Cut-free Sequent Calculus and Natural Deduction for the Tetravalent Modal Logic

Martín Figallo

doi:10.1007/s11225-021-09944-3

Abstract

AbstractThe tetravalent modal logic ($${\mathcal {TML}}$$ TML ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic$${{\mathcal {TML}}}^N$$ TML N ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.

Highlights

The class TMA of tetravalent modal algebras was first considered by Antonio Monteiro (1978), and mainly studied by I
The tetravalent modal logic T ML defined over Fm is the propositional logic F m, |=T ML given as follows: for every finite set Γ ∪ {α} ⊆ F m, Γ |=T ML α if and only if, for every A ∈ TMA and for every h ∈ Hom(Fm, A), {h(γ) : γ ∈ Γ} ≤ h(α)
Since our natural deduction system is strongly inspired by the cut-free sequent calculus SCT ML, one can likely expect normalization to hold for SCT ML

Summary

Introduction

The class TMA of tetravalent modal algebras was first considered by Antonio Monteiro (1978), and mainly studied by I. It is clear that in this setting the sentence P can only be true in the case where we have some information saying that P is true and we have no information saying that P is false, while it is false in all other cases (i.e., lack of information or at least some information saying that P is false, disregarding whether at the same time some other information says that P is true); that is, on the set {0, n, b, 1} of epistemic values this operator must be defined as 1 = 1 and n = b = 0 = 0 This is exactly the algebra that generates the variety of TMAs. Font and Rius [8] studied two logics related to TMAs. One of them is obtained by following the usual “preserving truth” scheme, taking {1} as designated set, that is, ψ follows from ψ1, .

Preliminaries

G Does not Admit a Cut-elimination Theorem

Cut-free Sequent Calculus for T ML

Some Applications of the Cut Elimination Theorem

Natural Deduction for T ML

Conclusions