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

Asked Oct 14 '18 at 20:03

Active Oct 14 '18 at 20:03

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I am trying to understand relevance logic. Here I prove $A\to (\neg A \to B)$ (a form of the Principle of Explosion) using ordinary natural deduction. Which inference(s) here are not allowed in relevance logic and why?

$A\space\space$ (assume)
$\neg A\space\space$ (assume)
$\neg B\space\space$ (assume)
$A\land\neg A\space\space$ (intro $\land$, 1, 2)
$\neg\neg B\space\space$ (intro $\neg$, 3, 4)
$B\space\space$ (elim $\neg\neg$, 5)
$\neg A \to B\space\space$ (intro $\to$, 2, 6)
$A\to (\neg A \to B)\space\space$ (intro $\to$, 1, 7)

asked Oct 14 '18 at 20:03

Dan Christensen

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See Greg Restall, Relevant and Substructural Logics, page 20 : "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 assumptions." – Mauro ALLEGRANZA Oct 14 '18 at 20:41
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@MauroALLEGRANZA What a bizarre restriction to place upon oneself! Thanks. – Dan Christensen Oct 15 '18 at 02:55
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@MauroALLEGRANZA Can $A\to (\neg A \to B)$ be proven by other means in relevance logic? How about $A \to (B \to A\land B$)? – Dan Christensen Oct 15 '18 at 03:28
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I do not think so: in Relevance Logic $¬p → (p → q)$ is not valid. – Mauro ALLEGRANZA Oct 15 '18 at 14:42
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@MauroALLEGRANZA What about $A \to (B \to A\land B)$? Can this really be rejected by serious scholars? – Dan Christensen Oct 16 '18 at 02:39

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

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