Re: The logic of atheism

Paul Holbach wrote:

> Paul Holbach wrote:
>
>> Or to put in the concise words of Patrick Grim:
>>
>> "Were there an omniscient being, what that being would know
>> would constitute a set of all truths. But there can be no set
>> of all truths and so can be no omniscient being."
>>
>> [Grim P. 2003. Logic and limits of knowledge and truth. /The
>> impossibility of God/. 381-407. Martin M, Monnier R, eds. New
>> York: Prometheus. (395+)]
>
> Also see:
>
> http://www.sunysb.edu/philosophy/faculty/pgrim/exchange.html

I think this passage from that exchange captures the essence of my
dialog with Aatu quite well:

> Your proposal, phrased with respect to a particular Cantorian
> argument, is to deny the existence of a diagonal property
> stipulated in the course of that argument.
>
> If such a strategy is to apply to offending Cantorian arguments
> systematically and in general, however, rather than being
> applied merely ad hoc on the whim of the wielder, we need
> a principle that will tell us which diagonal properties,
> propositions, truths, or conditions to reject. You don't, quite
> clearly, want to reject all diagonals.
>
> Here there is a nested pair of problems.
>
> The first is simply that we've been offered no satisfactory
> principle of such a sort, and I doubt very much that anyone will
> in fact be able to produce one.
>
> The second problem is deeper. There are of course deep
> affinities between the Cantorian results at issue here and
> certain aspects of the classical paradoxes. If those affinities
> hold, I think, we can bet that any principle that was proposed
> as an exhaustive condition of what diagonals to accept and what
> to reject would face a crucial and devastating test case
> constructed in its own terminology. If so, it's not merely that
> a comprehensive principle as to which diagonals to accept and
> which to reject hasn't in fact been offered. If important
> parallels hold, it's rather that no such principle could be
> offered.
>
> If we are in fact given no principle to guide a strategy of
> denying the diagonal, I think, such a strategy can only be
> applied in a manner bound to be rejected as unprincipled and ad
> hoc. If I'm right that no adequate principle can be given, of
> course, any such strategy will moreover be essentially and
> inescapably ad hoc.

That is, we see a repeated pattern in logic, from the Liar Paradox
through Cantor, Godel, and Turing, and all the way to my
own .sig, a pattern of the inherent nastiness associated with
self-referential negations--and that those very same negations are
inevitable when one bumps into the boundaries of a system. This
shouldn't be in the least bit surprising, as it's (seemingly) no
different from the nastiness associated with simple infinities.

In plain English, speaking of ``the ultimate /anything/'' is as
meaningless as speaking of ``the largest number.''

I think the difficulty in grasping the implications of
diagonalization, self-referential contradictions, and the like
is the same fundamental difficulty as arises when trying
to contemplate the infinite. I know /I/ still struggle with
conceiving ``infinity'' (even Cantor's Aleph-Null, let alone any
of the higher orders) as anything other than ``this really,
really, big, humongous, gigantic number.''

And, of course, that's just the problem. Linguistically,
``infinity'' isn't a /noun/--or, at least, it shouldn't be. More
properly, it's a /verb,/ describing the process which satisfies
the results of the operation that results in the infinite.

Humans do a fair job at arithmetic without the aid of tools...but
most of us fail miserably at comparable tasks when it comes to
calculus. Yet it's calculus that we need to answer questions about
omni-properties; arithmetic just won't do the trick. (I mean all
that metaphorically, of course.)

With a bit of practice, however, one can learn to spot the kinds
of situations where self-referential contradictions arise. Once
you get good at it, the Truth that there is no /ne plus ultra/
becomes strikingly obvious. And what is the modern theological God
but the /creme de la creme/ of the /ne plus ultra?/

It's in these kinds of matters that I suspect the Zen Buddhists
may have a leg up on those raised in Western cultures. A koan
would seem to be well suited to the kind of understanding
necessary here. I suspect that a sufficiently Enlightened monk
would appreciate a statement such as, ``There are no genuinely
universal propositions.''

Mr. Grim also addresses the point that you, Paul, and I have been
trying to make:

> A simple example helps. We can convince ourselves, I think, that
> the concept of round squares is an incoherent one. It is
> tempting to conclude on that basis that round squares don't
> exist. But this last position brings with it the well-known
> philosophical difficulties of negative existentials. Perhaps the
> apparent difficulties there are merely apparent. But at any rate
> we can avoid them by stopping with our first claim: that the
> concept of round squares is an incoherent one.
>
> Perhaps that is how we should phrase our positive conclusion
> here as well: the concept of omniscience is an incoherent
> concept, as is the notion of a totality of truth or of a
> proposition about all propositions. Having convinced ourselves
> that the notion of a proposition about all propositions is an
> incoherent one, we are tempted to conclude that no propositions
> are genuinely universal. The phrasing of this last position
> brings with it all the philosophical difficulties you point
> out. But perhaps we could avoid them, while still having a
> positive conclusion, by stopping with our first claim: that the
> concepts at issue are incoherent ones.
>
> My fallback and first love remains the pure form of the argument
> above, offered without positive conclusion. If a positive
> conclusion is demanded, this suggestion is perhaps worth a
> try. It must be added immediately, however, that we'll be able
> to take this suggestion seriously only if we're willing to give
> up a few things: at least (1) a Russellian treatment of definite
> descriptions and (2) the idea that simple predications somehow
> involve hidden quantifications. But it is perhaps time we gave
> up those anyway.

Or, in other words, ``Hey, it ain't /my/ problem that you worship
a married bachelor!''

Another choice quote:

> Omniscience is standardly glossed as being 'all-knowing' or
> 'knowing everything', precisely as its 'omni' would suggest. If
> there is no 'everything' of the relevant type to know, there can
> be no omniscience as standardly glossed.
>
> You suggest that we understand omniscience as 'a maximal
> degree of knowledge' or as 'maximal perfection with respect to
> knowledge' (above, p. 00041). (Isn't 'maximal perfection' a bit
> redundant?) But should it turn out that for any degree of
> knowledge there must be a greater, it would appear that there
> can be no 'maximal perfection' with respect to knowledge--and
> thus no omniscience as you suggest we understand it.

Cheers,

b&

--
EAC Memographer
BAAWA Knight of Blasphemy
``All but God can prove this sentence true.''

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