[hist-analytic] The status and relevance of "standard logic"

Danny Frederick danny.frederick at btinternet.com
Tue Nov 3 04:30:25 EST 2009

Hi Roger,

<<There are no "hidden axioms" involved
A type theory is not necessary>>

A distinction between language and metalanguage must be made in order to
avoid semantic paradoxes. This is a theory of linguistic types. Further
theories of linguistic types are also assumed. For example, the following is
a logical truth (in PL + identity):

Ex(x = a).

This amounts to the claim that every name designates. It is not possible to
say (truly) in first-order logic that Santa Claus does not exist. We have to
go to the metalanguage and say that 'Santa Claus' does not designate
anything, or 'Santa Claus' is not a name. Do you think that the claims that
Santa Claus is not a name, and that 'Nothing is identical to Santa Claus' is
ill-formed, are as obvious as parts of elementary mathematics? They seem to
me to be obviously false.

And can a theory of linguistic types be separated from a theory of logical
types? I don't think so.

Thus, there are a many dubious assumptions needed to make the system work.
And they are hidden because they are not explicitly stated as axioms of the
system: they are just assumed by the system builder.

You ask: how can it be that the question of how doubtful something appears
does not bear upon its truth?

Just consider some examples. Basic Law V was not doubtful to anyone before
Russell produced his paradox. All manner of moral and religious truths are
indubitable to fundamentalists. That simultaneity is relative to a
co-ordinate system was highly doubtful when Einstein proposed it (I think it
still is highly doubtful), yet it is nowadays generally accepted as true by
physicists and others.

<<nothing is beyond doubt and we have no absolute warrants of truth>>

This sounds as if it concedes my point. In fact, I think it does. But you
will disagree because you want to order doubts according to HOW doubtful
they are and because you want to link this (purely subjective) order to an
(objective) order of closeness to truth (or perhaps likelihood of truth).
But this cannot be done. I agree we can often say that one thing seems to be
more doubtful to us than another. But this is purely subjective: it has no
bearing on the question of truth (consider the previous list of examples and
others like them).



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