On Tue, Aug 07, 2007 at 10:18:38AM +0100, Barker, Sean (UK) wrote:
> Adrian, John, Pat
>
> Adrian is right about my concern is the effect on the real world.
> Although I haven't studied the history of logic, I'm hardly surprised
> that the various independent formulations of two valued logic are
> isomorphic to FOL (or at least homomorphic to some subset of FOL). (01)
That's probably not the best way to put it. FOL is not itself a fixed
structure like, say, the rational number structure, so it doesn't really
make sense to talk about it being isomorphic to anything. There are a
number of ways of approaching the issues, but for purposes here a
*logic* can be thought of as a (typically rather large and inclusive)
family of formal languages together with a semantic theory for those
languages. A semantic theory for a class of languages is simply a
systematic method of assigning semantic values to the members of the
various syntactic categories of those languages. An application of a
semantic theory to a particular language yields an *interpretation* of
the language, i.e., an assignment of appropriate semantic values to the
basic lexical items of the language. Appropriate semantic values are
then assigned by the theory to complex syntactic constructions of the
language -- notably, truth and falsity to sentences -- recursively in
terms of the semantic values of their component parts. (02)
What makes a logic *first-order* is a fairly technical matter, but it
consists essentially in the possession of two semantic properties,
compactness and the downward Löwenheim-Skolem property. These
properties are critically connected to the fact that FOL is *complete*,
that is, that it is possible to *axiomatize* logical consequence, i.e.,
the fundamental logical relation that holds between a set S of sentences
and a given sentence A when any interpretation that makes all the
members of S true also makes A true. It is (among other things)
completeness that makes FOL so important to automated reasoning, as it
means that, whenever A is a logical consequence of S in a first-order
logic, there is actually a *proof* of A from some (finite) subset of S.
(Alas, finding that proof is another kettle of fish entirely, which is
why we have more constrained, less expressive logics like description
logics and their ilk.) (03)
> However, if you were to start with a set of truth tables and
> systematically changed True to False and vice versa, and AND to OR and
> vice versa (though NOT remains the same) then you would apparently
> have the same system, even through True is now False. (04)
Well, that's certainly a rather perverse way to express a trivial fact
about propositional logic: You can violate the usual convention and let
1 represent Falsity instead of Truth and let 0 represent Truth instead
of Falsity, and then interpret "&" and "v" to express the truth
functions disjunction and conjunction, respectively, instead of
conjunction and disjunction, respectively, and you will still have a
propositional logic, albeit eccentrically interpreted. To characterize
this simple fact as "True is now False" is, at best, misleading and, at
worst, sophistry. (05)
> The industrial problem is not semantics "as a game played this way",
> but the behaviour of a system (=people + procedures + goals +
> materiel, including computers), and the difficulty is more in
> demonstrating that the logic is applicable to situation of interest. I
> expect you are familiar with supermarket mathematics where 1+1 = 1 (or
> buy one get one free), and possible the economic 'logic' that goes
> with such promotions. (06)
Yes, we can, for marketing purposes, pun the addition symbol. Indeed,
that people recognize it as a pun is why it is an effective
attention-getter. That does not mean that 1 plus 1 might be other than
2. I don't see any deep point here at all. (07)
> ... The problem is that the word "semantics" seems to cover a family
> of related meanings. (08)
As has been noted here a number of times. Kathy Laskey and Pat Hayes
(among others) have both written insightful posts on the topic
recently. (09)
> Perhaps we ought to qualify what we want to mean by different flavours
> of semantics. I will offer the following outrageous suggestion (enough
> to keep Pat raging for a couple of days :-))
>
> Semantics - the behaviour of a set of terms and operations in a
> system: (010)
Since terms don't behave, that's surely not a good choice. Moreover,
there is nothing necessarily semantic in the bare notion of an
operation, even an operation on terms. (011)
> Real world semantics: how people and machines behave in response to
> input; (012)
Well, there is a notion of "operational" semantics in the wind that
sounse something like this, but I'd hesitate to use "real world
semantics" for it, since that phrase, or something like it, is often
meant to indicate the *real* or *actual* meanings of terms in the world,
as opposed to their meanings in some model. (013)
> Logicians' semantics: A set of terms T some superset of {TRUE, FALSE}
> and a set of operations O:T -> T (014)
Surely that won't work, as your operations there only map terms to
terms; i.e., those are *syntactic* operations. You at least want the
terms to be mapped into your superset of {TRUE,FALSE}. That said, I
think it would be difficult to provide so succinct a definition. My
semi-formal account above seems to me a reasonable sketch of the general
notion in logic, but it too could be improved or altered depending on
one's purposes. (015)
> More seriously, I would find it useful if the professional
> philosophers, logicians and computer scientists could agree a taxonomy
> of the uses of "semantics", and then establish them by frequent use on
> the forum. (016)
For the logician's sense, the best thing for one to do is to spend a
little time learning the basics of first-order logic and model theory --
which in my view is a sine qua non for anyone who claims to be working
in formal ontology anyway. As for a simple general gloss, semantics is
the study of (formal or natural) linguistic meaning -- where, typically,
meaning is some sort of word/world connection. A semantic *theory* for
a class of languages is a systematic account of the meanings that the
expressions of a natural or formal language have, or can be assigned.
And the semantics of a given term just *is* its meaning (according to
some semantic theory, if the context warrants). (017)
Chris Menzel (018)
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