Alternative Axiomatizations of the Conditional System VC - Claudio E. A. Pizzi (Organon F, 2019)
The central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A >< A) and T: □A ⊃ A, which are subjoined to a basic principle named Int: (A ∧ B) ⊃ (♢A > ♢B). A hierarchy of extensions of the basic system V called VInt, VInt1, VInt1T is then construed and submitted to a semantic analysis. In Section 3 VInt1T is shown to be deductively equivalent to VC. Section 4 shows that in VC the thesis X1 is equivalent to X1∨: (♢A >< A) ∨ (♢¬A >< ¬A), so that VC is also equivalent to a variant of VInt1T here called VInt1To. In Section 6 both X1 and X1∨ offer the basis for a discussion on systems containing CS, in which it is argued that they cannot avoid various kinds of partial or full trivialization of some non truth-functional operators.