In this paper we study the admissible rules of intermediate logics. We establish some general results on extension of models and sets of formulas. ese general results are then employed to provide a basis for the admissible rules of the Gabbay-de Jongh logics and to show that these logics have finitary unification type.

Rybakov (1984a) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility. * Support by the Netherlands Organisation for Scientific Research under grant 639.032.918 is gratefully acknowledged.

Since the seminal work of Yarbus, multiple studies have demonstrated the influence of task-set on oculomotor behavior and the current cognitive state. In more recent years, this field of research has expanded by evaluating the costs of abruptly switching between such different tasks. At the same time, the field of classifying oculomotor behavior has been moving toward more advanced, data-driven methods of decoding data. For the current study, we used a large dataset compiled over multiple experiments and implemented separate state-of-the-art machine learning methods for decoding both cognitive state and task-switching. We found that, by extracting a wide range of oculomotor features, we were able to implement robust classifier models for decoding both cognitive state and task-switching. Our decoding performance highlights the feasibility of this approach, even invariant of image statistics. Additionally, we present a feature ranking for both models, indicating the relative magnitude of different oculomotor features for both classifiers. These rankings indicate a separate set of important predictors for decoding each task, respectively. Finally, we discuss the implications of the current approach related to interpreting the decoding results.

Refutation systems are formal systems for inferring the falsity of formulae. These systems can, in particular, be used to syntactically characterise logics. In this paper, we explore the close connection between refutation systems and admissible rules.We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely, we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay-de Jongh logics. We show that this gives a characterisation of these logics in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev's logic.

Any intermediate logic with the disjunction property admits the Visser rules if and only if it has the extension property. This equivalence restricts nicely to the extension property up to n.In this paper we demonstrate that the same goes even when omitting the rule ex falso quod libet, that is, working over minimal rather than intuitionistic logic. We lay the groundwork for providing a basis of admissibility for minimal logic, and tie the admissibility of the Mints-Skura rule to the extension property in a stratified manner.Keywords: admissible rules, minimal logic, disjunction property, extensions of Kripke modelsThe admissible rules of a theory are those rules under which the theory is closed. Derivable rules are admissible. For classical propositional logic, this is the whole story. For intuitionistic propositional logic (IPC) -and minimal logic -it is not.Friedman (1975, Problem 40) conjectured admissibility for IPC to be decidable, as has been confirmed by Rybakov (1984). De Jongh and Visser conjectured that the Visser rules form a basis of admissibility for IPC, that is to say, all admissible rules of IPC become derivable after adjoining the Visser rules. Rozière (1992) and Iemhoff (2001b) independently confirmed this. Again independently, Skura (1989) demonstrated that IPC is the sole intermediate logic that admits a restricted form of the Visser rules.At the Pisa Proof Theory workshop of 2012 George Metcalfe gave a tutorial on admissible rules. As has become standard practice, Metcalfe mentioned Lorenzen (1955) as the first place where admissible rules where studied an sich. Jan von Plato objected that Johansson (1937) already discussed them. Odintsov and Rybakov (2012) proved admissibility for minimal logic to be decidable. In this paper we lay the groundwork for studying all admissible rules of Johansson's minimal logic, with the eventual goal of providing an explicit basis of admissibility. This paper aims to provide uniformity to some of the literature regarding admissible rules for logics above minimal logic. We make several observations, many of which not elsewhere available in the generality stated here. Although this paper contains novel results, most notably the semantic characterization of admissibility for an adaptation of the rules studied by Skura, its main purpose is to provide a unified approach to the study of admissible rules over minimal logic.