Re: Tautologies Then and Now
From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/09/04
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Date: Thu, 09 Dec 2004 07:49:46 GMT
"Stephen Harris" <cyberguard1048-usenet@yahoo.com> wrote in message
news:AcTtd.40421$6q2.21793@newssvr14.news.prodigy.com...
>
>>> "paul" <paul8801@on-ramp.nl> wrote in message
>>> news:r58er0t5vm620c96lubbut8noob43023ue@4ax.com...
>>>> On Tue, 07 Dec 2004 23:58:40 GMT, "Stephen Harris"
>>>> <cyberguard1048-usenet@yahoo.com> wrote:
>>>>
>>> I think you are right about truth tables (unless there is something
>>> technical),
>
> There appears to be "truth functional proxies".
>
>> For tautologies there is a general method for showing intrinsic truth,
>> a truth table. There is no general method for showing the intrinsic
>> truth of valid statements. A general algorithm for proving a formula to
>> be valid is not possible. But I'm wondering if in a particular class of
>> cases, if there is a specific algorithm for proving formulae to be valid,
>> which would function in principle like a constrained truth table.
>
> Truth functional proxies.
>
>>
>
> As to using logic for reasoning about natural language and common sense.
>
> Talking about Trees and Truth-conditions
> Reinhard Muskens http://www.illc.uva.nl/j50/contribs/muskens/muskens.pdf
> Abstract
> "An attractive way to model the relation between an underspecified
> syntactic representation and its completions is to let the underspecified
> representation correspond to a logical description and the completions to
> the models of that description. This approach, which underlies the
> Description Theory of (Marcus et al. 1983) was integrated with a pure
> unification approach to Lexicalized Tree-Adjoining Grammars (Joshi et al.
> 1975, Schabes 1990) in (Vijay-Shanker 1992) and was further developed in
> the `D-Tree Grammars' (DTG) of (Rambow et al. 1995). We generalize
> Description Theory by integrating semantic information, that is, we
> propose
> to tackle both syntactic and semantic underspecification using
> descriptions.
> Our focus will be on underspecification of scope. We use a generalized and
> completely declarative version of the D-Tree formalism. Although trees in
> our set-up have surface strings at their leaves and are in fact very close
> to ordinary surface trees, there is also a strong connection with the
> Logical
> Forms (LFs) of (May 1977). We associate logical interpretations with these
> LFs using a technique of internalising the logical binding mechanism
> (Muskens
> 1996). The net result is that we obtain a Description Theory-like grammar
> in
> which the descriptions underspecify semantics. Since everything is framed
> in
> classical logic it is easily possible to reason with these descriptions.
>
> Internalising Binding
> How can we assign a semantics to the lexical descriptions in fig. 1? We
> must
> e.g. be able to express the semantics of n1 in terms of the semantics of
> n2,
> whatever the latter turns out to be, i.e. we must be able to express the
> result of quantification into an arbitrary context. In mathematical
> English
> we can say that, for any @, the value of allx@ is the set of assignments a
> such that for all b differing from a at most in x, b is an element of the
> value of @. We need to be able to say something similar in our logical
> language. The language must talk about meaning; it must talk about things
> that function like variables and constants, things that function like
> assignments, etc. The first will be called registers, the second states.
> Two
> primitive types are added to the logic: Pi and s, for registers and states
> respectively. We shall have variable registers, which stand proxy for
> variables and constant registers for constants. ...
>
> We have essentially mimicked the Tarski truth conditions for predicate
> logic
> in our object language and in fact it can be proved that, under certain
> conditions, we can reason with terms generated in this way as if they were
> **the predicate logical formulas they stand proxy for (see Muskens 1998).
>
> It should be stressed that the technique discussed here can be used to
> embed any logic with a decent interpretation into classical logic. For
> example, (Muskens 1996) shows that we can use the same mechanism to embed
> Discourse Representation Theory (DRT, Kamp & Reyle 1993). In a full
> version of this paper we shall also present a version of our theory based
> on DRT."
>
> SH: This indicates to me that if predicate logic can be embedded, then
> so can a simpler aspect, truth functional proxies, akin to propositional
> logic which would help logical reasoning about natural language input.
>
> I found researching this thread from vague memories quite interesting.
>
> "In Sentential Logic, we can prove an argument schema to be invalid by
> specifying a set of truth assignments to the sentential letters which
> results in true premises and a false conclusion; we thereby show that one
> line of the argument schema's truth table allows an interpretation having
> true premises and a false conclusion. In Predicate Logic, an argument
> >schema typically consists of sentence schemata which are not truth
> functional: quantifiers, not truth functional connectives, are the major
> operators of the typical "quantified
>
> *argument schemata." And quantifiers are not truth functional operators
> since they may represent an infinite number of individuals; the truth
> value of a quantified sentence schema is therefore not a function of the
> truth values of any _finite_ number of simple sentence schemata.
>
> *Nevertheless, we can test the validity of a quantified argument schema
> _indirectly_ by constructing and testing its _truth functional proxy for
> some_ (non-empty) _domain_ of a specified (finite) number of individuals;
> each of the premises, and the conclusion, in the original schema will be
> equivalent _in that domain_ to its truth functional counterpart in the
> proxy.
>
> Because it is comprised of truth functional sentence schemata, a proxy may
> be tested for validity by the short-cut method of truth value assignment,
> or by *means of a truth table.*
>
> And if a proxy proves to be invalid, it will provide a "recipe" for
> constructing an interpretation of the corresponding quantified argument
> schema into the same domain which will serve as a counter example, or
> refutation, to that argument schema. Thus, if the original quantified
> argument schema is valid, then _all_ of its corresponding proxies must
> also be valid. If _any one_ of the proxies corresponding to a quantified
> argument schema is invalid, then since it is therefore possible for the
> schema to have an interpretation into some domain under which its premises
> are true while its conclusion is false, the schema itself is invalid. Note
> that even though one particular corresponding proxy is valid, the original
> quantified argument schema might nevertheless be invalid: to be valid,
> _every_ corresponding proxy (for _every_ non-empty domain) must be valid."
> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
> Regards,
> Stephen
>
>
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