[FOM] Weak logic axioms

[Alex Blum]

	Some time ago under the present subject heading Michael Lee Finney wrote:
"...you could then prove
 (4)   (p & q -> r) -> (p -> r) v (q -> r)
which I thought that surely was invalid. But again, I appear to have been incorrect. I asked if these are classical theorems, and it turns out that they are amazingly easy to prove in classical logic (just turn entailment into disjunction and you are pretty much there)."
	 I would like to further comment on (4): 
There seems to be however no understandable way of reading (4) which makes it true. Bur if the disjunctions in the consequence are transposed, thus:
 '(-r->-p) v(-r->-q)' then (4) is understandable. The consequence may mistakenly be read as '(pvq)->r'. One may question (4) on the ground that that the antecedent does not imply either disjunct. That would be a mistake as well for we have the same with 
'p->(qv-q)'. 
	(4) is interesting for it indicates that Truth does not distribute over disjunction. For if 'T[p & q) -> r) -> (p -> r) v (q -> r)]' then 'T(p & q -> r) -> T[(p -> r) v (q -> r)]'. But the following is not true:
(4*)'T(p & q -> r) -> [T(p -> r) v T(q -> r)]'.

Alex Blum