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We present twodimensional tableau systems for the actuality, fixedly, and uparrow operators. All systems are proved sound and complete with respect to a twodimensional semantics. In addition, a decision procedure for the actuality logics is discussed. 

In a recent paper, Brian Rabern suggests a semantics for languages with two kinds of modality, standard Kripkean metaphysical modality as well as epistemic modality. This semantics presents an alternative to twodimensionalism, which was developed in the last decades. Both Rabern’s semantics and twodimensionalism are subject to a puzzle that Chalmers and Rabern (2014) call the nesting problem. I will investigate how Rabern’s semantics answers this puzzle. 

Graeme Forbes (2011) raises some problems for twodimensional semantic theories. The problems concern nested environments: linguistic environments where sentences are nested under both modal and epistemic operators. Closely related problems involving nested environments have been raised by Scott Soames (2005) and Josh Dever (2007). Soames goes so far as to say that nested environments pose the “chief technical problem” for strong twodimensionalism. We call the problem of handling nested environments within twodimensional semantics “the nesting problem”. We show that the twodimensional (...) 

This article surveys recent developments in the epistemology of modality. 

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic twodimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic twodimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two (...) 

We consider a naturallanguage sentence that cannot be formally represented in a firstorder language for epistemic twodimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the firstorder language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal. 

We present a sound and complete Fitchstyle natural deduction system for an S5 modal logic containing an actuality operator, a diagonal necessity operator, and a diagonal possibility operator. The logic is twodimensional, where we evaluate sentences with respect to both an actual world (first dimension) and a world of evaluation (second dimension). The diagonal necessity operator behaves as a quantifier over every point on the diagonal between actual worlds and worlds of evaluation, while the diagonal possibility quantifies over some point (...) 

The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...) 

Some central epistemological notions are expressed by sentential operators O that entail the possibility of knowledge in the sense that 'Op' entails 'It is possible to know that p'. We call these modalepistemological notions. Using apriority and being in a position to know as case studies, we argue that the logics of modal epistemological notions are extremely weak. In particular, their logics are not normal and do not include any closure principles. 

In this paper we present tableau methods for twodimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain firstorder quantifiers with identity, and all the formulas in the language are doublyindexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate (...) 

A glass couldn't contain water unless it contained H2Omolecules. Likewise, a man couldn't be a bachelor unless he was unmarried. Now, the latter is what we would call a conceptual or analytical truth. It's also what we would call a priori. But it's hardly a conceptual or analytical truth that if a glass contains water, then it contains H2Omolecules. Neither is it a priori. The fact that water is composed of H2Omolecules was an empirical discovery made in the eighteenth century. (...) 