Bulletin of the Section of Logic

The Phenomenology of Second-Level Inference: Perfumes in The Deductive Garden

David Makinson — 2020-12-30

We comment on certain features that second-level inference rules commonly used in mathematical proof sometimes have, sometimes lack: suppositions, indirectness, goal-simplification, goal-preservation and premise-preservation. The emphasis is on the roles of these features, which we call 'perfumes', in mathematical practice rather than on the space of all formal possibilities, deployment in proof-theory, or conventions for display in systems of natural deduction.

From Intuitionism to Brouwer's Modal Logic

Zofia Kostrzycka — 2020-12-30

We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

Empirical Negation, Co-Negation and the Contraposition Rule II: Proof-Theoretical Investigations

Satoru Niki — 2020-12-30

We continue the investigation of the first paper where we studied logics with various negations including empirical negation and co-negation. We established how such logics can be treated uniformly with R. Sylvan's CC_ωas the basis. In this paper we use this result to obtain cut-free labelled sequent calculi for the logics.

Length Neutrosophic Subalgebras of BCK=BCI-Algebras

Young Bae Jun — 2020-12-30

Given i, j, k ∈ {1,2,3,4}, the notion of (i, j, k)-length neutrosophic subalgebras in BCK=BCI-algebras is introduced, and their properties are investigated. Characterizations of length neutrosophic subalgebras are discussed by using level sets of interval neutrosophic sets. Conditions for level sets of interval neutrosophic sets to be subalgebras are provided.

Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations

Alexej P. Pynko — 2020-12-30

Basic results of the paper are that any four-valued expansion L₄ of Dunn-Belnap's logic DB₄ is de_ned by a unique (up to isomorphism) conjunctive matrix ℳ₄ with exactly two distinguished values over an expansion 𝔄₄ of a De Morgan non-Boolean four-valued diamond, but by no matrix with either less than four values or a single [non-]distinguished value, and has no proper extension satisfying Variable Sharing Property (VSP). We then characterize L₄'s having a theorem / inconsistent formula, satisfying VSP and being [inferentially] maximal / subclassical / maximally paraconsistent, in particular, algebraically through ℳ₄|𝔄₄'s (not) having certain submatrices|subalebras.

Likewise, [providing 𝔄₄ is regular / has no three-element subalgebra] L₄ has a proper consistent axiomatic extension if[f] ℳ₄ has a proper paraconsistent / two-valued submatrix [in which case the logic of this submatrix is the only proper consistent axiomatic extension of L₄ and is relatively axiomatized by the Excluded Middle law axiom]. As a generic tool (applicable, in particular, to both classically-negative and implicative expansions of DB₄), we also prove that the lattice of axiomatic extensions of the logic of an implicative matrix ℳ with equality determinant is dual to the distributive lattice of lower cones of the set of all submatrices of ℳ with non-distinguished values.