Strong Depth Relevance

Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these two properties. I also show how to use this fact to demonstrate that these logics seem to be constructive in previously unknown ways.

The fragment of Classical Logic that respects the Variable-Sharing Principle

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

Sette’s calculus P1 and some hierarchies of paraconsistent systems

The necessary condition for a calculus to be paraconsistent is that its consequence relation is not explosive. This results in rejection of the principle of ex contradictione sequitur quodlibet. In 1973, Sette presented a calculus, denoted as $P^1$, which is paraconsistent only at the atomic level, i.e. $\alpha $ and ${\sim }\alpha $ yield any $\beta $ if, and only if the formula $\alpha $ is not a propositional variable. The calculus has been viewed as one of the noteworthy paraconsistent calculi since then. The objective of this paper is to propose a new semantics for Sette’s calculus and present some hierarchies of the paraconsistent calculi, which are based on $P^1$. We demonstrate that $P^1$ is sound and complete with respect to the semantics and so are all the calculi under consideration.

Modal logics with hard diamond-free fragments

We investigate the complexity of modal satisfiability for a family of multi-modal logics with interdependencies among the modalities. In particular, we examine four characteristic multi-modal logics with dependencies and demonstrate that, even if we restrict the formulae to be diamond-free and to have only one propositional variable, these logics still have a high complexity. This result identifies and isolates two sources of complexity: the presence of axiom $D$ for some of the modalities and certain modal interdependencies. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multi-modal logics in a general setting.

A Note on the Relevance of Semilattice Relevance Logic

A propositional logic has the variable sharing property if φ → ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic (R+u) and its extension with an involution negation (R¬u) have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed.

Construcciones consecutivas ponderativas en español: focalización, correlativos y correlatos adjuntos

This paper deals with the complex sentences in Spanish formed by the correlation tanto(s)~tanta(s)... que (i.e. Comí tanto que me sentó mal ‘I ate so much that I felt bad’). In opposition to the majority of the previous analyses, which treat que as an adjacent of the quantified phrase with tanto, we will argue that tanto... que is not a functional group nor a maximal projection lexically generated, but a long-distance correlation that is semantically and pragmatically conditioned. In particular, we will defend that tanto is an evaluative scalar quantifier that behaves like correlatives do: it maximizes the meaning of its clause and is pragmatically significant (it marks contrastive focus by means of intensive accent and intonation and may give rise to an informative structure of topic-comment). In addition, the clause with que behaves like the typical correlates: it is always postposed to the correlative and it rejects recursive operations (like coordination). As occurs in the comparative correlatives from some Germanic languages and the relative correlatives in languages like Hindi, the clause with que is an adjunct of the first clause. Lastly, and following Dayal’s work on relative correlatives in Hindi (1995), we will suggest that tanto is a two-place quantifier linking a propositional variable, instantiated by the clause with que.

Different Arguments, Same Problems

I illustrate with three classical examples the mistakes arising from using a modal operator admitting multiple interpretations in the same argument; the flaws arise especially easily if no attention is paid to the range of propositional variables. Premisses taken separately might seem convincing and a substitution for a propositional variable in a modal context might seem legitimate. But there is no single interpretation of the modal operators involved under which all the premisses are plausible and the substitution successful.

Against Philo’s interpretation of conditional. The case of Aristotle´s thesis

There is an Aristotelian thesis that can be considered controversial. That is the thesis related to a denied conditional with only one propositional variable and in which, in addition, one of its clauses is also denied. While the thesis is not a tautology, people tend to accept it as true. Pfeifer’s approach can account for this fact. However, I try to show that this problem can also be explained from other alternative frameworks, in particular, from that of the mental models theory, that of López-Astorga based on the pragmatic phenomenon of conditional perfection, and that of the mental logic theory. Likewise, I indicate the difficulties regarding Aristotle’s thesis of the mental models theory and López-Astorga’s proposal, and conclude that the account of the mental logic theory is the strongest alternative to Pfeifer’s explanation and that what is clearly obvious is that conditional should not be materially interpreted.http://dx.doi.org/10.15304/ag.35.2.2542

Given the Principal Disjunctive Normal Form of Simple Proposition Formula

The Lord of the formula of disjunctive normal form is an important content in propositional logic. When the propositional variable is large, the paper gives a formula for the main disjunctive normal form of simple method.