What does nihilism tell us about modal logic?

Philosophical Studies https://doi.org/10.1007/s11098-024-02166-y What does nihilism tell us about modal logic? Christopher James Masterman1 Accepted: 5 May 2024 © The Author(s) 2024 Abstract Brauer (Philos Stud 179:2751–2763, https://doi.org/10.1007/s11098-022-01793-7, 2022) has recently argued that if it is possible that there is nothing, then the correct modal logic for metaphysical modality cannot include D. Here, I argue that Brauer’s argument is unsuccessful; or at the very least significantly weaker than presented. First, I outline a simple argument for why it is not possible that there is nothing. I note that this argument has a well-known solution involving the distinction between truth in and truth at a possible world. However, I then argue that once the semantics presupposed by Brauer’s argument is reformulated in terms of truth at a world, we have good reasons to think that a crucial semantic premise in Brauer’s argument should be rejected in favour of an alternative. Brauer’s argument is, however, no longer valid with this alternative premise. Thus, plausibly Brauer’s argument against D is only valid, if it is not sound. Keywords Nihilism · Modal logic · Possible worlds · Truth at a world · Propositions 1 Introduction Let metaphysical nihilism be the view that absolutely nothing exists—no concrete objects, properties, propositions, states of affairs, and so on. According to Brauer (2022), this view is false, but it could have been true. In fact, Brauer argues that the very possibility of this view has startling consequences for the logic of metaphysical modality: if metaphysical nihilism is even possible, then the correct modal logic for metaphysical modality cannot be any stronger than the logic D. Here, I argue that Brauer’s argument is unsuccessful; or at least significantly weaker than presented. In short, I argue that if we take care to formulate our modal semantics, then we have good reasons to think that Brauer’s argument is valid only if it is unsound. * Christopher James Masterman 1 Faculty of Philosophy, University of Cambridge Raised Faculty Building, Sidgwick Avenue, Cambridge CB3 9DA, UK 13 Vol.:(0123456789) C. J. Masterman Here’s how I proceed. First, I outline Brauer’s (2022) argument as it is presented (Sect. 2). I then present some simple arguments for why, contrary to Brauer’s claims, metaphysical nihilism is not possible (Sect. 3). I show that these arguments most plausibly suggest that we should carefully formulate our modal semantics in terms of the notion of truth at a world, not truth in a world (Sect. 4). Therefore, I reformulate Brauer’s argument in terms of this notion (Sect. 5), but then argue that the reformulated argument is plausibly not sound—I outline a comparative case for why we should accept a competing semantic premise governing possibility over the crucial semantic premise in Brauer’s argument (Sects. 6–8). 2 Brauer’s argument against D First, I present Brauer’s argument against D and some preliminary principles presupposed in his argument. Brauer begins by considering the following claim. Nothing exists. (N) (N) is not true; but Brauer maintains that (N) could have been; or at least, the possibility of (N) is a live enough one that we should be interested in what follows from its possibility (2022, 2751). Brauer names the claim that (N) is possible: It is metaphysically possible that nothing exists. (PN) The core of Brauer’s paper then purports to establish that if (PN) holds, the correct logic for metaphysical modality cannot be as strong as D. What then is the logic D? The language of D is the language of propositional modal logic (L𝜌) which we define recursively as follows from a countably infinite stock of propositional letters p1 , p2 , p3 , …, with the usual abbreviations for the logical connectives ∧, →, and ↔. 𝜙 ∶= pi | ¬pi | pi ∨ pj | □pi | ◊pi We define the logic D as an extension of K. The logic K consists of the set of theorems (⊢ 𝜙) resulting from the following axiomatic base and rules of inference, where 𝜙 and 𝜓 are both well-formed formulae of L𝜌. (PL) (MP) (NR) (K) (OP) If 𝜙 is a tautology, then ⊢ 𝜙. If ⊢ 𝜙 and ⊢ 𝜙 → 𝜓 , then ⊢ 𝜓 . If ⊢ 𝜙, then ⊢ □𝜙. □(𝜙 → 𝜓) → (□𝜙 → □𝜓) is an axiom. □𝜙 ↔ ¬◊¬𝜙 is an axiom. D extends K with the following axiom scheme. (D) □𝜙 → ◊𝜙 is an axiom. 13 What does nihilism tell us about modal logic? That is, the logic D is defined here to be set of theorems resulting from the axioms of K and (D) by zero or more applications of the inference rules of K. How does Brauer shows that if (PN) is true, D is not the correct logic for metaphysical modality? There are two crucial ideas at play. First, I take it that Brauer’s argument is assuming, as I think is correct, that a necessary condition for a modal logic being correct for metaphysical modality is that the theorems of such a logic should be true simpliciter, provided the modal operators are interpreted in terms of metaphysical modality, the logical connectives are interpreted as standard, and we uniformly replace sentences of the formal language with sentences of natural language. Let’s call such a uniform substitution and interpretation of a theorem a metaphysical instance, or instance for short, e.g., □ (Sally is a cat) → ◊ (Sally is a cat) is an instance of the distinctive axiom—and theorem—of D, whereas □(Sally is a cat) → ◊ (Sally is fat) is not—in the latter case, 𝜙 is not uniformly replaced in the theorem. If D is the correct logic for metaphysical modality, then the former is true. Of course, we should also want it that if D is the correct logic for metaphysical modality, then the former is logically true, see (Salmon, 1989, 29). However, for our purposes we focus only on the following simpler necessary condition for a logic L being correct for metaphysical modality. If L is correct for metaphysical modality, then any instance of an L-theorem is true (L) I take it that little of this needs much motivation—if certain theorems of some logic L are false when appropriately interpreted, then L cannot be the correct logic for the modality involved in that interpretation. The second crucial part of Brauer’s argument against D is showing that if (PN) is true, there are false instances of theorems of D and thus, by (L), D cannot be the correct logic for metaphysical modality. Adopting a possible worlds semantics for metaphysical modality in which possible worlds are ersatzist maximally consistent states of affairs (2022, 2752), Brauer notes that it’s possible that nothing exists, i.e., (PN) is true, only if there is a consistent state of affairs S and it is true relative to S that (N). Brauer then considers what must also be true in S: Since nothing exists, there cannot be any states of affairs, and in particular, there cannot be any maximal consistent states of affairs. That is, in S it is true that there are no possible worlds, and thus nothing is possible (Brauer, 2022, 2754) 13 C. J. Masterman I (...truncated)