[FOM] Weak logic axioms

Michael Lee Finney michael.finney at metachaos.net
Sat Feb 25 15:34:41 EST 2012

I was mulling over the axioms in some of the depth relevance logics,
and thought of two axioms which I have never seen mentioned or
discussed anywhere.

First, consider the axioms

   (1)   (p -> q) & (p -> r) -> (p -> q & r)
   (2)   (p -> r) & (q -> r) -> (p v q -> r)

these (normally) allow us to prove

   (A)   (p -> q) & (p -> r) <-> (p -> q & r)
   (B)   (p -> r) & (q -> r) <-> (p v q -> r)

and are usually seen as making statements about conjunction and
disjunction. They are present in most weak logics.

But then, I thought about

   (3)   (p -> q v r) -> (p -> q) v (p -> r)

and I thought "that isn't valid". But then I looked at it more and
found that it does appear to be valid. Then I thought that it might
only be valid if entailement did not allow contraposition because 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). They
also satisfy the BN4 semantics.

However, these allow you to prove

   (C)   (p -> q v r) <-> (p -> q) v (p -> r)
   (D)   (p & q -> r) <-> (p -> r) v (q -> r)

which, when taken with (A) and (B)

   (A)   (p -> q & r) <-> (p -> q) & (p -> r)
   (B)   (p v q -> r) <-> (p -> r) & (q -> r)
   (C)   (p -> q v r) <-> (p -> q) v (p -> r)
   (D)   (p & q -> r) <-> (p -> r) v (q -> r)

looks very much like left and right distruction principles of
entailment over conjunction and disjunction.

I am not sure what practical use (3) and (4) have, but they appear to
be valid and I cannot see any path to proving them using anything in
any of the major relevance logics. Nor do I see any obvious connection
to less desirable properties such as distribution, permutation and
contraction. But, even if they do not have a direct practical use, if
they establish fundamental properties of the entailment operator
perhaps they should be present.

Does anyone have any insights into these? Should these be added as
axioms to the relevance logics? How would these affect the various
relevance logics? Can they be proven in any of the relevance logics
(and if so, how)?

Michael Lee Finney
michael.finney at metachaos.net

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