ONT Re: Model Theory

[Jon Awbrey <jawbrey@oakland.edu>]

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

JC: My meagre mail space is getting filled with redundant information.

Ah, but some would say that redundancy is the very essence of information.

JA: 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?

JC: By predicative I mean the standard usage that all of the defined terms
    can be isolated on the right hand side, and are in principle eliminable.

So you order things like so -- Definiens (Defining) = Definiendum (Defined) --?

I must confess to being of two minds about this issue.
One part of my brain, washed in the cool, clear, fresh
water ponds that are filled by that fountation of Lethe
that we call Mathematics, is oblivious to the idea that
there is any other kind of definition, as all recursions
are supposed to be eliminable by Beth's theorem, that is,
if I remember or ever got it right, but still, I do hold
out hope that I might someday discover the fountation of
"synthetic a priori" (SAP) knowledge -- so there you have
my confession of heresy.  I would not even think of asking
another to echo that dangerous sentiment, but let me ask:
To what do you oppose this distinction of "predicative"?
Is it only "impredicative" or something more synthetic?

JC: By content, I mean, minimally, the capture of a collection of concrete
    (i.e. non-abstract) instances that are picked out by the concept whose
    content is under study.

So then, you more contented with a definition of this sort,
that proposes to define "content" in terms of itself?

JC: What I see in your recent posts is a haze of syntactic
    redundancy with no attention to the content.

Haze is often in the eye of the beholder.

JC: A set of WFFs (an abstraction of abstractions) is inconsistent if and only if
    they are necessarily false, i.e., there is no way to assign contents (give an
    interpretation) such the every member of the set comes out true.  To make this
    non-abstract, a collection of conditions must be defined that are real possibilities,
    and they must be capable of being true together, and they can serve as an interpretation
    of the WFFs.  If there is nothing that fits the bill, then either all of the WFFs have no
    such interpretation, or at least some do, and the conjunction of these has no interpretation.

Can you give me a source, other than yourself, for this definition of inconsistency?

JC: Corollaries:  Nothing is inconsistent unless it has no content in this sense.
    If something has no content in this sense, then either some non-redundant
    component is meaningless, or else it is inconsistent.

JC: A few other points: q is true in A entails A |= q, and vice versa, and A satisfies q.
    With Tarski's definition of truth, A satisfies q entails that A |= q, and thus q is true in A.

JC: A is inconsistent entails that A satisfies nothing and that nothing
    satisfies A.  None the less, if A is inconsistent, for all q, A |-- q.
    Hence, another good definition of inconsistent would be A is inconsistent
    if and only if, for all q, A |-- q, but it is not the case that A |= q.
    I call such cases semantically defective.  The liar paradoxes are primary
    cases, but there are lots of others.  This is why I say that inconsistency
    is fundamentally a semantic notion (though this is obscured in many approaches
    to logic).

JC: Furthermore, the various paradoxes arise because of a
    predilection to assume that all WFFs have interpretations,
    or content.  This is not always true, as the above definition
    makes immediately apparent.  We can discover that something
    we thought to be meaningful is not meaningful.

I agree that it is important to note that not all signs denote.
However, my analysis of the Liar suggests to me that it is simply
a matter of assuming a false statement, from which anything follows.

JC: I've argued before that the very basis for taking Goedel's theory seriously requires
    a presumption that logical statements are meaningful (have a content).  One can get
    results that don't rely on this presumption, e.g., for Post Canonical Systems,
    and these can be applied to show the limits of logical completeness, but this
    only matters if logical completeness and incompleteness matter.

JC: There's a lot more, concerning why logic matters, related to
    the way Prop Calc captures truth completely and decidably,
    but I'll stop here.

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