tag:blogger.com,1999:blog-8137988136860941398.post6011792267161622092..comments2017-10-03T11:14:35.545-07:00Comments on Sprachlogik: Identity Expressed with One-Place PredicationTristan Hazehttp://www.blogger.com/profile/18008340011384137776noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-8137988136860941398.post-89982050647335623362012-04-10T03:31:50.073-07:002012-04-10T03:31:50.073-07:00this is what i had in mind when i said reference w...this is what i had in mind when i said reference was second order, or something of that sort. i don't know if you remember. ;)anselfirnoreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-80536383825804894232012-03-23T05:55:41.844-07:002012-03-23T05:55:41.844-07:00Sorry for the late reply. I don't think that i...Sorry for the late reply. I don't think that <i>is</i> making the same moves. You need to specify a relation in your example ('loved' in 'a-loved'), but in the case of identity I can just say: let the extension of 'is T', where 'T' is a term, be {T}.<br /><br />(This is basically just a repetition of what I said in my second last comment in the paragraph beginning 'But I don't need to...'. I think it shows the answer to your question - 'Couldn't we make the same moves for an arbitrary relation?' - to be 'no'.)Tristan Hazehttp://sprachlogik.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-43132376281519487962012-03-17T08:37:45.380-07:002012-03-17T08:37:45.380-07:00That sounds fine to me, but if that's your att...That sounds fine to me, but if that's your attitude then I'm not sure how to take your response about how identity is special. Couldn't we make the same moves for an arbitrary relation? The extension of the one-place predicate "a loves ..." is the set of things which are a-loved.Jeffhttps://www.blogger.com/profile/01407714989987462536noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-79393847543521534562012-03-16T19:24:22.412-07:002012-03-16T19:24:22.412-07:00Jeff,
Thanks for the comment. I've been think...Jeff,<br /><br />Thanks for the comment. I've been thinking about it for a while now.<br /><br />When writing the post, I also thought about the fact that there is no very explicit ban on multiple objects being in the extension of a term-containing predicate. Note however that I do say 'let the <i>sole</i> member of' a term-containing predicate be the contained term's referent. So no model conforming to my specifications will give a term-predicate an extension containing multiple objects.<br /><br />Now it may be that the best way to define 'sole', or 'only', involves the notion of identity, but I'm not sure that's any sort of problem for what I'm saying. Firstly, the metalinguistic use of identity in a definition of 'sole' or 'only' could itself be construed much as I construe it in the object language. Secondly, and perhaps more to the point, I'm not trying to give a reductive analysis of the notion of identity.<br /><br />I'd be interested to know what you think of this defense.Tristan Hazehttp://sprachlogik.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-62313888334871097072012-03-14T08:33:01.633-07:002012-03-14T08:33:01.633-07:00I don't describe it that way.
Good point—sorr...<i>I don't describe it that way.</i><br /><br />Good point—sorry for the sloppy reading.<br /><br /><i>I can just say 'Let T's referent be in the extension'.</i><br /><br />Well, just saying that isn't quite going to work, since it doesn't rule out that other things besides T's referent might also be in the extension. So you'd want to say: let T's referent and <i>only</i> T's referent be in the extension. I'm used to analyzing this "only" in terms of identity: if x is in the extension then x is identical to T's referent. But maybe you could come up with an alternative approach. I'd be interested to see it worked out.<br /><br />Similarly, if the extension of '<a>' is a set, then you can specify it as { the extension of 'a' }, and again identity doesn't come up explicitly. I'm used to analyzing this set notation in terms of identity, too: x is in {y, z} iff x = y or x = z. But maybe you could explain things differently, and I'd also be interested to see those details worked out.Jeffhttps://www.blogger.com/profile/01407714989987462536noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-37496880649465428432012-03-02T18:26:37.599-08:002012-03-02T18:26:37.599-08:00Jeff,
Thanks for commenting. I'm not sure if ...Jeff,<br /><br />Thanks for commenting. I'm not sure if I understand your question, or precisely what currying amounts to for two-place relations in a first-order language.<br /><br />Suppose you wanted to do the same thing with 'loves'/'L', with curly brackets this time: you'd have to say something like: for all terms T, let {T} be a predicate, with everything which loves T's referent in its extension.<br /><br />But I don't need to do that in the case of identity: I don't need to say 'Let everything which is identical to T's referent be in the extension'. I can just say 'Let T's referent be in the extension'. So this doesn't seem like just another case of "currying a relation", if I understand the procedure rightly.<br /><br />On your last point: I agree that it would be misleading to describe anything in the above post as 'identity as a one-place predicate', for just the reason you give. I don't describe it that way.<br /><br />Anonymous,<br /><br />That all seems right, yes. I accept that I could have just changed the semantics while keeping the syntax the same. I think changing the syntax in this way indicates the changed semantics. So yes, you could say it's to prevent people seeing it as a relational form out of habit. (I should say though, in case it seems otherwise, that I don't particularly want anyone to adopt my notation.)Tristan Hazehttps://www.blogger.com/profile/18008340011384137776noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-87626639193600406982012-03-02T06:34:53.909-08:002012-03-02T06:34:53.909-08:00OK. I was pointing out that the syntactic differen...OK. I was pointing out that the syntactic difference is negligible, since in the syntactic sense there's nothing that makes putting a single symbol between two variables a better way of representing a relation than using two symbols. It's the semantics we use that make our relation symbols relation symbols. But I can see now that you are actually more interested in an aspect of the semantics.<br /><br />Still, though, there are two aspects to the semantics. One is purely extensional - associating formulas with the models (and variable assignments) that satisfy them. You don't claim to be making any change to this aspect, because < a >b and a=b are equivalent. The other is about how to decide compositionally whether a particular model stands in that relation to a particular formula, and that's the part that makes use of an identity relation or your new identity predicates. You are making a change here, as you point out, but the introduction of the symbols < > is fairly superfluous, because you could just decide that our procedure for evaluating a=b in a model will be to interpret a= as the predicate with extension {a} and then check if b is an element (i.e. to interpret a= as you plan to interpret < a >). So I'm guessing that you think there is something further to be gained from changing the syntax. Is it just to prevent logicians from thinking of = in terms of a relation out of habit?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-54261012959107346702012-03-02T04:02:49.250-08:002012-03-02T04:02:49.250-08:00What you've done here looks to me like a speci...What you've done here looks to me like a special case of the general operation of currying a function: replacing many-argument operations with one-argument higher-order operations. For example, the two-place addition function can be replaced with a one-place function add, where add(x) is a function that takes each number y to x+y.<br /><br />The fact that it's general doesn't make it uninteresting. (It's a neat trick, and one that is often used in computer science to simplify function representations.) But it does raise the question: is there anything special about identity that makes you think its curried form is particularly illuminating?<br /><br />Also, I think that describing this as "identity as a one-place predicate" is a bit misleading. Being-identitical-to-Jeff is a one-place predicate. But *identity*, in general, in your account is represented by the operation ⟨·⟩. That isn't a one-place predicate; it's something that takes terms to one-place predicates—a kind of higher-order function symbol.Jeffhttps://www.blogger.com/profile/01407714989987462536noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-37007384720996841842012-03-01T20:28:46.056-08:002012-03-01T20:28:46.056-08:00Thanks for the comments!
Shawn,
As I have constr...Thanks for the comments!<br /><br />Shawn,<br /><br />As I have construed things, occurences of variables inside term-containing predicates <i>can</i> be bound by (first-order) quantifiers. (Or can you see a problem arising with this which I've missed?<br /><br />Anonymous,<br /><br />You're right that, syntactically, that's all the difference amounts to. The point is that when we do things this way, identities get classified as one-place predications, and their semantics does not explicitly involve any identity relation. I find this instructive.<br /><br />At the end of your main comment, you use 'talk about' in a way I'm not quite clear about. In one sense, I'm happy to admit that formulae like '< b >a' "talk about a relation", in saying something which is equivalent to the relational expression 'a = b'. But in at least two other senses, such formulae <i>don't</i> talk about a relation: they don't refer to any relation, nor to they contain a relation-symbol.Tristan Hazehttps://www.blogger.com/profile/18008340011384137776noreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-4100644804124590112012-03-01T19:52:46.703-08:002012-03-01T19:52:46.703-08:00That should begin 'if you can form < t >...That should begin 'if you can form < t > even where...', and the third-from-last line should end with '... the only difference between < x > y ...'.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-2979608054027671752012-03-01T19:49:10.042-08:002012-03-01T19:49:10.042-08:00If you can form even where t is or contains a var...If you can form even where t is or contains a variable, I don't see that you've changed anything. What's the difference between and the string t=? They both form atomic formulas when concatenated with a term, and quantification needs to work the same for both, as Shawn says, if your language is to be first order. I don't think the difference between writing an expression with two symbols, as $x$y, and writing it with one, as x$y - and this is the only difference between y and x=y - is enough to ensure that the expression isn't being used to talk about a relation, as opposed to a one-place predicate.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8137988136860941398.post-68477051361343158862012-03-01T18:48:54.689-08:002012-03-01T18:48:54.689-08:00You say, "In place of '∃x (x = x)', w...You say, "In place of '∃x (x = x)', we write '∃x(〈x〉x)'". Could you explain this a bit more? You say you treat "〈a〉" as a predicate, but if so, the position between the angle brackets is not open for quantification. It would require some second-order resources to quantify into that. This might not be a bad thing since many people think identity is a second-order notion anyway.Shawnnoreply@blogger.com