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For one of the most well-known many-valued logics FDE, there are several semantics, including the star semantics by Richard Routley and Valerie Routley, the two-valued relational semantics by Michael Dunn and the four-valued semantics by Nuel Belnap. The last semantics inspired Yaroslav Shramko and Heinrich Wansing to introduce the trilattice SIXTEEN\(_3\). In this article, we offer two alternative semantical presentations for SIXTEEN\(_3\), by applying the Routleys’ semantics and the Dunn semantics. Based on our new semantics, we discuss related systems with less truth values, as well as the relation to FDE-based modal logics.

One of the simple approaches to paraconsistent logic is in terms of three-valued logics. Assuming the standard behavior with respect to the “classical"values, there are only two possibilities for paraconsistent negation, namely the negation of the Logic of Paradox and the negation of Sette’s logic P \(^1\). From a philosophical perspective, the paraconsistent negation of P \(^1\) is less discussed due to the lack of an intuitive reading of the third value. Based on these, the aim of this paper is to fill in the gap by presenting a semantics for P \(^1\) a la Jaśkowski which sheds some light on the intuitive understanding of Sette’s logic. A variant of P \(^1\) known as I \(^1\) will be also discussed.

In this paper, we investigate an extended first-order Belnap-Dunn logic with classical negation. We introduce a Gentzen-type sequent calculus FBD+ for this logic and prove theorems for syntactically and semantically embedding FBD+ into a Gentzen-type sequent calculus for first-order classical logic. Moreover, we show the cut-elimination theorem for FBD+ and prove the completeness theorems with respect to both valuation and many-valued semantics for FBD+.

We motivate and develop an extension of Nelson’s constructive logic N3 that adds a counterfactual conditional to the existing setup. After developing the semantics, we will outline how our account will be able to give a nice analysis of natural language counterfactuals. In particular, the account does justice to the intuitions and arguments that have lead Alan Hajek to claim that most conditionals are false, but assertable, without actually forcing us to endorse that rather uncomfortable claim.