ONT Re: Model Theory

[Jon Awbrey <jawbrey@oakland.edu>]

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| O swear not by the moon, th'inconstant moon
| that monthly changes in her circled orb,
| Lest that thy love prove likewise variable.
|
| 'Romeo & Juliet', 2.1.151-153.

John Collier wrote (JC):
Jon Awbrey wrote (JA):

JA, quoting Chang & Keisler:

    | We shall say that @S@ is 'inconsistent'
    | iff we have @S@ |- q for all sentences q.
    | Otherwise, we say that @S@ is 'consistent'.

JC: I am not going to answer this one, Jon.
    I have other work to do.

JA: I am guessing that you did not intend your remark to serve
    as an example of a self-referent performative contradiction,
    so I will just thank you for participating in this experiment
    as long as you did.  To sum it up from where I stand and look
    at it, we have established that usage differs in regard to the
    concept of consistency, even among those thinkers who have been
    blessed to flourish in the light of Twentieth Century Mannerism,
    that some people care about usage and its differentiations, that
    is, so long as they have nothing better to do, and that is enough
    for my purposes, since it releases our pro-academic inhibitions
    against thinking for ourselves, and, well, allows us to begin
    thinking for ourselves, which means, most likely, that we may
    find no point of convergence within our finite lifetimes,
    however much hope we may have that there might be one,
    somewhere, somewhen, somehow.  So goes my Canon in D.

JC: Jon,  The definitions make no difference whatsoever to logic, as long as
    they are introduced predicatively.  I see no reason to spend time with this.
    They can however be misleading. Consider, if you will, the reference of the
    term "inconstant".

JA: The escape viscosity of logic from definition makes me swoon,
    So hie me away to look up "impredicate" in my well-worn rune,
    But why do I get a suggesture that I have just been e-mooned?
    And here I imagined that you would appreciate the definition
    Of formless forms of constituent inconsistency that elide to
    Draw any distinction at all in a space of syntactic instants,
    Posing a form of constancy that is constant all too constant.

JC: Jon, I can't understand what you said here, but it is
    well known that predicative definitions add no content.

What is to be gained here?  I started by looking at a family of very simple logical systems,
variously known as Alpha, Propositional Calculus, Sentential Logic, Zeroth Order Logic (ZOL).
Matters of conceptual clarity, practical efficiency, or plain and fancy taste aside for now,
they all amount to so many ways of talking about the same formal objects, namely, lattices
of propositions or boolean functions.  I am scoping these logical e-coli in order to get
a sense of the dimensions of syntax and semantics, proof and models, in a simple case.
We all know that the world gets hairier beyond this frame, but if we cannot achieve
clarity here then what hope do we have in a larger domain?  I have my preferences
that differ from C&K in various places, but I brought them in as pretty standard,
because you seem to be under the impression that I have never cracked open
a 20th Century logic book.  The definition of inconsistency here is pretty
standard stuff.  Maybe you can tell me what you mean by "predicative" and
by "content" as I sense that it will be futile to go to my shelves again.
While I'm asking, why would any of it be relevant to this level of logic?
And what does the reference of the term "inconsistent" mean to you?

Jon Awbrey

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