[FOM] Weak logic axioms

[Michael Lee Finney]

This is conceptually no different, and again is dependent on
primeness.

For any T(p v q), if the logic is prime then either T(p) or T(q) and
so T(p) v T(q). If the logic is not prime then you can have T(p v q)
without either T(p) or T(q).

Most logics are prime. Certainly, classical logic is prime. If
Aristotle's argument is going to work you need to specify a logic
which is not prime.

A positive, weak logic with conjunction and disjunction, but neither
distribution nor negation is probably not prime. There could always be
some other axioms that ensure primeness, but not too likely.

The problem of primeness is like that of entailment. If you don't have
one or more axioms for disjunction that constrain the case when p and
/ or are not distinguished then you may not have primeness. In
general, if you have an operator you want to say what happens for
distinguished values. That is what most axioms do. However, you leave
the system underspecified unless you also find a way to say what
happens for undistinguished values. That is frequently dependent on
the properties of your logic.

In particular, p v ~p essentially says that either a value is
distinguished or its negation is distinguished - in the presence of
primeness.

Neither (p v ~p) nor (p -> q) v (q -> p) ensure primeness. Their
effects are dependent on primeness. The arguments that you are trying
to make all assume the lack of primeness.

> I do see now that the argument for the non-distribution of truth
> over disjunction from 'T(p & q -> r) -> T[(p -> r) v (q -> r)]' to 
> the hard to understand 'T(p & q ->> r) -> [T(p -> r) v T(q ->
> r)]'.will not work. For 'T[(p -> r) v (q -> r)]' is no less hard to understand.
> But we still have Aristotle's argument from fatalism that 
> 'T(pv-p)' does not imply 'TpvT-p'.
> Alex Blum