What Is the Correct Logic of Necessity, Actuality and Apriority?

Abstract

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 two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional 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 arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. §1. Introduction. Epistemic two-dimensional semantics as proposed by David Chalmers, e.g., in Chalmers (2004), provides an account of meaning that allows a possible world semantics of necessity as well as apriority. The notions of necessity and apriority intended here are those distinguished by Kripke (1972); the first is sometimes called metaphysical necessity. They can roughly be paraphrased by saying that necessary is what could not have failed to be the case, and a priori is what can be known in an a priori way. In Fritz (2013), a propositional modal logic with operators for necessity, actuality and apriority is defined which captures the relevant ideas of epistemic two-dimensional semantics. In particular, a class of relational structures is defined, and it is argued that it represents the evaluation of sentences according to epistemic two-dimensional semantics, and that therefore, the logic characterized by this class captures the relations of the three modalities according to epistemic two-dimensional semantics. Epistemic two-dimensional semantics is a controversial theory, and philosophers who do not want to commit themselves to it can't justify the correctness of this logic using its semantics. This raises the question whether there is a way of arguing for its correctness that is not based on epistemic two-dimensional semantics. The main aim of this paper is to outline such an argument.

The Logic of Sequence Frames

“…In particular, that such logics provide us with a logical analysis of metaphysical necessity, actuality, and a priori knowledge is now an important facet of prominent philosophical views in philosophy of language, epistemology, and metaphysics. Logics developed with the purpose of shedding light on a priori reasoning and its relation with the modal notions of necessity and actuality arguably originated with Davies & Humberstone (1980), and have recently been investigated by Restall (2012), Fritz (2013, 2014), Fusco (2020), and others. Furthermore, the semantic treatment of epistemic and indexical terms in the works of Evans (1979), Kaplan (1989), and Chalmers (2004, 2014) make essential use of elements present in two-dimensional modal logics, while Weatherson (2001) and Wehmeier (2013) adopt similar two-dimensional semantics to provide formal treatments of both subjunctive and indicative conditionals.…”

“…The goal of this paper is to generalize two-dimensional modal logics with actuality operators to any finite dimension . To be more precise, the logics investigated here are generalizations of the logic for epistemic two-dimensional semantics studied by Fritz (2014) under the name , which in a certain sense is the same logic as those investigated proof-theoretically in Restall (2012) and Lampert (2018)—although the logics in Lampert (2018) contain first-order quantifiers. The formal language defined here contains several modal operators, one for each dimension, including indexed boxes, , as well as actuality operators, .…”

“…There is, however, a subtle difference in the semantic clause for @ when we move to a full-blooded two-dimensional modal logic 6 . In this case, formulas are evaluated against pairs of possible worlds, , where w is taken intuitively as a counterfactual world and v as a world considered as actual (or an epistemic scenario , such as in Fritz (2014)), and so the set of evaluation points W in the frames can now be defined as the Cartesian product of some nonempty underlying set of worlds, say, S . This means that rather than having a single fixed actual world as a part of the frame, any world can be actual as long as it occupies the second coordinate of the pair of worlds relative to which a formula is evaluated.…”

“…That is, the actual world is held fixed while the counterfactual world varies. Additionally, two-dimensional modal languages are equipped with a diagonal necessity operator, interpreted intuitively as an apriority operator, which is sometimes taken as a primitive operator in the language (Restall, 2012; Fritz, 2014; Lampert, 2018), or defined in terms of other operators (Davies & Humberstone, 1980). Here we notate it as , and its corresponding accessibility relation will be , which in turn relates a pair to any pair such that .…”

Natural Deduction for Diagonal Operators

“…In addition to Davies and Humberstone's axiomatic system, there is a hypersequent system in Restall (2012) and an (equivalent) axiomatic system in Fritz (2013Fritz ( , 2014 designed to capture some of the core formal aspects of versions of twodimensional semantics. These systems involve the usual modal operators, a primitive diagonal (apriority) operator, and an actuality operator.…”