Pat Hayes wrote: (01)
>>> Why do you assume that 'rains' is logically false? I have no idea
>>> what its actual logical truth-value is; (02)
Of course. (03)
>>
>> because I take
>>
>> (that (rains))
>>
>> to be a proposition that it rains, without any indexicals, that is,
>> that it rains everywhere, at all times.
>
> I don't think that is how it should be understood. (04)
Why? As John says, that's my choice. Perhaps if I chose otherwise,
(that (rains)) would be false. Why can't I decide that 'rains' means
that it rains at all times and in all places? This, of course, is
inessential. (05)
> After all, that can
> be stated explicitly:
>
> (forall (c)(ist c (that (rains)))) (06)
Well. Again, this says that in every context, the proposition that it
rains (whatever it means, according to my wish) is context-true in that
context. It is not about raining (whatever 'rains' means). (07)
I admit that talking about context-dependent truth of sentences sounds
more natural to me than talking about truth-independent context-truth of
propositions. This may be an unfortunate mindset. (08)
I am also suspicious about the distinction between a proposition's being
non-contextually true a proposition's being true in all contexts. What
is the difference between being true independently of context and being
true whatever context is chosen? (09)
If (010)
(dead osama) (011)
"asserts that Osama-Bin-Laden simply has a property, without any
qualification as to time or circumstances", then (012)
(ist t1 (that (dead osama))) (013)
asserts that c and (that (dead osama)) are ist-related, without any
quantification as to time or circumstances. If you add a bit of
ontology and interpret ist as a is-true-in relation between a context
and a proposition, then the above asserts that the proposition that
osama simply has a property is-true-in c. It is still about osama's
simply having a property (it is about the proposition about osama's
simply having a property), only that viewed from the perspective of a
context. It is about the proposition (that (dead osama))'s simply being
related to c. (014)
>> Or how should it be understood?
>
> I have no idea. You wrote it, not me :-). (015)
Sure. But you have the idea that (that (rains)) should not be
understood as the proposition that it always rains -- how come? I wrote
it, not you. (016)
> In IKL, a relation with no
> arguments is itself a proposition, so one could write it without the
> "that":
>
> (forall (c)(ist c rains)) (017)
Just a syntactic feature? (018)
>
>>
>>> but in any case that would be irrelevant to its *contextual* truth,
>>> which is modelled in IKL by the ist relation. "True throughout an
>>> interval" and "standing in the ist relation to an interval" are just
>>> two ways to say the same thing. Or perhaps, if you feel that truth at
>>> a time is something fundamental, by all means say that the
>>> ist-formulation is IKL's way of modelling or describing or
>>> representing the notion of truth at a time.
>>
>> But are we not taking a round here, saying that (ist c p) *does* mean
>> that p *is true* in c, even if it is false?
>
> My point is that truth in a noncontextual classical logic such as FOL or
> CL or IKL - logical truth - on the one hand; and 'truth at a time' or
> 'truth in a context', on the other, are two *distinct* notions. They
> have different meanings and different logics, and support different
> notions of interpretation and of proof. They can be related in various
> ways, which are well-understood, but they are not the same notion. In
> particular, if I can warp the English for a moment and write L-truth for
> the first notion, contextual truth is *not* L-truth at a time or L-truth
> in a context, since it is quite literally meaningless to relativize
> L-truth to a context. So if we look at a simple English present tense
> sentence such as "gusty winds exist", it can be understood in two
> different ways: as uttered in the present (and perhaps a particular
> place, ie with an implicit 'now, here') and referring to it, or as
> uttered timelessly. (019)
But what would 'gusty winds exist' mean uttered timelessly? What would
'2+2=4' mean uttered timelessly? What would it mean in the real world? (020)
> And these are not the same kind of meaning. When
> setting out to formalize meanings, one has to make a choice whether to
> attempt to model the eternal interpretation or the tensed (more
> generally, contextual) interpretation. One gets a different logic in
> each case. (021)
> So, now, armed with this distinction: IKL, being a classical logic,
> takes the first route. Truth in IKL is understood non-contextually. IKL
> sentences are not asserted "in" a context of any kind. They are simply
> asserted. The English that they model, if that can even be spoken of
> coherently, is only that of 'eternal' sentences in the simple
> grammatical present tense, but understood as tenseless in meaning. There
> is no implicit "here, now" indexical lying behind the meaning of any IKL
> sentence. So, to model the meaning of English sentence forms which *are*
> indexical or contextual in this way in IKL, it is necessary to put in
> these contextual parameters explicitly, to "de-contextualize" the
> logical form. To represent the meaning of the New Mexico road sign
> (omitting the modality for now to keep it simple) "Gusty winds exist" in
> IKL, one would write not
>
> (exist (x)(and (Gusty x)(Wind x)))
>
> but something more like
>
> (exist (x)(and (Location x herePlace)(Time x nowTime)(Gusty x)(Wind x)))
>
> (There are also several other ways, and this is something of a
> caricature, but you get the idea.)
>
> So far IKL is indistinguishable from any classical FOL: one has to
> 'de-contextualize' any context-dependent assertion or indexical content
> to represent its 'context-relative' meaning in any noncontextual logical
> framework. But IKL also has the ability to refer to propositions, so
> this gives us yet another way to encode the intended meaning of a
> contextual assertion, one which is superficially very similar to the way
> it is represented in a contextual logic. We can in fact mirror the
> context-logical syntax almost exactly. Context logic has a basic
> expression form
>
> a: (ist C <sentence>)
>
> eg. (ist Present (exists (x) (and (gusty x)(wind x))) )
>
> where 'ist' is a logical symbol on a par with the connectives and
> quantifiers, one with a meaning fixed by the context-logical semantics.
> As it stands this is not well-formed in IKL, since it seems to state a
> relation between a thing and a sentence. But we can mirror this in IKL
> with the closely similar expression
>
> [a]: (ist C (that <sentence>) )
>
> eg. (ist C (that (exists (x) (and (gusty x)(wind x)))) )
>
> which is syntactically legal: it states that a relation called "ist"
> (which is now simply a relation symbol, not a special logical form)
> holds between C and the proposition (that <sentence>). What it *means*
> however is essentially aribitrary in IKL. In particular, it has no
> IKL-sanctioned *logical* connection with truth in IKL. But we can agree
> to give it a meaning, and it is useful to do so.
>
> Consider the translation from a to [a] to be an embedding of context
> logic into IKL syntax. Now, it turns out that this translation
> completely mirrors the context logic. That is, A entails B in context
> logic just when [A] logically entails [B] in IKL. (Actually this is not
> quite true: the syntactic embedding needs to also perform a mapping on
> the names used opaquely inside contexts, so is more complex than
> described here: see the IKL guide for details.) (022)
Yes, this is an interesting bit of the guide. (023)
> In context logic, "ist" defines a context-relative notion of truth. In
> IKL, "ist" is simply a relation which holds between C and the
> proposition (that <sentence>) precisely when <sentence> is true-in-C in
> the context logic. (024)
But true-in-c in context logic means, as far as I get it, 'the
proposition which is the meaning of <sentence> in c is true'. This is
the same -- the only -- notion of truth, the L-truth, in the logic.
What is contextual, is the mapping from a sentence to a proposition. No
need for different notions of truth. (025)
> I tend, perhaps confusingly, to use this terminology
> when referring to the IKL translation.
>
> (There is some history here which may be relevant. IKL was developed in
> order to provide an interlingua for a variety of formalisms, two of
> which used contexts and three did not. It was therefore necessary for us
> to go into some detail concerning the various notions of truth and
> satisfiability involved. John McCarthy, who introduced context logic
> some time ago but did not work out the underlying theory in detail,
> insisted that the sentence form inside 'ist' should be regarded as
> denoting a proposition, and 'ist' thought of as a relation between
> contexts and propositions, and that contexts were simply whatever
> satisfied whatever context theory one had in place. All of which lay
> behind our design of IKL. Having introduced propositions as objects,
> however, we discovered, rather to our own surprise, that this rendered
> the rest of the context logic irrelevant: one can express the entire
> context-logical framework within a classical logic which can describe
> propositions. (It also seems, although we have not yet fully
> investigated this, to provides a new way of resolving the classical
> paradoxes.) As classical logics are far better understood than
> contextual logics, this seems like a useful technical advance.)
>
> I believe you may feel that truth is often, or perhaps always, best
> understood as being relative to a context, (026)
Depends on what you now mean by 'truth'. The truth of a proposition,
no; this was one of my primary questions on this thread, and we seem to
have agreed here that propositions have fixed truth values. The truth
of a sentence, yes, is relative to a context. (027)
> so that truth-in-a-context is
> an objective matter and context logic is the most appropriate formal
> vehicle for meaning. (028)
Yes, proviso that 'truth-in-a-context' means truth of a sentence in a
context. (029)
My suspicion about non-contextual truths is that (030)
a) when it comes to propositions, all are non-contextually true (or
false), so it makes no sense to talk about their truth in a context; (031)
b) when it comes to sentences, all are true (or false) only in a context. (032)
A context logic addresses the latter; a non-contextual logic (fol, say)
addresses the former; IKL seems to me an approach to mix these by means
of contextualizing propositions while claiming it is not contextual. (033)
> Although this position is widely held, I myself do
> not agree: I understand the tensed use of the present tense as referring
> to the actual present time, so that "It is raining now" spoken at 3.30
> pm on the 21 March 1998 is exactly and precisely synonymous with the
> tenseless assertion "Raining at 3.30 pm on the 21 March 1998".
> Contextually asserted and indexical assertions are, on this view, simply
> incomplete, and have part of their logical meaning implicit. Most of
> what is said in natural language does not express a determinate
> proposition until it has been rendered non-contextual by having all its
> implicit indexicality resolved, and at that point it can perhaps be
> rendered into a classical logic, but not before. (034)
Agree. (035)
> Hope this helps. (036)
I appreciate that you make that much effort to answer my (...) questions
-- thanks. Whether this helps, I am not sure yet. If not, surely the
fault is mine. (037)
vQ (038)
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