Against the Additive View of Imagination

According to the additive view of sensory imagination, mental imagery often involves two elements. There is an image-like element, which gives the experiences qualitative phenomenal character akin to that of perception. There is also a non-image element, consisting of something like suppositions about the image's object. This accounts for extra-sensory features of imagined objects and situations: for example, it determines whether an image of a grey horse is an image of Desert Orchid, or of some other grey horse. The view promises to give a simple and intuitive explanation of some puzzling features of imagination, and, further, to illuminate imagination's relation to modal knowledge. I contend that the additive view does not fulfil these two promises. The explanation of how images come to be determinate is redundant: the content constituting the indeterminate mental images on which the view relies is sufficient to deliver determinate images too, so the extra resources offered by the view are not required. When applied to modal epistemology, the view either delivers implausible results or offers no distinctive insight. Since the view is sold on its explanatory merits, and since these are dubious, the additive view should be rejected.     

Axiomatizing hybrid products: How can we reason many-dimensionally in hybrid logic?

The aim of this paper is to propose a two-dimensional hybrid logic in order to formalize inferences containing both spatial and temporal propositions. The semantic idea behind the proposal is to name both horizontal and vertical lines in a 2D-plane by two kinds of nominals. This is a generalization of the idea of naming a point in one-dimensional hybrid logic. I give an axiomatization of the proposed two-dimensional hybrid logic and show that it enjoys a general completeness result (called pure completeness) with respect to product Kripke frames. Moreover, in order to capture $T\times W$-frames studied by R.H. Thomason (1984), I introduce the notion of a dependent product frame, which enables us to represent the dependence of space over time. I also give a complete axiomatization of this dependent two-dimensional hybrid logic, and, as a corollary, reveal that a hybridization of $T\times W$-logic enjoys strong completeness.     

On Dynamic Topological and Metric Logics

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f² (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q ⁺. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.     

Gentzen Calculi for the Existence Predicate

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.     

Identical Twins,Deduction Theorems,and Pattern Functions: Exploring the Implicative BCSK Fragment of S5

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives.     

Expressivity of Second Order Propositional Modal Logic

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.     

Some Embedding Theorems for Conditional Logic

We prove some embedding theorems for classical conditional logic, covering ‘finitely cumulative’ logics, ‘preferential’ logics and what we call ‘semi-monotonic’ logics. Technical tools called ‘partial frames’ and ‘frame morphisms’ in the context of neighborhood semantics are used in the proof.     

A general method for proving decidability of intuitionistic modal logics

We generalise the result of [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34] on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in monadic second order logic. In [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34], such constraints apply to one relation at a time. We modify their proof to obtain decidability for constraints involving several relations. Now we can use this result to prove decidability of multi-modal modal logics where conditions on accessibility relations involve more than one relation. Our main application is intuitionistic modal logic, where the intuitionistic and modal accessibility relations usually interact in a non-trivial way.     

Pritchard Versus Pritchard on Luck

This paper argues for a particular account of luck by comparing two distinct versions of the modal account of luck that have been provided by Duncan Pritchard (2005, 2014). More specifically, it argues that there are three respects in which Pritchard’s earlier modal account of luck is preferable to his later account: it accounts better for the fact that luck comes in degrees, it includes a significance condition, and it better acknowledges the subjective nature of luck. The paper then discusses two consequences of the points it makes for epistemology: an alleged pragmatic encroachment, and a particular view on the relation between knowledge, luck, and justification.