>Pat Hayes wrote:
>
>>>>Why do you assume that 'rains' is logically false? I have no idea
>>>>what its actual logical truth-value is;
>
>Of course.
>
>>>
>>>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.
>
>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? (01)
Sorry. Yes, of course you can. You can in fact state it as an axiom
in IKL, if you wish: (02)
(iff (rains)(forall (s)(ist (that (rains s))))) (03)
or more radically (04)
(= rains (that (forall (s)(ist (that (rains s))))) ) (05)
But it is not a *logical* truth. That is, this choice is not
necessary, or forced upon one by the inherent structure of contexts
or propositions. One could just as well, and just as consistently,
assert (06)
(iff (rains)(exists (x)(and (madeOf x cheese)(Orbits x Earth)))) (07)
> This, of course, is
>inessential.
>
>>After all, that can be stated explicitly:
>>
>>(forall (c)(ist c (that (rains))))
>
>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. (08)
Right. Bearing in mind that as far as IKL is concerned, "p is
context-true in c" is just an English gloss for "the relation ist
holds between c and p" (09)
> It is not about raining (whatever 'rains' means). (010)
Well, it is about it to the extent that it determines it's ist-relation to c. (011)
>
>I admit that talking about context-dependent truth of sentences sounds
>more natural to me than talking about truth-independent context-truth of
>propositions. (012)
I agree, which is why I tend to slip into that way of talking myself.
But the formalism treats it the second way. The surprising fact (and
it is a fact) is that this works: that is, that (modulo the
complication regarding opaque names) the two readings coincide. (013)
> This may be an unfortunate mindset.
>
>I am also suspicious about the distinction between a proposition's being
>non-contextually true a proposition's being true in all contexts. (014)
It sounds as though you would like the above equivalence to be
logically true. IKL could have been constructed in this way; but to
do so would have required that 'ist' be given a logical status, and
not be a mere relation between contexts and propositions. That would
have made the actual logic more complicated, and made it into a
species of context logic. Whereas our mandate (and scholarly
interest) was in keeping it as simple as we could, and we wanted to
avoid making our interlingua into a context logic. (015)
> What
>is the difference between being true independently of context and being
>true whatever context is chosen? (016)
Er... they have nothing to do with one another? What is the
difference between a duck and a camel? (017)
>
>If
>
>(dead osama)
>
>"asserts that Osama-Bin-Laden simply has a property, without any
>qualification as to time or circumstances" (018)
But that way of phrasing it is misleading, since it suggests that
having a property is something that CAN be qualified by time or
circumstance. IKL does not admit that notion of qualification as a
logically meaningful kind of expression. This sentence in IKL means
simply that osama has a property. (019)
In any case, time and circumstance are not enough. One has to allow
other kinds of contextual qualification (or at least, the context
logicians claim this), including spatial locations (It is true in
France ...) beliefs (Joe believes ...), fictional narratives (In the
Conan Doyle novels, ...), provenances (According to Hoyle, ...) and
even purely 'formal' kinds of "context" such as "It is consistent
with set S13 of sentences that ..." (020)
.
>, then
>
>(ist t1 (that (dead osama)))
>
>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. (021)
Right. And that is the IKL way of modelling what would be stated in a
context logic as the claim that osama has a property is contextually
true in c. (022)
> 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. (023)
But what does this 'viewing from a perspective' mean, logically? How
shall we represent this in a formalism? Are we obliged to go beyond
classical logic to do an adequate job of encoding this kind of
meaning? Many have claimed that this is inevitable, and that the
presence of such perspectives or contexts means that classical logic
is inadequate. The formal adequacy of the IKL encoding of such forms
suggests however that this assumption is wrong, or at any rate needs
to be re-examined critically. (024)
> It is about the proposition (that (dead osama))'s simply being related to c.
>
>>> Or how should it be understood?
>>
>>I have no idea. You wrote it, not me :-).
>
>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. (025)
See above. (026)
>
>>In IKL, a relation with no arguments is itself a proposition, so
>>one could write it without the "that":
>>
>>(forall (c)(ist c rains))
>
>Just a syntactic feature? (027)
Yes, essentially. It is way that the use/mention distinction applied
to propositions comes out in the IKL syntax. (028)
>
>>
>>>
>>>>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.
>
>But what would 'gusty winds exist' mean uttered timelessly? (029)
What it says: gusty winds exist. In fact, they do, so it is true. (030)
> What would '2+2=4' mean uttered timelessly? (031)
I take it be uttered timelessly, usually. I certainly don't usually
mean it say, two plus two equals four *now*. (032)
> What would it mean in the real world? (033)
That discussion would take us into the foundations of mathematics and
mathematical philosophy, and Id rather not venture there. Im quite
happy to be a robust Platonist for ontological purposes. So, I will
claim that 2+2=4 is simply timelessly true in the real world. (034)
>
>
>>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.
>
>>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.)
>
>Yes, this is an interesting bit of the guide.
>
>>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.
>
>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'. (035)
true *in c*. (036)
> 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. (037)
Oh no, it really does assume a context-dependent truth. Meanings can
change, but they also can remain the same but truth-values change. I
agree, your notion would not require such a radically different logic. (038)
>
>>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,
>
>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. (039)
Ah, I see. No wonder we are having trouble communicating. The primary
authorities on the context logic we were referring to insisted on the
following, when we asked them what 'ist' meant: (040)
1. The syntactic form is "ist( <context-name> <sentence>)" (Not the
IKL form, note)
2. The first argument denotes a context. What a context is, is not
specified in the logic. Anything that satisfies the logical axioms
can count as a 'context'.
3. The second 'argument' denotes a proposition, but the proposition
it denotes may be dependent on the context denoted by the first
argument. That is, the mapping from sentences to propositions is
itself contextual, and so opaque. Or at any rate it may be. In this
respect, ist(c ...) acts rather like a modality with c as a parameter.
4. The truth-value of a proposition may itself be different in
different contexts. That is, it may be that all the names in the
language refer in the same way in two contexts c1 and c2, and still
it be the case that ist(c1 p) and ist(c2 p) differ in truth-value. (041)
When pressed on what exactly they meant by a proposition, we were not
able to get a clear answer, other than it is clear that a sentential
syntactic form is able to denote one. About the only universally
agreed axiom was the distribution of ist over conjunction, which has
to be stated as an axiom schema or an inference rule in a first-order
context logic: (042)
(forall (c)(if (ist c (and <P> <Q>)) (and (ist c <P>)(ist c <Q>)) )) (043)
but in IKL is an axiom: (044)
(forall (c p q)(if (ist c (that (and (p)(q)))) (and (ist c p)(ist c q)) )) (045)
>
>>so that truth-in-a-context is an objective matter and context logic
>>is the most appropriate formal vehicle for meaning.
>
>Yes, proviso that 'truth-in-a-context' means truth of a sentence in a context.
>
>My suspicion about non-contextual truths is that
>
>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;
>
>b) when it comes to sentences, all are true (or false) only in a context.
>
>A context logic addresses the latter (046)
The context logics we were dealing with claimed both kinds of
contextual relativity, and addressed both. The IKL translation uses
IKL ist to handle the former (relativity of truth to context) and the
contextual-naming device to address the latter (relativity of
sentence meaning to context). (047)
>; 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.
>
>>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.
>
>Agree.
>
>>Hope this helps.
>
>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. (048)
If someone of your obvious insight can be muddled by our exposition,
we have some more explanatory work to do. (049)
Pat (050)
>
>vQ (051)
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