Wednesday, 21 August 2013

Which One of These Logical Theses Does Not Hold for Relevant Logics?

Which One of These Logical Theses Does Not Hold for Relevant Logics?

I write in Polish notation and have included fully infixed notation here
also which indicates parsing order.
For every relevant logic simplification fails:
Simplifcation: CpCqp or [p$\rightarrow$(q$\rightarrow$p)]
I have a proof that from Syllogism, Commutation, Conjunction-Out Left, and
Conjunction-in I can deduce CpCqp, given detachment also.
Syllogism: CCpqCCqrCpr or
{(p$\rightarrow$q)$\rightarrow$[(q$\rightarrow$r)
$\rightarrow$(p$\rightarrow$r)]}
Commutation: CCpCqrCqCpr or
{[p$\rightarrow$(q$\rightarrow$r)]$\rightarrow$[q$\rightarrow$(p$\rightarrow$r)]}
Conjunction-Out Left: CKpqp or [(p$\land$q)$\rightarrow$p]
Conjunction-In: CpCqKpq or {p$\rightarrow$[q$\rightarrow$(p$\land$q)]}
I highly doubt relevant logics don't have CCpqCCqrCpr or CKpqp. Does
CpCqKpq not hold for some relevant logics, and CCpCqrCqCpr not hold for
others? Or do they both fail for all relevant logics? Or does only one of
them not hold? If so, which one?

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