From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (à la Scott) for formulating sound deduction rules.

Powers, Possibilities, and Time

Of equally fundamental importance to the current debate over causal powers are its Megaric consequences, the connection between powers and modality. One of the central motivations for adopting a powers ontology is said to be the support causal powers provide for grounding and explaining alternative possibilities. Calvin Normore provides a robust defence of this idea by defending the deeper thesis that time makes a difference for modalities because the existence of powers at a time impose formal constraints on the structural conditions governing the accessibility relation between the actual and the possible. Some alternative states of affairs are not genuine possibilities in the actual world, he argues, because of the powers that obtain in the actual world. Normore, moreover, roots his defence of this thesis in the medieval debate between Scotus and Ockham over whether what is possible is possibly actual (Scotus maintained ‘no it need not be’, whereas Ockham maintained ‘yes it had to be’).

A Kotas-Style Characterisation of Minimal Discussive Logic

In this paper, we discuss a version of discussive logic determined by a certain variant of Jaśkowski’s original model of discussion. The obtained system can be treated as the minimal discussive logic. It is determined by frames with serial accessibility relation. As the smallest one, this logic can be treated as a basis which could be extended to richer discussive logics that are obtained by varying accessibility relation and resulting in a lattice of discussive logics. One has to remember that while formulating discussive logics there is no one-to-one determination of discussive logics by modal logics. For example, it is proved that Jaśkowski’s logic D 2 can be expressed by other than S 5 modal logics. In this paper we consider a deductive system for the sketchily described minimal logic. While formulating the deductive system, we apply a method of Kotas that was used to axiomatize D 2 . The obtained system determines a logic D 0 as a set of theses that is contained in D 2 . Moreover, any discussive logic that would be expressed by means of the provided model of discussion would contain D 0 , so it is the smallest discussive logic.

THE TEMPORAL LOGIC OF TWO DIMENSIONAL MINKOWSKI SPACETIME IS DECIDABLE

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, x can access y if an electromagnetic or (slower than light) mechanical signal could be sent from x to y. We use Prior’s tense language of F and P representing causal accessibility and its converse relation. We consider two versions, one where the accessibility relation is reflexive and one where it is irreflexive. In either case it has been an open problem, for decades, whether the logic is decidable or axiomatisable. We make a small step forward by proving, in each case, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable and that the complexity of each problem is PSPACE-complete.A consequence is that the temporal logic of intervals with real endpoints under either the containment relation or the strict containment relation is PSPACE-complete, the same is true if the interval accessibility relation is “each endpoint is not earlier”, or its irreflexive restriction.We provide a temporal formula that distinguishes between three-dimensional and two-dimensional Minkowski spacetime and another temporal formula that distinguishes the two-dimensional case where the underlying field is the real numbers from the case where instead we use the rational numbers.

Useful Four-Valued Extension of the Temporal Logic KtT4

The temporal logic KtT4 is the modal logic obtained from the minimal temporal logic Kt by requiring the accessibility relation to be reflexive (which corresponds to the axiom T) and transitive (which corresponds to the axiom 4). This article aims, firstly, at providing both a model-theoretic and a proof-theoretic characterisation of a four-valued extension of the temporal logic KtT4 and, secondly, at identifying some of the most useful properties of this extension in the context of partial and paraconsistent logics.

Four-valued modal logic: Kripke semantics and duality

Along the lines of recent investigations combining many-valued and modal systems, we address the problem of defining and axiomatizing the least modal logic over the four-element Belnap lattice. By this we mean the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued. Our main result is the introduction of two Hilbert-style calculi that provide complete axiomatizations for, respectively, the local and the global consequence relations associated to the class of all four-valued Kripke models. Our completeness proofs make an extensive and profitable use of algebraic and topological techniques; in fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach.

Fair Division of a Graph

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases wherethe underlying graph has simple structure, and/or the number of agents---or, less restrictively, the number of agent types---is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.

Access granted to Zombies

In his ?Access Denied to Zombies?, Gualtiero Piccinini argues that the possibility of zombies does not entail the falsity of physicalism, since the accessibility relation can be understood so that even in S5 system for modal logic worlds inaccessible from our world are allowed (in the case in which the accessibility relation is understood as an equivalence rather than as universal accessibility). According to Piccinini, whether the zombie world is accessible from our world depends on whether physicalism is true in our world, which is something that cannot be answered in a non-question-begging way. In order to show this, he recalls a well known strategy of making a parody of the zombie argument. After pointing out that Piccinini?s strategy of parodying the zombie argument renders his former strategy, based on the distinguishing between the two notions of accessibility, redundant, I recall the two ways of handling parodies of the zombie argument. In addition, I argue that persisting on the distinction between accessibility understood as an equivalence and universal accessibility in dealing with the zombie argument relies upon accepting modal dualism (a view that there are two spaces of possibilities rather than one), which is something usually dismissed for methodological reasons (simplicity in particular). Given that Piccinini has not provided new arguments neither in favour of modal dualism nor in favour of parodying the zombie argument, the conclusion he infers remains unsupported by the premises he uses.

Internal past, external past, and counterfactuality: evidence from Japanese

This paper argues that there are two different types of counterfactuality, which are overtly represented in Japanese by adding the past either to the main verb or to the modal. In one pattern where the modal takes scope over the past, the counterfactuality cannot be canceled. Along the lines of Iatridou’s (2000) and Ogihara’s (2008) analyses, I propose that the past is a modal past and it directly indicates the counterfactuality. In another pattern where the past takes scope over the modal, the counterfactuality can be canceled. Appealing to the ideas provided in Condoravdi 2002 and Ippolito 2003, 2006, I suggest that the past is temporal and it expresses that there was a past time when the proposition could still be true. The accessibility relation is de?ned in the past. The counterfactuality is obtained by the conversational implicature.