[FOM] FTGI1:Classical Propositional Calculus

[Sean C Stidd]

Some general questions about this:

> 2. Mathematics appears to be crippled if we don't allow all five of 
> these. The removal of some of these is more crippling than others.

Is the need for these extra connectives pragmatic or of deeper
foundational significance (or both)? We learn about all the propositional
connectives being 'reducible' to neither-nor on our grandfather's knee,
Sheffer's sole logical constant. Not only these five but everything
mentioned in (1) except the equivalence-relation can be derived from the
Sheffer stroke. And even if you don't like the SS, you appear to be able
to remove three of the five (as long as one doesn't eliminate negation,
or all but negation and if and only if). Is there any foundational
significance to the 'crippling' here mentioned?

> 6. If we remove "if then", then it appears that this is so crippling  
> as to make mathematics undoable in the practical sense. Of course, 
> we 
> can replace "if A then B" with "notA or B", or with "not(A and 
> notB)". But this is too horrible to contemplate using. The following 
> question naturally arises.
> 
> 7. Is there any formal reason for why removing if then is 
> particularly crippling? One immediate "answer" is that any 
> equivalent 
> of if then in terms of not, or, and, iff, must involve the use of 
> two 
> distinct connectives from these remaining four. But can this answer 
> be improved?

At first glance it would seem that there cannot be any formal reason,
since we could go through a proof and replace every instance of "if A
then B" with "(not A) or B" without altering the proof in any other way
at all. At least this seems to be the case in classical natural deduction
where you can infer any proposition from a contradiction. The
introduction and elimination rules translate straightforwardly in this
case. (And semantically speaking they are equivalent in meaning, so
outside proof theory the case would seem to be weaker.)

Nonetheless, many mathematicians, scientists, and philosophers have
recoiled at identifying logic's 'implies' or 'if then', which is
equivalent to (not A) or B, with the meanings of implication-statements
in their theories. There is a long philosophical literature on this
issue, and Russell discusses it already in his Introduction to
Mathematical Philosophy, pp. 152-4 (and I think even before this). His
position is that while the validity of an implication-inference solely
relies on the truth of p and the truth of (not p) or q, such inferences
are only useful when (not p) or q is known on grounds independent of a
knowledge of p's falsehood or a knowledge of q's truth, since in either
of those cases the implication is superfluous for inference. Russell's
idea seems to be then that there is 'more' in 'implies' in ordinary usage
not for any logical reason, but because in the cases where we rely on
'implies' most crucially we seem to have knowledge of some kind of more
primitive connection between the two propositions that entails the
implication without entailing the falsehood of p or the truth of q.

There is a long literature in philosophy trying to spell out the nature
of this 'stricter' implication-like connection in logical terms,
beginning I think with C. I. Lewis; here quantificational and modal
apparatus is often employed.

> We now come to the question as to just what not, and, or, if then, 
> iff, mean.

A lot of the issues in the middle part of this post point to a classic
problem in the philosophy of logic, mentioned e.g. by Sheffer in his
review (in Mind) of Principia Mathematica. This is 'the logocentric
predicament': there is, trivially, no explanation of logical concepts
which does not in some sense depend on logic, because all explanations
presuppose logic. While this provides a nice stumper to use against
social constructionists - if someone says 'logic is just a matter of
institutionalized power-relations', you can reply by saying 'well, if you
use institutionalized power-relations to explain logic, you are committed
to the use of logic in your explanation; so what you've shown at best is
that logic is reducible to institutionalized power-relations plus logic,
a much less interesting claim, you must admit' - it's a very difficult
thing for scientifically minded people, who are explainers by nature, to
accept.

Perhaps the best we can do in the case of 'explaining' the most
fundamental logical relations is exhibit the various logical systems in
interesting ways that allow us to grasp 'at a glance' various structural
features of them which we might not be aware of in our ordinary use of
logical notions. The truth-tables certainly are one tool for doing this.