General information in relevant logic

This paper sets out a philosophical interpretation of the model theory of Mares and Goldblatt (The Journal of Symbolic Logic 71, 2006). This interpretation distinguishes between truth conditions and information conditions. Whereas the usual Tarskian truth condition holds for universally quantified statements, their information condition is quite different. The information condition utilizes general propositions. The present paper gives a philosophical explanation of general propositions and argues that these are needed to give an adequate theory of general information.     

On Permutation in Simplified Semantics

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993) concerning the modelling conditions for the axioms of assertion A → ((A → B) → B) (there called c6) and permutation (A → (B → C)) → (B → (A → C)) (there called c7). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections.     

Weakly higher order cylindric algebras and finite axiomatization of the representables

We show that the variety of n-dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski’s quasi-projective relation algebras. Supported by the Hungarian National Foundation for Scientific Research grant T73601.     

The logic of how-questions

Philosophers and scientists are concerned with the why and the how of things. Questions like the following are so much grist for the philosopher’s and scientist’s mill: How can we be free and yet live in a deterministic universe?, How do neural processes give rise to conscious experience?, Why does conscious experience accompany certain physiological events at all?, How is a three-dimensional perception of depth generated by a pair of two-dimensional retinal images?. Since Belnap and Steel’s pioneering work on the logic of questions, Van Fraassen has managed to apply their approach in constructing an account of the logic of why-questions. Comparatively little, by contrast, has been written on the logic of how-questions despite the apparent centrality of questions such as How is it possible for us to be both free and determined? to philosophical enterprise.¹ In what follows I develop a logic for how-questions of various sorts including how-questions of cognitive resolution, how-questions of manner, how-questions of method, of means, and of mechanism.     

Some Supervaluation-based Consequence Relations

In this paper, we define some consequence relations based on supervaluation semantics for partial models, and we investigate their properties. For our main consequence relation, we show that natural versions of the following fail: upwards and downwards Lowenheim–Skolem, axiomatizability, and compactness. We also consider an alternate version for supervaluation semantics, and show both axiomatizability and compactness for the resulting consequence relation.     

Neighborhoods for Entailment

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the neighborhood semantics meets the Routley–Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable.     

Extended Quantum Logic

The concept of quantum logic is extended so that it covers a more general set of propositions that involve non-trivial probabilities. This structure is shown to be embedded into a multi-modal framework, which has desirable logical properties such as an axiomatization, the finite model property and decidability.     

Deontic Interpreted Systems

We investigate an extension of the formalism of interpreted systems by Halpern and colleagues to model the correct behaviour of agents. The semantical model allows for the representation and reasoning about states of correct and incorrect functioning behaviour of the agents, and of the system as a whole. We axiomatise this semantic class by mapping it into a suitable class of Kripke models. The resulting logic, KD45_n ^i-j, is a stronger version of KD, the system often referred to as Standard Deontic Logic. We extend this formal framework to include the standard epistemic notions defined on interpreted systems, and introduce a new doubly-indexed operator representing the knowledge that an agent would have if it operates under the assumption that a group of agents is functioning correctly. We discuss these issues both theoretically and in terms of applications, and present further directions of work.