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ALTERNATIVE AXIOMATIZATIONS OF THE CONDITIONAL SYSTEM VC

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Pizzi C. E. A.
Organon F. medzinárodný časopis pre analytickú filozofiu
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2019
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vol. 26
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issue 3
427 – 445
EN
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.
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