Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (shrink)

It is demonstrated that Hamblin's 15 tense theorem does not only hold for temporal logic with linear time but also for branching time. Furthermore three other theorems with finitely many tenses are shown.

Based on a notion of "companions to stit formulas" applied in other papers dealing with astit logics, we introduce "choice formulas" and "nested choice formulas" to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of dstit with multiple (...) agents each of which corresponds to a semantic condition concerning the number of possible choices for agents. (shrink)

This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gö;del-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting-Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok-Esakia-Theorem is proved for this embedding.

We study a class of formulas generalizing the weak law of the excluded middle and provide a characterization of these formulas in terms of Kripke frames and Brouwer algebras. We use these formulas to separate logics corresponding to factors of the Medvedev lattice.

This chapter contains sections titled: Introduction Modal propositional Logics (MPLs) Modal Quantificational Logics(QMLs).

We present our personal view on W.J. Blok's contribution to modal logic.

Several philosophers—including C. S. Peirce, William James, Hilary Putnam and Crispin Wright—have proposed various versions of the notion that truth is an epistemic ideal. More specifically, they have held that a proposition is true if and only if it can be fixedly warranted by human inquirers, given certain ideal epistemic conditions. This paper offers a general critique of that idea, modeling conceptions of ideality and fixed warrant within the semantics that Kripke developed for intuitionistic logic. It is shown that each (...) of the two plausible notions of fixed warrant faces difficulties and that, moreover, “truth” defined in terms of either of them is distressingly dependent upon one’s conception of idealized inquiry and perhaps also upon one’s standards of warrant. (shrink)

What an intuitionist may refer to with respect to a given epistemic state depends not only on that epistemic state itself but on whether it is viewed concurrently from within, in the hindsight of some later state, or ideally from a standpoint “beyond” all epistemic states (though the latter perspective is no longer strictly intuitionistic). Each of these three perspectives has a different—and, in the last two cases, a novel—logic and semantics. This paper explains these logics and their semantics and (...) provides soundness and completeness proofs. It provides, moreover, a critique of some common versions of Kripke semantics for intuitionistic logic and suggests ways of modifying them to take account of the perspective-relativity of reference. (shrink)

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...) methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. (shrink)

There are many parallels between the role of possible worlds in modal logic and that of times in tense logic. But the similarities only go so far, and it is important to note where the two come apart. This paper argues that even though worlds and times play similar roles in the model theories of modal and tense logic, there is no tense analogue of the possible-worlds analysis of modal operators. An important corollary of this result is that presentism cannot (...) be the tense analogue of actualism. (shrink)

This paper examines the logic of fictions within fictions. I argue that consistently nested consistent fictions must have certain formal characteristics. The most important is that they form a tree structure. Depending on one’s theory of fictional objects, additional constraints may apply regarding the appearance of a fictional object in two or more fictional universes. The background motivation for the paper is to use iterated fiction operators as a tool for making sense of iterated modal operators; I conclude by noting (...) briefly where the results about nested fictions can, and where they cannot, be extended to nested possible worlds. (shrink)

It has been argued that actualism – the view that there are no non-actual objects – cannot deal adequately with statements involving iterated modality, because such claims require reference, either explicit or surreptitious, to non-actual objects. If so, actualists would have to reject the standard semantics for quantified modal logic (QML). In this paper I develop an account of modality which allows the actualist to make sense of iterated modal claims that are ostensibly about non-actual objects. Every occurrence of a (...) modal operator involves the stipulation of a possible world, and nested modal operators require stipulation of nested possible worlds. I provide an actualistically acceptable (AA) semantics for QML wherein the nesting relation is irreflexive and intransitive and forms a tree. Despite these restrictions, AA models can beshown to be sound and complete for a wide variety of modal logics. (shrink)

In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...) modal logic. At the conceptual and philosophical level, the translation provides a classical interpretation of the meaning of the logical operators of various non-distributive propositional calculi. Technically, it allows for an effortless transfer of results, such as compactness and the Löwenheim-Skolem property and it opens up new directions for deducing properties of substructural logic systems by establishing appropriate transfer theorems. (shrink)

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...) a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

The paper investigates interpretations of propositional and firstorder logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for (...) intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators. (shrink)

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...) to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given. (shrink)

The present PhD dissertation aims to examine the relation between modality and change in Aristotle’s metaphysics. -/- On the one hand, Aristotle supports his modal realism (i.e., worldly objects have modal properties - potentialities and essences - that ground the ascriptions of possibility and necessity) by arguing that the rejection of modal realism makes change inexplicable, or, worse, banishes it from the realm of reality. On the other hand, the Stagirite analyses processes by means of modal notions (‘change is the (...) actuality of what is virtual insofar as it is virtual’). In other words: to grasp what change is, one has to resort to the modal idiom of potentialities, while the fact that there is change is indicative of the fact that nature is full of modal properties. -/- Aristotle’s modal and kinetic realism finds a negative in the figure of the Megaric Diodorus Kronus. The polemical situation of Greek philosophy has indeed the dialectical advantage of not opposing the Aristotelian position to a sui generis straw man. Both in its reduction of modal judgements to temporal quantifications and in its dissolution of the reality of the state-of-being-in-motion in favour of a cinematographic view of processes, the philosophical figure of reductionist antirealism embodied by Diodorus constitutes an alternative that takes the opposite view of Aristotle’s realism. -/- The present study is structured as a discussion between these two positions. In the face of Diodorus’ antirealist challenges, Aristotle articulates modal and kinetic considerations: the answers he gives to Diodorus’ puzzles thus provide valuable insight into his metaphysics. Moreover, the examination of Aristotle’s and Diodorus’ metaphysics, because of their insight and originality, will not fail to interest the philosopher concerned with the metaphysical foundations of modern physics, insofar as the entanglement between modalities and processes is nowadays at the core of mechanics (phase spaces, path integrals, etc.). (shrink)