# Are there logics without modus ponens?

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.

• Is implication meaningful without modus ponens? What should be an alternative rule? Perhaps $a \to b$ and $b\to c$ give $a\to c$? – md2perpe Jul 29 '17 at 19:44

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.
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: $\phi \vdash \psi$.