Information completeness in Nelson algebras of rough sets induced by quasiorders

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.Comment: 15 page

Applying abstract algebraic logic to classical automata theory : an exercise

In [4], Blok and Pigozzi have shown that a deterministic finite au- tomaton can be naturally viewed as a logical matrix. Following this idea, we use a generalisation of the matrix concept to deal with other kind of automata in the same algebraic perspective. We survey some classical concepts of automata theory using tools from algebraic logic. The novelty of this approach is the understand- ing of the classical automata theory within the standard abstract algebraic logic theory

Some modal and temporal translations of generalized basic logic

We introduce a family of modal expansions of Łukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal Łukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna’s poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Łukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter

Proofs of valid categorical syllogisms in one diagrammatic and two symbolic axiomatic systems

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn't produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz's drafts on logic contained sufficient ingredients to prove them by an algebraic method -- which we call the Leibniz-Cayley (LC) system -- without having to make use of the more expressive and complex machinery of first-order quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method -- which we call the McColl-Ladd (ML) system -- employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz's historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic.Comment: 66 pages, 9 figures (some of which include subfigures), 5 tables (one of which includes 2 subtables). A cut-down version of this article, which removes the discussion on diagrammatic logic with Euler diagrams, was submitted to the "History and Philosophy of Logic" journal with a different titl