57 research outputs found
A fundamental non-classical logic
We give a proof-theoretic as well as a semantic characterization of a logic
in the signature with conjunction, disjunction, negation, and the universal and
existential quantifiers that we suggest has a certain fundamental status. We
present a Fitch-style natural deduction system for the logic that contains only
the introduction and elimination rules for the logical constants. From this
starting point, if one adds the rule that Fitch called Reiteration, one obtains
a proof system for intuitionistic logic in the given signature; if instead of
adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a
proof system for orthologic; by adding both Reiteration and Reductio, one
obtains a proof system for classical logic. Arguably neither Reiteration nor
Reductio is as intimately related to the meaning of the connectives as the
introduction and elimination rules are, so the base logic we identify serves as
a more fundamental starting point and common ground between proponents of
intuitionistic logic, orthologic, and classical logic. The algebraic semantics
for the logic we motivate proof-theoretically is based on bounded lattices
equipped with what has been called a weak pseudocomplementation. We show that
such lattice expansions are representable using a set together with a reflexive
binary relation satisfying a simple first-order condition, which yields an
elegant relational semantics for the logic. This builds on our previous study
of representations of lattices with negations, which we extend and specialize
for several types of negation in addition to weak pseudocomplementation; in an
appendix, we further extend this representation to lattices with implications.
Finally, we discuss adding to our logic a conditional obeying only introduction
and elimination rules, interpreted as a modality using a family of
accessibility relations.Comment: added topological representation of bounded lattices with
implications in Appendi
A unified relational semantics for intuitionistic logic, basic propositional logic and orthologic with strict implication
In this paper, by slightly generalizing the notion of 'proposition' in
'Propositional Logic and Modal Logic - A Connection via Relational Semantics'
by Shengyang Zhong, we propose a relational semantics of propositional language
with bottom, conjunction and imlication, which unifies the relational semantics
of intuitionistic logic, Visser's basic propositional logic and orthologic with
strict implication. We study the semantic and syntactic consequence relations
and prove the soundness and completeness theorems for eight propositional
logics
Morphisms and Duality for Polarities and Lattices with Operators
Structures based on polarities have been used to provide relational semantics
for propositional logics that are modelled algebraically by non-distributive
lattices with additional operators. This article develops a first order notion
of morphism between polarity-based structures that generalises the theory of
bounded morphisms for Boolean modal logics. It defines a category of such
structures that is contravariantly dual to a given category of lattice-based
algebras whose additional operations preserve either finite joins or finite
meets. Two different versions of the Goldblatt-Thomason theorem are derived in
this setting
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