# Are there any known deficits of "relevant logic"?

The principle of explosion is the law of classical logic and similar systems of logic, according to which any statement can be proven from a contradiction. Some early formal systems like Frege's Begriffsschrift contained hidden contradictions in the basic axioms, but the fundamental problem doesn't vanish even if the basic axioms are free from contradictions (=consistent). The problem is that it's all too easy for a human to make a mistake leading to a contradiction.

One strategy how this problem is addressed in everyday live is to be suspicious with respect to chains of reasoning which wander too far off from the topic at hand. This strategy could be called the relevance principle.

This problem also motivated the development of different systems of paraconsistent logic. It seems to me that relevance logic is one of these systems, even so I admit that its main goal is to avoid the paradoxes of material and strict implication. From the texts I read about relevance logic, I conclude that it's quite successful at formalizing the relevance principle (via the variable sharing principle). However, what I miss so far are investigations whether there are important and relevant theorems that can't be proved if the relevance principle is adopted.

What makes me uncomfortable about the relevance principle is that complex numbers can be used to prove some statements about natural numbers which can be very difficult to prove without them. So I wonder whether there exist good justifications for the relevance principle. However, this is also a more direct question about the existing systems of relevance logic and their propositional variable sharing principle. Their proof theory seems to be designed with the goal that implications violating the variable sharing principle can't be proved, without compromising the ability to prove implications satisfying the variable sharing principle.

However, I didn't find any indications whether these goal were achieved. Will I find such indications (or even proofs) if I read more thorough expositions of relevance logic, or is there something wrong with my expectations that such indications (or proofs) should be given?

• Relevance logic (RL) tries to avoid the paradoxes of material/strict implication - also, it tries to exclude inferences like The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not. which are classically valid, but are ordinarily counterintuitive. In this sense they are 'closer' to "everyday uses of logic reasoning". Instead, you seem to assume that RL is a "strategy" to avoid "long and complicated chains of reasoning" (?)… Why would you think that? Could it be that you misunderstood RL?
– DBK
Apr 16, 2012 at 0:00
• The use of quotation marks here vexes me greatly. Apr 16, 2012 at 2:49
• @stoicfury I rewrote the question without quotation marks. I also tried to make more definite statements. For the remaining vague statements, I added short explanations how the vagueness arose. Apr 16, 2012 at 8:10
• @DBK Of course it is possible that I misunderstood RL. However, I think it is more likely that i didn't succeed in writing a clear question, especially in explaining what I mean by the relevance principle. I clarified in the question now that relevance logic is quite successful at formalizing the relevance principle, at least in my opinion. But my question is whether the relevance principle itself can be justified (for example as a special case of Occam's razor). Apr 16, 2012 at 8:19
• @DBK An investigation whether the variable sharing principle affects the provability of "important and relevant theorems" is exactly what I'm looking for. However, in order to better capture the essence of the relevance principle, the variable sharing principle must be accompanied by modifications of the proof theory (similar to linear logic) to avoid irrelevant propositional variables creeping in. But these modifications make it less obvious that all "important and relevant theorems" can still be proved. Apr 16, 2012 at 10:27

Not sure if problems you are wondering about exists.

Maybe (A-> B) -> ((A -> (A->B)) , (Converse Contraction) is one of them if you think it should be valid that is. (it isn't valid in E and R)

Do you find (A-> B) -> ((B -> A) -> (A -> B)) paradoxical?

The lenght of a proof depends on which axioms and rules you use Hilbert style syatems are always longer than Gentzen style systems. to which system are you refering?

I think the problems are more in the field of:

Which relevant logic do you mean in the first place? E, T, R , Ack or even another one? I was just reading that even in E there are some paradoxes. (Entailment volume 1 par 14.6 aptly named "paradox regained)

What is negation in the first place? (my current score is that there are 12 different versions) Is the disjunctive syllogism valid ? (Q, P v ~Q => P)

Problems with the Conjunction (there is a difference between ((P&Q) -> R) and (P -> (Q -> R)) don't ask me what.

• Thanks, both "(A->B)->((A->(A->B))" and "(A-> B) -> ((B -> A) -> (A -> B))" satisfy the variable sharing principle, so these are definitively the type of problems I was asking about. So all I have to do now is convince myself that they are really not valid in E and R, and then try to understand whether this is actually intended or not. I knew that there exist many different versions of relevant logic like E, T, R, or ..., but I didn't fully understand why. Apr 27, 2012 at 15:18
• "(A-> B) -> ((B -> A) -> (A -> B))" Is valid "(A->B)->((A->(A->B))" is invalid Apr 27, 2012 at 17:09
• I'm not so sure that Gentzen style systems always have shorter derivations than Hilbert style derivations. In any Hilbert system with a rule X(x, y), x|-y, condensed detachment is a derivable rule of inference making the (CCCqprCpr, CqCpr) into a complete derivation. I'm not entirely clear what you mean by a Gentzen style proof (I'd guess proof trees... or do you mean any natural deduction style proof?), but is the Gentzen style proof really that short? Jun 14, 2013 at 18:03

According to the answer to this question, relevant logic+PA, although successfully proving its own consistency by finatary means, something not possible in first order logic+PA, it fails to show that there are integers that are not quadratic residues. This is a fairly standard piece of number theory to lose, if we are interested in only conservative extensions of PA; however it could be taken to mean a very different kind of arithmetic is possible under relevant logic.

One of the problems with relevance logic is that it is not monotonic