Analyticity, Balance and Non-admissibility of $$\varvec{Cut}$$ Cut in Stoic Logic

Abstract

Abstract.This paper shows that, for the Hertz-Gentzen Systems of 1933 (without Thinning), extended by a classical rule T 1 (from the Stoics) and using certain axioms (also from the Stoics), all derivations are analytic: every cut formula occurs as a subformula in the cut's conclusion. Since the Stoic cut rules are instances of Gentzen's Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a "relevance criterion" and of two "balance criteria", and hence (in contrast to one of Gentzen's 1933 results) that a particular derivable sequent has no derivation that is "normal" in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T 1 rule and the instances of Cut with just two antecedent formulae in the first premiss.

Indirect Proof and Inversions of Syllogisms

“…This gives us a direct deduction of A from Γ, Π − (P, Q) as required. The Lemma is one of the Stoic rules of inference, the first thema, T 1, for which see [1] or [2]. Proof.…”