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(Follow-up to Which line(s) of my proof of $A\to (\neg A \to B)$ are not allowed in relevance logic? )

Theorem: $A\to (B\to (A\land B))$

Proof:

  1. $A\space\space$ (assume)
  2. $B\space\space$ (assume)
  3. $A\land B\space\space$ (intro $\land$ 1,2)
  4. $B \to (A\land B)\space\space$ (intro $\to$ 2, 3)
  5. $A\to (B\to (A\land B))\space\space$ (intro $\to$ 1, 4)

Apparently the inference on line 3 is banned by relevance logic. Commenting on the same issue (as I understand it) in a similar proof, Greg Restall writes:

We must do something to the rule for conjunction introduction to ban this proof. The required emendation is to only allow conjunction introduction when the two subproofs have exactly the same open assumption.

-- Relevant and Substructural Logics, page 20

My question: Why must we do anything to the rule for conjunction? Does the usual form of the rule result in any logical inconsistencies? Given the stunning successes of classical logic in just about every field of human endeavour, it seems to me that only logical inconsistencies could justify tinkering with the basic rules of logic in this way. Also, has relevance logic been used by any mathematicians to any significant degree?

Dan Christensen
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  • Aren't relevant logicians mathematicians? :P (Honestly, the one I feel like relevance would be suspicious of, if any, is (4).) – Noah Schweber Oct 16 '18 at 16:53
  • @NoahSchweber I thought they existed in more or less separate, parallel universes. Is that not the case? – Dan Christensen Oct 16 '18 at 16:59
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    Is stuff like this not mathematics? I think we have to distinguish between three things: the study of relevant logic in mathematics, the philosophical arguments for/against/around relevant logic, and relevant mathematics (that is, mathematics done according to relevant logic). The first is clearly mathematics, while the second - who knows? As to the third, in contrast with intuitionism/constructivism I don't know of anyone who actually uses relevance as their whole ambient logic, but that doesn't mean they don't exist. – Noah Schweber Oct 16 '18 at 17:08

1 Answers1

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Why must we do anything to the rule for conjunction?

In Relevance Logic, the conditional symbol, $\to$, has a weaker meaning than in classical logic.   The claim $A\to B$ is only considered true when $B$ relevantly follows from $A$.   (Recollect the perannial questions of "Why is $A\to B$ true then $A$ is false, since surely $B$ cannot follow from $A$ when $A$ isn't even true?" Relevance Logic tries to capture that intuition that there is more to conditionality than material implication.)

As such, conjunction inherits this weaker meaning.   In Relevance Logic, the claim $A\land B$ is true exactly when both $A$ and $B$ are true and they have the same relavance (follow from the same assumptions).

Does the usual form of the rule result in any logical inconsistencies?

Internally.   If you don't restrict conjunction then you can prove $A\to (B\to A)$ when $B$ is not relevant to $A$, which should not be valid in Relevance Logic.   The system should not do what we want it to not do.

Given the stunning successes of classical logic in just about every field of human endeavour, it seems to me that only logical inconsistencies could justify tinkering with the basic rules of logic in this way.

The universe does not always subscribe to classical logic (re: Quantum Logic).   Exploring alternate proof systems at least provides the tools to think flexibly about such things.

Graham Kemp
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  • $\neg A \to (A \to B)$ is counter intuitive so why not invent a form of logic that makes it impossible? Is that it? Then impress people that you actually do some math with one hand tied behind your back like this. It does sound like fun, but maybe they didn't know that it isn't really so hard to prove from a few self-evident rules of logic: Intro $\land$, Elim $\neg \neg$, Intro $\to$ and Intro $\neg$. See my own formal proof at http://www.dcproof.com/ImpliesLines3-4.html – Dan Christensen Oct 17 '18 at 03:15
  • Also, I think you will find that propositional logic is consistent (i.e. free of any internal contradictions) without these onerous new restrictions on Intro $\land$. – Dan Christensen Oct 17 '18 at 03:15