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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 $\big(p\rightarrow(q\rightarrow p)\big)$

I have a proof that from Syllogism, Commutation, Conjunction-Out Left, and Conjunction-in I can deduce $CpCqp$, given detachment also. You may see the August 19th answer here for details.

Syllogism: $CCpqCCqrCpr$ or $\Big((p\rightarrow q)\rightarrow\big((q\rightarrow r)\rightarrow(p\rightarrow r)\big)\Big)$

Commutation: $CCpCqrCqCpr$ or $\Big(\big(p\rightarrow(q\rightarrow r)\big)\rightarrow\big(q\rightarrow(p\rightarrow r)\big)\Big)$

Conjunction-Out Left: $CKpqp$ or $\big((p\land q)\rightarrow p\big)$

Conjunction-In: $CpCqKpq$ or $\Big(p\rightarrow\big(q\rightarrow(p\land q)\big)\Big)$

I highly doubt relevant logics don't have $CCpqCCqrCpr$ or $CKpqp$ when the have a conjunction connective. 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?

Mohsen Shahriari
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Doug Spoonwood
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1 Answers1

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In (most) relevant logic Syllogism and Communitation are Theorems, but for the conjunctions theorems it all gets a bit complicated, in most $CKpqp$ and $CKpqq$ are theorems but $CpCqKpq$ is not.

Instead of $CpCqKpq$, $CKCpqCprCpKqr$ is a theorem (this is also a theorem in classical logic) but you could argue that the last is not an conjunction-in theorem at all.

To get a real conjunction in rule, most relevance logics add an inference rule $\dfrac{\vdash p \quad \vdash q} {\vdash Kpq}$ but this is an inference rule not a theorem.

Mohsen Shahriari
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Willemien
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  • So, if we think about the rules that The Detachment Theorem hints at for "CKCpqCprCpKqr"... specifically KCpqCpr => CpKqr, {KCpqCpr, p} => Kqr, the first rule involves an exchange of a conjunction for a conditional. So, I'd call the first a conjunction-conditional exchange rule. The second has a conjunction in the left part of the rule suggested, and a conjunction on the right part of the rule suggested. So, I'd call that a conjunction-conjunction exchange rule. Neither of which produces a conjunction when you didn't have one before. The production of a connective you didn't have before.. – Doug Spoonwood Sep 10 '13 at 21:56
  • seems characteristic of an "in" or "introduction" rule. So, I definitely would argue that CKCpqCprCpKpr is not a conjunction-in theorem at all. I do see it as plausible to think of it as a conjunction-conjunction exchange theorem due to the second rule I've referred to above, but as the case with many other conditionals, the theorems along with The Detachment Theorem say more than that. – Doug Spoonwood Sep 10 '13 at 21:59