Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy is (...) that, unlike the existing substantialism approach from Field (1980), the new strategy naturally generalizes to theories formulated in terms of state space. (shrink)

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified (...) only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages. (shrink)

Current orthodoxy in modal logic and metaphysics has it that actuality is non-contingent in the following sense: for all p, if actually, p, then necessarily, actually, p. Call this thesis (Actuality) Necessitism and its negation (Actuality) Contingentism. Thus, according to Contingentism, there is at least one proposition p which is actually true but which could have been actually false. In another paper, one of us (Glazier 2023) has recently defended Contingentism. The present paper explores the logic of actuality under Contingentism. (...) After informally introducing Contingentism and outlining what we take to be the main motivations for accepting it, we consider how contingent actuality should be taken to be. We argue that the nature, or essence, of actuality imposes certain limits on the extent to which actuality could have been different, and these then inform our view of the logic of actuality. Turning to the formal study of the logic of contingent actuality, we specify a formal language and a possible world semantics for that language and distinguish a number of natural notions of validity and consequence to which it gives rise. We present axiomatizations of the different resulting logics and examine the variety of iterated modalities in our systems. We establish soundness and completeness results. (shrink)

Modal Meinongianism provides the semantics of sentences involving intentional verbs Priest. To that end, Modal Meinongianism employs a pointed non-normal quantified modal logic model. Like earlier Meinongian views Modal Meinongianism has a characterisation principle, that claims that any condition whatsoever is satisfied by some object in some world. Recently, Everett has proposed an argument against QCP that, if successful, gives rise to problems identical to those Russell raised for Naïve Meinongianism, namely that it allows for true contradictions, and allows us (...) to define anything into existence. Everett claims that the ordinary meanings of “actual” license an inference pattern, such that if an object satisfies Actual A at some world, then that object satisfies A in the actual world. Given that actual world is the designated point of evaluation for truth simpliciter, QCP would fall prey to Russell’s criticisms. As opposed to Everett, I argue that, even if we grant Everett the assumption that “actual” is a modal indexical that rigidly refers to the actual world, it does not conform to the inference pattern above. This is because when an object satisfies Actual A at some world, this alters the assertoric force of “actual”, because “actual” is interpreted in the scope of some modal or intentional operator. I also explain that Everett’s proposed example carries existential commitment because the problematic noun-phrase occurs outside the scope of a modal or intentional operator. (shrink)