## Notre Dame Journal of Formal Logic

### An Intriguing Logic with Two Implicational Connectives

Lloyd Humberstone

#### Abstract

Spinks introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in Section 1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two implications in the more familiar language of modal logic. In Sections 2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and--after a digression (in Section 4) on the intuitionistic analogue of the material in Section 3--consider them together in Section 5, closing with a discussion in Section 6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent Sections 1-6 from becoming overly distended.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 41, Number 1 (2000), 1-40.

Dates
First available in Project Euclid: 29 July 2002

https://projecteuclid.org/euclid.ndjfl/1027953481

Digital Object Identifier
doi:10.1305/ndjfl/1027953481

Mathematical Reviews number (MathSciNet)
MR1915129

Zentralblatt MATH identifier
1005.03026

#### Citation

Humberstone, Lloyd. An Intriguing Logic with Two Implicational Connectives. Notre Dame J. Formal Logic 41 (2000), no. 1, 1--40. doi:10.1305/ndjfl/1027953481. https://projecteuclid.org/euclid.ndjfl/1027953481

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