Logica Universalis

Makarov, Ilya

doi: 10.1007/s11787-015-0117-9pmid: N/A

The article deals with finding finite total equivalence systems for formulas based on an arbitrary closed class of functions of several variables defined on the set {0, 1, 2} and taking values in the set {0,1} with the property that the restrictions of its functions to the set {0, 1} constitutes a closed class of Boolean functions. We consider all classes whose restriction closure is either the set of all functions of two-valued logic or the set T a of functions preserving $${a, a\in\{0, 1\}}$$ a , a ∈ { 0 , 1 } . In each of these cases, we find a finite total equivalence system, construct a canonical type for formulas, and present a complete algorithm for determining whether any two formulas are equivalent.

Citkin, Alex

doi: 10.1007/s11787-015-0116-xpmid: N/A

The goal of this paper is to generalize a notion of quasi-characteristic inference rule (by using finite partial algebras instead of finite subdirectly irreducible algebras) in the following way: with every finite partial algebra we associate a (multiple-conclusion) rule, and study the properties of these rules. We prove that any equivalential logic can be axiomatized by such rules. We further discuss the correlations between characteristic rules of the finite partial algebras and canonical rules. Then, with every algebra we associate a set of characteristic rules that correspond to each finite partial subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular quasi-characteristic rules.

Peterson, Clayton

doi: 10.1007/s11787-014-0111-7pmid: N/A

This paper provides an analysis of contrary-to-duty reasoning from the proof-theoretical perspective of category theory. While Chisholm’s paradox hints at the need of dyadic deontic logic by showing that monadic deontic logics are not able to adequately model conditional obligations and contrary-to-duties, other arguments can be objected to dyadic approaches in favor of non-monotonic foundations. We show that all these objections can be answered at one fell swoop by modeling conditional obligations within a deductive system defined as an instance of a symmetric monoidal closed category. Using category theory as a foundational framework for logic, we show that it is possible to model conditional normative reasoning and conflicting obligations within a monadic approach without adding further operators or considering deontic conditionals as primitive.

Odintsov, Sergei; Rybakov, Vladimir

doi: 10.1007/s11787-014-0110-8pmid: N/A

Our paper aims to investigate inference rules for Nelson’s logics and to discuss possible ways to determine admissibility of inference rules in such logics. We will use the technique offered originally for intuitionistic logic and paraconsistent minimal Johannson’s logic. However, the adaptation is not an easy and evident task since Nelson’s logics do not enjoy replacement of equivalences rule. Therefore we consider and compare standard admissibility and weak admissibility. Our paper founds algorithms for recognizing weak admissibility and admissibility itself – for restricted cases, to show the problems arising in the course of study.

Francez, Nissim

doi: 10.1007/s11787-015-0118-8pmid: N/A

I show that in the context of proof-theoretic semantics, Dummett’s distinction between the assertoric meaning of a sentence (it’s meaning when viewed as “stand alone”) and its ingredient sense (its meaning when viewed as a constituent of an embedding sentence) can be seen as a distinction between two proof-theoretic meanings of a sentence: 1. Meaning as a conclusion of an introduction rule in a meaning-conferring natural-deduction proof system. 2. Meaning as a premise of an introduction rule in a meaning-conferring natural-deduction proof system. The effect of this distinction on compositionality of proof-theoretic meaning is discussed.