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Since Kripke, philosophers have distinguished a priori true statements from necessarily true ones. A statement is a priori true if its truth can be established before experience, and necessarily true if it could not have been false according to logical or metaphysical laws. This distinction can be captured formally using twodimensional semantics. There is a natural way to extend the notions of apriority and necessity so they can also apply to questions. Questions either can or cannot be resolved before experience, (...) 

This article surveys recent developments in the epistemology of modality. 

In order to block controversial predictions of 2D semantics (The Nesting Problem), Chalmers and Rabern (2014) propose adding an additional constriction called “the liveness constraint” in definitions of epistemic modals. Without this constraint, all scenarioworld pairs counterfactual to a scenarioworld pair considered as actual in a 2D matrix for a contingent a priori proposition ϕ appear problematic for 2D semantics. This is because, although it is false that ϕ in such pairs, it is a priori true that ϕ. I consider (...) 

This paper investigates and develops generalizations of twodimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional ndimensional modal logics and show them to be decidable by means of a systematic tableau construction. 

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 (...) 

According to the modal account of propositional apriority, a proposition is a priori if it is possible to know it with a priori justification. Assuming that modal truths are necessarily true and that there are contingent a priori truths, this account has the undesirable consequence that a proposition can be a priori in a world in which it is false. Epistemic twodimensionalism faces the same problem, since on its standard interpretation, it also entails that a priori propositions are necessarily a (...) 

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 (...) 

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, 210–224 2014) call the nesting problem. I will investigate how Rabern’s semantics answers this puzzle. 

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. 

Twodimensional semantics, which can represent the distinction between a priority and necessity, has wielded considerable influence in the philosophy of language. In this paper, I axiomatize the dagger operator of Stalnaker’s “Assertion” in the formal context of twodimensional modal logic. The language contains modalities of actuality, necessity, and a priority, but is also able to represent diagonalization, a conceptually important operation in a variety of contexts, including models of the relative a priori and a posteriori often appealed to Bayesian and (...) 

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 (...) 

Presenting the first comprehensive, indepth study of hyperintensionality, this book equips readers with the basic tools needed to appreciate some of current and future debates in the philosophy of language, semantics, and metaphysics. After introducing and explaining the major approaches to hyperintensionality found in the literature, the book tackles its systematic connections to normativity and offers some contributions to the current debates. The book offers undergraduate and graduate students an essential introduction to the topic, while also helping professionals in related (...) 

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. 

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 chapter provides a general overview of the issues surrounding socalled semantic monsters. In section 1, I outline the basics of Kaplan’s framework and spell out how and why the topic of “monsters” arises within that framework. In Section 2, I distinguish four notions of a monster that are discussed in the literature, and show why, although they can pull apart in different frameworks or with different assumptions, they all coincide within Kaplan’s framework. In Section 3, I discuss one notion (...) 

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. 

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 (...) 

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. (...) 