# What is Ackermanns 'admissibility of gamma' in relevance logic?

Various surveys I've looked at relevance logic mention Ackermanns admissibility of gamma rule. I've not come across the term in logic before, what is it, how is it used and is it also important outside of relvance logic?

## 1 Answer

Ackermann's Rule Gamma is the rule that from "├── A->B" and "├── A", one may infer "├── B," where "->" denotes the material conditional. (Equivalently, from "├── ~A V B" and "├── A", infer "├── B.")

Note that this is distinct from modus ponens or disjunctive syllogism; this is the rule that if ~A V B is a theorem and A is a theorem then B, too, is a theorem. Such a distinction is clear in, e.g., Rule Necessitation in modal logics; from "├── A," infer "├── L A," where "L" denotes necessity. This clearly doesn't mean that a theory is closed under necessitation, merely that all theorems are necessary.

• That is suprising! I'd have called your first example modus ponens, and the second disjunctive syllogism. But you're saying that these two names are actually synonyms for the second example. But this is purely I laymans view, I haven't studied philosophical logic. – Mozibur Ullah Oct 5 '13 at 19:04
• @MoziburUllah The thing is that there's this distinction between an external and an internal logic. If there's no deduction theorem in the usual form; they are not "the same" logic. Basically you can have admissible rules that are not derivable (although that doesn't happen in classical logic, which is complete). The explanation above while easier to grok for modal logics; it's rather incomplete for relevance logics... – Fizz Mar 31 at 16:32
• The issue (IIRC) is that gamma is admissible but not derivable in R. Somewhat more useful examples of failure of the usual deduction theorem in these slides: www2.cs.cas.cz/~cintula/slides/AAL-4.pdf For intuitionist logic the problem of which rules are admissible but not derivable has been studied more intensely irif.fr/~roziere/admiss/mscs93.pdf – Fizz Mar 31 at 16:35
• Somewhat more recent/relevant slides on this math.unibe.ch/e19399/e19400/e292892/e293040/… They confirm what I said above about gamma in R (p. 14). – Fizz Mar 31 at 16:52