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Re: Logical formalisms



 >At 09:01 PM 7/28/99 , Vern Crisler wrote:
 >
 >I was wondering when our great systematizer was going
 >to enter the fray.

Sorry, it's just little ol' me.

 >It must first be established that it is even possible for
 >there to be "one counterexample" to the law of contradiction.

In classical logic, this possibility cannot be established.
But that's the whole point; in a non-classical logic, this
provision is stipulated in the definition of the formal
language, and the rest of the analogs to a classical formal
language are revised accordingly.

 >I'm not sure that a "logic" that can be interpreted both
 >as referring to actual chaos or to apparent paradox really
 >would represent an advance. :-)

Currently, the natural-language descriptions of actuality
provided in the Scriptures are taken by some to represent
apparent conceptual paradoxes; by others, they are taken
to denote actual chaos and are therefore rejected.  [That's
why we consider apologetics worthwhile.]  So, a formal
language that retains these literary and denotative nuances
wouldn't necessarily constitute a conceptual advance, but
might count as a useful alternative method of formalizing
the systematic theology that we derive from the Scriptures.

 >But these "apparent" paradoxes are only meaningful
 >against a background of traditional logical laws.

No, these apparent paradoxes are only meaningful against
the backdrop of God's absolute, comprehensive, and wholly
consistent knowledge of the creation.  And a relevant
subset of that knowledge, available to us in a quasi-system
embodied in scripture and abstracted into systematic
theology, may be expressed in a variety of useful ways
including classical formalism.

We can rely on "traditional logical" formulas to express
our systematic theology, but then isolate and bracket out
those areas where classical entailment is thwarted by our
ignorance of some indispensible premises.  Or, we can
rely on less traditional logical formulas that handle the
problem of our finitude and its concomitant epistemic
mystery in other ways.  Take your pick.

 >Moreover, restricting or redefining logic to allow for
 >apparent chaos might be taken by critics to be an
 >immunizing strategy, and would fair no better, I suppose,
 >than simply admitting we lack a complete knowledge of the
 >the subject domain.  The latter, though it might be
 >criticized as an immunizing strategy, too, -- at least
 >leaves rationality intact.

The former leaves rationality intact, too.  The difference
between the options I've discussed is not the retention or
rejection of rationality.  The difference is [1] the
expression of rationality in terms that require silence or
underarticulated distinctions at mysterious or apparently
paradoxical junctures, versus [2] the expression of
rationality in terms that incorporate and operate on
markers that stand for these silences or underarticulated
distinctions.  And while you rightly note that [2] is just
as open to charges of "immunization" or "band-aid theology"
as [1] is, [2] at least has the advantage of more easily
permitting algorithmic manipulation of the systematic
theology.

 >Well, I don't know how one can do that without actually
 >doing "violence to rationality."  :-)

Then perhaps you should set aside the Aristotle and
Maritain for a while, and look into some logics that seek
to express (rather than reject) rationality by way of
a systematic dismissal of bivalence and existential import.

:)

 >>And Christian Theists may refrain from truthvaluating some
 >>propositions, on the basis that we lack both scriptural
 >>warrant and the possibility of otherwise knowing enough to
 >>valuate -- trusting that God knows enough to rationalize
 >>the recondite.
 >
 >This, of course, does not in itself require that we adopt
 >a "different" or "deviant" logic.

Of course not.  I'm not suggesting that free logics (the sort
I have in mind) are *required* in general.  I'm simply trying
to point out their usefulness *if we wish to paraphrase* our
systematic theology into a readily manipulable formal system
that admits of less potential for errors of biblically
unwarranted rationalism than classical formalism does.

 >>My words above should *not* be taken as a blanket
 >>endorsement of non-classical logics.
 >
 >Assuming that a "non-classical logic" is even a form of logic.

Assuming this?  No.  There's a substantial literature on the
question of whether a free logic, for example, is consistent,
complete, and otherwise adequate for the usual tasks to which
we put the propositional calculus.

 >>But isn't Van Til's scriptural and Reformational
 >>point that the *tools* of rationalism, however necessary
 >>in appropriate contexts for suitable purposes, do not
 >>dictate the limits of what we seek to express?
 >
 >I don't think Van Til can be brought in as a witness against
 >classical rationality (as if there could be any other kind). :-)

I'm not putting Van Til to such a use.

 >But he does oppose any kind of "positivist" reading of
 >classical rationality.

Yes, among other abuses of the norms of reason.

 >>Appropriate non-classical formalisms should be considered
 >>as fitting vehicles for communicating *the same theology*
 >>in a more efficient and less confusing way.
 >
 >Well, I'd like to see an example that doesn't "confuse"
 >matters any more than they already might be.  If you think
 >appeals to mystery are "misunderstood," on the classical
 >basis, what makes you think that appeals to extended or
 >deviant "logics" will somehow make the situation better?

I think this for the same reason that I think a steady and
level discussion of transcendental argumentation is more
efficient and less confusing [and therefore more productive]
than a rhetorically dense advocacy session on the same topic
might be.  My guess is that those who have experienced both
the "yes, we can!" approach and the "how exactly can we?"
approach find benefit in each, but find the latter more
directly and efficiently conducive to understanding and
mastery.  Ditto for a formalism that captures our systematic
theology, gaps and all, in a readily manipulated format.

But hey, not everyone likes formally efficient symbols and
algorithms.  I'm all for dense semiotic systems such as
literature, law, and poetry, too.  In fact, they have pride
of place among my own preferences.  After all, God chose to
communicate in the latter mode, not the former.

Cheers,

David Byron
david.byron@aya.yale.edu


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