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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