Every Berman’s variety \ which is the subvariety of Ockham algebras defined by the equation \ and \) determines a finitary substitution invariant consequence relation \. A sequent system \ is introduced as an axiomatization of the consequence relation \. The system \ is characterized by a single finite frame \ under the frame semantics given for the formal language. By the duality between frames and algebras, \ can be viewed as a \-valued logic as it is characterized by a (...) distributive lattice of \ elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system \ is established for \. The Craig interpolation property of \ is shown proof-theoretically utilizing \. (shrink)

A deterministic weakening \ of the Belnap–Dunn four-valued logic \ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \ is formalized as a sequent calculus. The completeness and decidability of \ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \ is described, and found to be a subvariety of Berman’s (...) variety \. Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \ is the logic of a nine-element distributive lattice with a negation. Moreover, \ is embedded into \ by Glivenko’s double-negation translation. (shrink)

The minimal weakening \ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of \ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of \. These logics have the finite model property and they are decidable.

The language of Belnap–Dunn modal logic \ expands the language of Belnap–Dunn four-valued logic with the modal operator \. We introduce the polarity semantics for \ and its two expansions \ and \ with value operators. The local finitary consequence relation \ in the language \ with respect to the class of all frames is axiomatized by a sequent system \ where \. We prove by using translations between sequents and formulas that these languages under the polarity semantics have the (...) same expressive power on the level of frames with the language \ under the relational semantics for classical modal logic. (shrink)

Kripke’s Fregean quantification logic FQ fails to formalize the usual first-order logic with identity due to the interpretation of the conditional operator. Motivated by Kripke’s syntax and semantics, the three-valued Fregean quantification logic FQ3 is proposed. This three valued logic differs from Kleene and Łukasiewicz’s three-valued logics. The logic FQ3 is decidable. A sound and complete Hilbert-style axiomatic system for the logic FQ3 is presented.