SUO: Rules of Inference, Rules of Entailment
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Jay Halcomb wrote:
>
> When thinking about notions of inference and entailment (and when reading
> various authors), it's also important to keep in mind considerations about:
>
> 1) Inference (or entailment) from infinite sets
> of assertions vs. that from finite sets.
I will be a little bit slow on the uptake here,
as the last time that I used the word "entailment"
it was in the context of so-called "relevance logics",
in which I am no longer any variety of a true-believer.
As it happens, the word that I used to use for this was "consequence", as in:
| We say that [phi] is a 'consequence' of [Sigma], in symbols [Sigma] |= [phi],
| iff every model of [Sigma] is a model of [phi].
|
| C.C. Chang & H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973,
| page 11.
Where was I? Oh yes, infinity ...
I was going to break in here and ask you whether you really think that
we "fallible and mortal finite information creatures" (FAMFIC's) will
really have much to do with "infinite sets of assertions" (ISOA's) --
but, on second thought, I can see that we are well on our way ...
> 2) Inference (or entailment) from sets of assertions which have free
> variables vs. that from sets of assertions without free variables.
>
> Logical systems differ in these regards.
> Results about entailment and provability
> will vary with them, in general.
>
> Jay
So I guess that all that remains of my initial reserve about
the way that we have been cast into viewing this distinction
is that I believe I can see a subtle bias at work here if we
treat model theory merely as a study of the relation of such
what-you-may-call either "consequences" or "entailments" to
the various sets of sentences that condition or entail them.
Now, casting model theory into that form or into that mold
does create a nice, all too nice symmetry with proof theory,
but it has the rather more insinuous effect of making it seem
like the only real function of model theory is to imitate the
creations of proof theory, in lieu of what is more likely true,
the other way round, the role of proof being to parallel truth.
Just My Take On The Matter,
Jon Awbrey
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