example, one can define a sort continuum of
> granularity with a nearly 1:1 relationship of sentences to
> propositions at one end and Montague-like propositions at the other.
Yes, and with my approach, you can have the full range from exactly 1:1
(not just nearly) between sentences and propositions up to a version
that is isomorphic to (not just like) Montague's.
> This suggests the idea of a variety of normal forms depending the
> one's preferred granularity.
Yes. I agree that we need this option. And if anyone has forgotten
or forgotten to save my old definition, I'd be happy to dust it off and
resend it to these mailing lists again.
>> And another famous source of complication comes from de re propositions. Is
>> this a proposition: the person standing behind you is female? If not, why
>> not: if so, how could it possibly be expressed in a formalism?
> On a fine-grained approach, what proposition the sentence "The person
> standing behind you is female" expresses is going to depend on context
> and on your intentions as the speaker. If you are using the
> description as a mere tag to pick out a certain individual (so that
> the correctness of the description doesn't really matter), then (on
> the "Russellian" view, at least) you are expressing a singular
> proposition containing the person in question as a constituent (what
> you're calling a de re proposition, I take it), the proposition itself
> the result of a certain sort of predication operator that takes an
> n-place relation and n individuals as arguments.
My response to Pat is somewhat different from Chris's, which I don't
really disagree with. But I would say that if your formal language
is FOL, you can't say "The person standing behind you is female"
or "This sentence is false" because you have no way to express context
dependent or indexical terms in FOL. If your formal language is CGs,
which do support indexicals, then I would say that the equivalence
classes are not defined over any CGs which contain indexicals.
That means that you must first resolve the indexicals to some
contextually defined individual before you can apply the equivalence
axioms. For Pat's example, that would be to resolve "you" to some
individual, say Bill, and then to resolve "the person standing
behind Bill" to some other individual, say Mary. The result is
that you have the proposition containing the sentence "Mary is female."
> If you're using the description to pick out whoever it is who
> satisfies it, then there are a couple of options. One is simply to
> eliminate the description by means of a standard Russellian analysis
> (There is one and only one person x such that x is standing behind you
> and x is female), which then can be taken to be denoting a complex
> "quantified" proposition (arising from a quantification operator from
> simpler propositions). The other is to introduce a primitive
> "description" operator that takes a property P to a property Q that
> holds of m just in case m is the only P, and carry out the analysis Q.
> Both approaches are discussed in the relevant literature.
My only objection to this comment is that I completely agree with
C. S. Peirce, who said that Russell's view of semantics was
"incredibly naive" and "superficial to say the least". Peirce
wrote a book review of Russell's 1903 _Principles of Mathematics_
which he compared unfavorably to Lady Victoria Welby's little book
_What is Meaning?_ That review started a lifelong correspondence
between Peirce and Lady Welby, and it may have been the reason why
Russell omitted all mention of Peirce in the preface to the
Principia Mathematica -- where he gave a glowing endorsement of
Frege. I believe that Frege and Peirce were both much better
logicians than Russell, although I would say that Peirce was the
more insightful philosopher.
>> >(2) I have not yet gotten the sense that there is a consensus on the nature
>> >of propositions, i.e. the axioms that characterize them.
> This is owing in part to the fact that different conceptions seem
> appropriate for different contexts. However, if we agree on a
> particular conception, e.g., the idea of very fine-grained, singular
> propositions with a structure that reflects the sentences that express
> them, then there should be a very plausible set of axioms for this
> conception (e.g., among others in this case, [Pa] = [Qb] iff P=Q and
> a=b). But in other contexts we might want Montaguovian propositions,
> and hence we'll need very different first principles. But we can
> axiomatize them as well.
>> The finest minds in Western civilisation havn't come to a consensus
>> on this in hundreds of years. We should be very sceptical of a
>> committee of even the *very best* computer scientists claiming it
>> has a 'standard'.
> Agreed, it would be sheer hubris to propose *the* standard account of
> propositions. What might more modestly be hoped for, however, is for
> several conceptions of proposition to be isolated and, drawing upon
> existing literature and powerful new formal techniques, theories
> corresponding to each of these conceptions to be made available as
> separate ontologies.
I agree with both Pat and Chris, but I also believe that we need
a standard. My recommendation is to define Proposition as an
equivalence class of sentences in the formal language of your choice.
Then I would leave it up to the user to choose which equivalence
relation seems best for a particular application. I will provide
people with my preferred axiomatization in terms of ex-con logic.
But anyone who prefers a different one can replace my axioms with
theirs. This is very much in the spirit of offering a library
of subroutines, object classes, or whatever, but giving
users the option of replacing any one of the items with another
version they may prefer. Two people who want to share information
would simply name which axiomatization they assume.
John