tag:blogger.com,1999:blog-6432111.post4995996925364693716..comments2017-04-19T13:03:57.282+01:00Comments on Theories 'n Things: Paracompleteness and credences in contradictions.Robbie Williamshttp://www.blogger.com/profile/02081389310232077607noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6432111.post-51291526159050110652008-03-25T19:00:00.000+00:002008-03-25T19:00:00.000+00:00Hi there,Thanks for these! (BTW I left a comment a...Hi there,<BR/><BR/>Thanks for these! (BTW I left a comment at the post you link to, too). <BR/><BR/>I'm pretty sure that, in the light of the examples you and others have been raising, I'm going to have to rethink my terminology. (Hooray for blogs helping to clarify thinking...) .<BR/><BR/>It's clear to me what one principled position is---the person who thinks of many-valued models (describable classically) just playing the same role that a certain kind of classicist might think 2-valued models play. This character thinks that "definitely" and "weak negation" and the like are all available as extensional operators, with no qualms or funny business. <BR/><BR/>Obviously there are many properties of arguments that you might be interested in if you're going for this setting---the K3-valid arguments have one nice property (preserving perfect truth) the LP-valid arguments have another (avoiding falsity). There's not obviously a good answer to the question what "the" logic of this setting should be. But if I were to choose, I'd be inclined to pick the "no drop in truth value" logic, since it seems to me that given certain assumptions about the appropriate doxastic attitudes to borderline (or "half-true") cases, this will be the logic in terms of which we can spell out what combinations of attitudes are coherent. <BR/><BR/>I'm also clear about an opposite extreme: where someone totally dispenses with "intended interpretations" (or anything in the intended-interpretation role), and just focuses on the logic. The logic isn't many-valued in any interesting sense---it's just a set of valid arguments that can be characterized via a certain technical device, many-valued semantics. I've got a clear sense of what's *not* a legitimate move against someone who adopts that stance---e.g. just because we can write down a truth table for exclusion negation or the tertium operator, we can't assume that these are coherent concepts. <BR/><BR/>But that leaves a great range of intermediate cases where people want to regard the model theory as more than an algebraic device, but don't think there's any one classical model that can be "intended". And a general thought is that we somehow need to do the reasoning about the non-classical models. <BR/><BR/>Examples of the intermediate position I guess include Priest's use of non-classical set theory for doing models, perhaps the Weir ideas just mentioned; and perhaps some of the proposals that Williamson discusses and criticizes in the degree theory chapter of his Vagueness book. And Beall and Field both talk about some more-than-algebraic role for model-theoretic constructions---so perhaps they've got an intermediate position, in the end. <BR/><BR/>I'm not sure I understand everything that is involved in the intermediate positions; whereas I'm very comfortable that I understand the two extreme positions I started with. What prompted this set of posts was my realization that the model-theory-as-algebraic-device option was available, and pretty attractive, and could be totally utterly different---not even approximately the same as---views which e.g. think that borderline sentences have an intermediate degree of truth. <BR/><BR/>What I need to do now is think about a non-classical model theory. There seem to be lots of options, i.e. retaining classical models, but giving up on excluded middle for the predicate "is the intended model". Or going non-classical with the models themselves (say by viewing them as sets characterized by a non-classical set theory---maybe that's a Priestian view). And there seem lots of other options. Those seem very different. E.g. if only the predicate "is an intended interpretation" is non-classical, and the underlying set theory is classical, then classical model theory still gives us the logic. If the models themselves are non-classical, then we'll have to examine arguments that this or that argument is valid, to see if they employ dubious classical reasoning. <BR/><BR/>What I'm inclined to think is that prior to doing this, we really need to know what a theory of the "intended model" is supposed to *do* for us. One thing classicists sometimes do is have an axiomatic specification of the intended interpretation, and from that derive canonical T-sentences for the language. You might think of that as underpinning a theory of understanding (understanding=knowing the canonical T-sentences) or you might think of them as giving a theory of what the representation properties of expressions are. If so, we've got at least one clear success-conditions for a non-classical model theory---allow the derivation of T-sentences. But it's *not* clear to me that this is what people are after. <BR/><BR/>So: lots more to think about! And lots of reading to do...Robbiehttp://www.blogger.com/profile/02081389310232077607noreply@blogger.comtag:blogger.com,1999:blog-6432111.post-18342496727311914512008-03-22T00:42:00.000+00:002008-03-22T00:42:00.000+00:00Oh and one other thing:Alan Weir considers a parac...Oh and one other thing:<BR/><BR/>Alan Weir considers a paracomplete theory in which validity is defined in a non-standard way. His reasons? He wants to take the model seriously -- indeed he requires a conditional that has a deduction theorem. He specifically doesn't wish to have a "no interpretation" account. He's what (I think) you'd call a "folklore nonclassicist". <BR/><BR/>In the end, Weir's truth predicate is transparent (T[A] and A are substitutable in all non-opaque contexts.) <BR/><BR/>See Weir's "Naive Truth Theory and Sophisticated Logic" in JC's _Deflationism and Paradox_ volume.cotnoirhttp://cotnoir.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-6432111.post-26914078204375270512008-03-20T14:13:00.000+00:002008-03-20T14:13:00.000+00:00Hi Robbie,Wrt you last comment about validity, the...Hi Robbie,<BR/><BR/>Wrt you last comment about validity, there are some interesting options here. For instance, there is a (somewhat) common definition for validity in K3 in which an argument A |- B is valid just if v(A) is less than or equal to v(B). This fits more naturally with the "degree of truth" view of the semantic values. I've got a post on what this logic might look light, and some related logics in the area. You might like to check it out: http://cotnoir.wordpress.com/2008/02/01/between-lp-and-k3/<BR/><BR/>Also, wrt to Field's view about degrees of belief, things become even stranger when we start thinking about conditionals. It's certainly available, I suppose, to a "no interpretation" paracomplete theorist to hold the same line for conditionals as well -- especially since Field's conditional collapses into the material in LEM-satisfying contexts. Field seems to think that the degree of belief in a conditional can diverge even more radically from the truth value. He thinks the Ramsey-Lewis stuff is plausible, and tries to drive a wedge between "conditional assertion" and "assertion of a conditional". There are some pretty complicated issues here with embedded conditionals that I don't really understand. But anyway, I bring this up only to point to the fact that Field may have some independent reason to divorce degree of belief in a proposition from it's semantic value.cotnoirhttp://cotnoir.wordpress.com/noreply@blogger.com