[FOM] Weak logic axioms

[Arnon Avron]

This has very little to do with FOM. Still, I'll answer your question
in short: by adding either (3) or (4) to the relevance logic R you get
what is known as the semi-relevant logic RM (which has Sugihara's
matrix as a characteritic matrix). This system does not have the
variable-sharing property, and so is not considered to be a relavance
logic. Usually it is axiomatized by adding
to R the axiom A->BvB->A (It is easy to see that each of (3) and (4)
are equivalent to this axiom).

Arnon Avron

On Sat, Feb 25, 2012 at 03:34:41PM -0500, Michael Lee Finney wrote:
> 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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