Organon F

Volume 26, August 2019, Issue 3, Pages 427–445

ISSN 2585-7150 (online) ISSN 1335-0668 (print)

Research Article | Special issue on Reflecting on the Legacy of C.I. Lewis: Contemporary and Historical Perspectives on Modal Logic

Alternative Axiomatizations of the Conditional System VC

Claudio E. A. Pizzi


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.


Conditional logic, centering condition, trivialization, modal collapse.

Author(s) and affiliation(s)


Claudio E. A. Pizzi


University of Siena (Emeritus)


Dipartimento di Scienze Sociali, Politiche e Cognitive, University of Siena, via Roma 56, 53100 Siena, Italy


About this article


17 September 2018


17 May 2019

Published online

24 August 2019


Institute of Philosophy of the Slovak Academy of Sciences

Institute of Philosophy of the Czech Academy of Sciences

Cite as


Pizzi, C.E.A. (2019). Alternative Axiomatizations of the Conditional System VC. Organon F, 26(3), 427–445.


Pizzi, Claudio E.A. 2019. "Alternative Axiomatizations of the Conditional System VC." Organon F 26 (3): 427–445.


Pizzi, C.E.A. (2019). Alternative Axiomatizations of the Conditional System VC. Organon F, 26(3), pp. 427–445.

Copyright information

© Claudio E. A. Pizzi

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