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The question doesn't go beyond the title.

And I don't mean logics that merely just don't have it as a primitive rule - I'm interested in logic where you can't actually use it.

I've searched around and looked at the more exotic logics that I know, but all use modus ponens. Are there logic that do have implication but go without that rule?

But maybe I'm confused and if I take away that tool to syntactically go from knowing $P$ to knowing $Q$, then there isn't anything to $\to$ left.

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  • $\begingroup$ Is implication meaningful without modus ponens? What should be an alternative rule? Perhaps $a \to b $ and $b\to c $ give $a\to c $? $\endgroup$ – md2perpe Jul 29 '17 at 19:44
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It was observed by Lewis & Langford that modus ponens fails in Lukasiewicz 3-valued logic. What they did not make clear is how it fails and why it should fail. It fails because Lukasiewicz allows conditionals to have the third logical value. Unrestricted use of modus ponens would allow the use of doubtful premises (a and a→b) to draw a conclusion b that is false. This is not and should not be valid reasoning. It seems to have escaped their attention that the deficiency can be repaired by restricting conditionals to those that are definitely true: (a & L(a→b)) → b is a tautology.

The suggestion of ((a→b) & (b→c)) → (a→c) also fails for the same reason and can be repaired the same way.

Modus ponens as well as most other rules of inference, such as (P $\rightarrow $ P) also fail in Kleene's 3-valued logic. This is because the conditional used in it assigns the third logical value whenever the antecedent and consequent both have that value.

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You can take some logic without implication, the logic dont have MP. The Belnap Logic 4, for example. This logic also dont have theorem, only infereces of type: $\phi \vdash \psi$.

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