Stoic Sequent Logic and Proof Theory

Abstract

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variablesharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

Copernicus and Axiomatics

“…However, there was no homogeneity on the nature of principles even at a more general level. Notably, Aristotle's definition of postulates does not coincide with the Euclidean use, and Proclus, reprising Geminus (first century BCE), differs from other Greek philosophical schools (Bobzien 1996;Acerbi 2010;Acerbi 2013;Bobzien 2019).…”

Naozaj Frege plagizoval stoikov?

“…Bobzien však nie je v hodnotení tejto možnosti konzistentná. Nemôžeme považovať len za kuriozitu fakt, že v práci o stoickej logike ako sekventovej logike (Bobzien, 2019) to pre prípad Gentzenovej a stoickej logiky už nepovažuje vôbec za fascinujúce, keďže tvrdí, že stoická teória dôkazu má hlavné znaky Gentzenovho sekventového kalkulu: Gentzenovi od stoikov nezávislý objav zhodného systému logiky zo záhadných dôvodov praje, ale Fregemu nie. Nevysvetľuje však prečo.…”

“…Ako vysvetľujúcu odpoveď sme predložili rekonštrukciu úsudku o nesúmerateľnosti uhlopriečky so stranou štvorca v stoickej logike (2006, 297 a n.). Sama Bobzien rekonštruuje stoickú logiku len ako sekventový propozičný kalkul (Bobzien 2019), na rozdiel od našej rekonštrukcie stoickej logiky ako predikátovej logiky.…”

Demonstration and the Indemonstrability of the Stoic Indemonstrables

“…If so, it 28 See e.g. Ierodiakonou, 2006, 1.1 ;Bobzien 2019, p.244 fn.21. Mueller, 1979, p.205 and Hitchcock 2006 Gentzen 1934, 193. is not strictly correct when Frede (1974, 128) and Łukasiewicz (1935, 117) assert that the indemonstrables were called indemonstrables because their role is that of axioms.…”

“…30 There is also (ii) no need for the 30 It has been suggested that strictly speaking in Stoic sequent logic, the anapodeiktoi are not analysable either. Bobzien argues that Stoic analysis is such that once an axiom has been reached no further rule can be applied (Bobzien 1996(Bobzien , 2019. Matching first and second indemonstrables are then not analysable into each other.…”