Re: Tautologies Then and Now

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/09/04 

Date: Thu, 09 Dec 2004 07:32:48 GMT

>> "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.

Regards,
Stephen