Re: Name the thesis: "Formal sentences capture informal ones"
From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 02/03/05
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Date: Thu, 03 Feb 2005 18:45:47 GMT
<tchow@lsa.umich.edu> wrote in message
news:41fbeb5c$0$580$b45e6eb0@senator-bedfellow.mit.edu...
> The Church-Turing thesis is familiar to many people, largely because it
> has been widely discussed both in textbooks and in popular science
> >writing.
> Having a name helps, too.
>
> There is an analogous thesis that is relevant to logic and the foundations
> of mathematics:
>
> (*) Formal sentences (in PA or ZFC for example) adequately express
> their informal counterparts.
>
Since I think "quantification theory" is a synonym for (FOPL) First
Order Predicate Logic, then there appear to be anomalies residing
in translating from natural language, which I take to be the vehicle
of informal logical statements about mathematics, and mapping all
such translations onto formal sentences which are identical in
meaning to the informal sentence which is to be represented formally,
or vice-versa.
Perhaps I am confusing your statement with Logicism. Or with
Feferman's position:
SF: "Though Tarski's consideration only of the question
"whether mathematical notions are logical notions" and not
of "whether mathematical truths are logical truths" appears
at first sight to be a reasonable one, it is not clear that
the two can be separated so neatly. For, any argument one
way or the other about the first question must necessarily
invoke assumptions about various properties of the notions
involved, and those lead one into the second question."
SH: Quantum theory has one standard underlying mathematical
formalism. Yet, there are eight major interpretations about
what that formalism describes about reality; interpretations
which I would think would be classified as informal and which
are not unique and seemingly contradictory in some cases, so
the formal statement is not enough to distinguish between them.
It is my impression that a "thesis" has no counter-examples,
so that what you are describing is more accurately a guideline?
http://www.fordham.edu/gsas/phil/klima/NLN.htm
Gyula Klima: Approaching Natural Language via Mediaeval Logic
I. ANOMALIES OF A PARADIGM
Mismatch of syntax
"As is well-known, natural language sentences of evidently the same
syntactic structure are represented by formulae of quantification
theory of entirely different structure, while the same formula may
have different "readings", expressible by natural language
sentences of widely different syntax.
Regarding these discrepancies, of course, one might say that
there is no justifiable need of a strict correspondence between
the syntactic structure of natural language sentences and the
formulae representing them. After all, a logical semantics, which
is to be a general semantics for all kinds of human languages,
should precisely disregard accidental grammatical features of
particular natural language expressions, and hence also the
delusive grammatical structure of natural language sentences in
general. All that is required for correspondence is that the
formula should state correctly the truth conditions of the
sentence which it represents, since it is only these truth
conditions that determine the logical relations of sentences
among each other.
Along these lines, mismatch of syntax may be made to appear
entirely harmless, by making a distinction between logical
form on the one hand, and grammatical form on the other,
placing much confidence in the capability of quantification
theory to express the former, and thereby justifiably ignoring
the latter.
Unrepresentable sentences
There is, however, a further set of anomalies, which comes as
a fatal blow to this interpretation of the relationship between
quantification theory and natural languages. For, as it turned
out, some apparently simple quantified sentences of natural
languages are demonstrably unrepresentable in first order
quantification theory in the sense that no first order formula
is able to give their correct truth conditions.[3]
As is well-known, examples of such sentences are those containing
the determiners 'most' or 'more than half of', and so on. But if
there are no formulae giving the correct truth conditions of such
sentences, then quantification theory is simply unable to supply
their logical form, and so the above-mentioned rationale for
drawing the distinction between logical and grammatical form
breaks down with these sentences. ...
Intensional and intentional contexts
To be sure, the above-mentioned "anomalies" may be considered as
such only because they pose problems to quantification theory
that everyone feels it should handle but cannot. It was clear
from the beginning that there are large portions of natural
language reasonings that simply fall outside the authority of
quantification theory, namely those involving intensional
contexts. Nevertheless, Frege's relegation of modal notions to
the sphere of psychology notwithstanding, logicians have been
working on expanding formal logic even to these contexts.
Possible worlds semantics produced interesting results concerning
modal notions and still seems to have some resources concerning
tensed modal contexts. However, in virtue of the coarse-grained
character of intensions available in possible worlds semantics,
several intentional contexts, namely those created by attitude
verbs, seem to defy analysis in terms of these intensions."[5]
Looking forward to another learning opportunity,
Stephen
Condensed from Sol Feferman:
'David Lewis (IHO) makes a persuasive case that we do
have an independent grasp of plural quantification that
doesn't have to be explained in terms of second-order
quantification, though there appears to be an asymmetry
between existential plurals (natural) and universal
plurals (not natural) in English.'
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