🚀 go-pugleaf

In article <1992Nov23.111320.17317@news.Hawaii.Edu>
lady@uhunix.uhcc.Hawaii.Edu (Lee Lady) writes:

>In article <1992Nov21.115346.17656@husc3.harvard.edu>
>zeleny@husc10.harvard.edu (Michael Zeleny) writes:

MZ:
>>Just because the correctness of your definition of synonymy is not
>>amenable to confirmation by (narrowly construed) empirical means, does
>>not imply that there is no matter of fact about it.  In particular, one
>>failing of the possible-worlds approach consists in its identification
>>of proof-theoretically equivalent propositions.  This is unproblematic
>>in situations which call for the study of such deductively closed
>>domains as doxastic commitment; however even in such cases, full
>>generality will be sacrificed to the implicit assumption that the
>>underlying Boolean algebra of propositions must be atomic.

LL:
>I may be an outsider to this discussion, but it's quite clear to me that
>a whole lot of smileys have been omitted here.  As near as I can
>determine, this paragraph is some sort of self-parody.  I like the way it
>sucks me in, starting with statements which almost seem to have some
>meaning -- at first, the absurdity of confirming the correctness of a
>definition by empirical means doesn't really hit one -- and it almost seems
>that I vaguely perceive what it is you're saying.  
>
>And the you build up to the grand finale statement that 
>>full generality
>
>[of what? in what sense? Of course the joke is on the poor reader who
>believes that there actually is some referrent for the word "generality"
>but that he is to dense to find it.] 
>
>>     will be sacrificed to the implicit assumption 
>
>[Another put on:  You see, reader, the reason you didn't see this
>assumption  [assumed by whom?]  is that it's implicit, and you just weren't
>smart enough to notice it!] 
>
>>that the underlying Boolean algebra of propositions must be atomic.  
>
>[Aha!  You see, dear reader, this is very deep stuff, having to do with
>highly abstuse concepts such as Boolean algebras, and that's why you
>can't understand it.]  
>
>In fact, I can't imagine any context except highly contrived ones in
>which the boolean algebra of propositions would ever be atomic.  
>
>
>I can only guess that the point you want to make is that attempts to
>develop a theory of meaning by invoking model theory and the model-theoretic
>concepts of syntactic truth and semantic truth ... are only useful as venues
>to display one's erudition and engage in one-upmanship.  
>
>(Incidentally, in model theory a sentence is said to be syntactically true 
>if it can be proved from the axioms.  A sentence is semantically true if
>it is true in every model, i.e. in every "interpretation."  This
>definition was paraphrased in this discussion by saying that the
>statement is true "in all possible worlds."  A mathematical theory is
>called "complete" is every semantically true statement is syntactically
>true, i.e. "every true statement is provable.")  
>
>(Sorry if I'm spoiling everyone's fun by explaining what the words
>actually mean.)

Thank you for playing to the audience, Lady.  I deeply regret my
scandalous presumption of referring to mathematical objects without your
express permission, and appreciate your selfless glossing of my true
meaning.  Allow me to return the favor by offering a brief explanation of
my original point; feel free to stop me whenever things get too
complicated for you.

Roughly speaking, Montagovian semantic theoriess are characterized by
their identification of propositions with sets of possible worlds.
Accordingly, for each proposition associated with a set A of cardinality
\alpha, there exist \alpha many atoms 0 -< {a} < A, where a is an element
of A.  Consequently, the Boolean algebra of propositions turns out to be
atomic, which is an undesirable constraint for a general theory of
meaning.  For just like your own enlightened self, I can't imagine any
context except highly contrived ones, in which the boolean algebra of
propositions would ever be atomic.

As for the merits of attempts to develop a theory of meaning "by invoking
model theory and the model-theoretic concepts of syntactic truth [sic]
and semantic truth", I am afraid that you are right again, as your own
example of blithering incomprehension ought to convince any reasonable
individual that theory of meaning is impossible.

LL:
>You did remind me of an interesting question is mathematical thinking,
>though.  It is commonplace for mathematicians to say that two particular
>theorems are equivalent to each other.  This is definitely a statement
>about syntactic equivalence, i.e. that either of the two can be proved by
>using the other.  On the other hand, by basic logic any two
>syntactically true theorems are in fact syntactically equivalent.  So is
>it possible to assign any somewhat precise meaning to the statement that
>two theorems are equivalent, or is this just an impossible vague
>assertion, comparable to the assertion that certain things are
>"interesting"?   (What this has to do with cognitive science is more than
>I can figure out.)  

This is a mess.  When you say that e.g. Zorn's lemma is equivalent to
Zermelo's axiom of choice, what you are asserting is exactly the
syntactical equivalence of these propositions in ZF, i.e. 
		ZF |- ZL <=> AC.  

This is as precise as it can be.  On the other hand, one might be
interested in the notion of semantic equivalence, or synonymy, whereupon
it is reasonable to assume that ZL is not synonymous with AC, insofar as
it would be possibile to believe in the truth of just one of them, so
that a more finely grained notion of equivalence will be called for.
Thus one might rule that two propositions are synonymous iff they are
interderivable by \lambda-conversion, or, equivalently (assuming Church's
thesis), iff their equivalence is demonstrable by effective computation.
This notion of synonymy (Church's Alternative (1)) assumes computational
omniscience of the cognizer, in the same way the aforementioned Montague
possible-worlds semantics (as well as Church's Alternative (2), which has
the merit of avoiding the atomicity constraint) makes a stronger
assumption of his logical omniscience.  Finally, a stronger criterion of
synonymy (e.g. Church's Alternative (0)) will allow only the interchange
of primitively synonymous constants.  Naturally, all of this can be
relativized to pragmatic contexts via appropriate treatment of deixis.
Given an empty list of primitive synonyms, and a sufficiently restrictive
notion of context, Swann's theory of absolute semantic irreproducibility
will result.

>--
>It is a poor sort of skepticism which merely delights in challenging
>those claims which conflict with one's own belief system.  
>                                                          --Bogus quote 
>lady@uhunix.uhcc.hawaii.edu         lady@uhunix.bitnet

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."