- rule of inference
- Lewis Carroll raised the Zeno-like problem of how a proof ever gets started. Suppose I have as premises (1)
*p*and (2)*p*→*q*. Can I infer*q*? Only, it seems, if I am sure of (3) (*p*&*p*→*q*) →*q*. Can I then infer*q*? Only, it seems, if I am sure that (4) (*p*&*p*→*q*& (*p*&*p*→*q*) →*q*) →*q*. For each new axiom (N) I need a further axiom (N + 1) telling me that the set so far implies*q*, and the regress never stops. The usual solution is to treat a system as containing not only axioms, but also rules of inference, allowing movement from the axioms. The rule modus ponens allows us to pass from the first two premises to*q*. Carroll's puzzle shows that it is essential to distinguish two theoretical categories, although there may be choice about which theses to put in which category.

*Philosophy dictionary.
Academic.
2011.*