3,842 research outputs found
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants
Proof Theory of Finite-valued Logics
The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics
Adding an Implication to Logics of Perfect Paradefinite Algebras
Perfect paradefinite algebras are De Morgan algebras expanded with a
perfection (or classicality) operation. They form a variety that is
term-equivalent to the variety of involutive Stone algebras. Their associated
multiple-conclusion (Set-Set) and single-conclusion (Set-Fmla) order-preserving
logics are non-algebraizable self-extensional logics of formal inconsistency
and undeterminedness determined by a six-valued matrix, studied in depth by
Gomes et al. (2022) from both the algebraic and the proof-theoretical
perspectives. We continue hereby that study by investigating directions for
conservatively expanding these logics with an implication connective
(essentially, one that admits the deduction-detachment theorem). We first
consider logics given by very simple and manageable non-deterministic semantics
whose implication (in isolation) is classical. These, nevertheless, fail to be
self-extensional. We then consider the implication realized by the relative
pseudo-complement over the six-valued perfect paradefinite algebra. Our
strategy is to expand such algebra with this connective and study the
(self-extensional) Set-Set and Set-Fmla order-preserving logics, as well as the
T-assertional logics of the variety induced by the new algebra. We provide
axiomatizations for such new variety and for such logics, drawing parallels
with the class of symmetric Heyting algebras and with Moisil's `symmetric modal
logic'. For the Set-Set logic, in particular, the axiomatization we obtain is
analytic. We close by studying interpolation properties for these logics and
concluding that the new variety has the Maehara amalgamation property
Independence and abstract multiplication
We investigate the notion of independence, which is at the basis of many,
seemingly unrelated, properties of logic like Rational Monotony in
non-monotonic logics, and interpolation theorems
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