Isaac Wilhelm

What, ontologically speaking, are best system laws? Should the laws of the best system account be identified with sentences, or propositions, or perhaps something else? These questions differ from more familiar questions about which criteria we should use to pick out the best system laws: regardless of whether laws are picked out using simplicity, informativeness, tractability, predictability, perfect naturalness, or whatever, there remains the question of what, ontologically, the best system laws are. But very little has been written on this issue; and what has been written is often ambiguous. So in this paper, we present and evaluate three ontological options for best system laws: taking them to be sentences, unstructured propositions, and structured propositions. Surprisingly, the two most cited options—sentences and unstructured propositions—face significant problems. Defenders of the best system account might want to take seriously the idea that best system laws are structured propositions.

I use typicality to formulate and defend a functional characterization of chance. According to the characterization, called ‘Typicality-Based Chance’, to be a chance is to be typically approximated by possible frequencies, where the details of the approximation are expressed by the law of large numbers. Among its other virtues, Typicality-Based Chance supports an attractively empiricist answer to a persistent question in the literature—often raised by those who view typicality with suspicion—concerning what justifies the mathematical measures used to express facts about typicality.

Mathematical measures are often used to express facts about what is typical — about what the vast majority of cases, in other words, are like. Many different measures can be used to express typicality facts, however. So exactly what justifies choosing one measure over another? In this paper, I propose an empiricist answer to this question. Roughly put, the answer only appeals to observations of frequencies, along with the standard scientific assumption that the results of experiments are not quasi-miraculous, atypical flukes.

Many metaphysical theories of identity, existence, and so on, are formulated using higher-order languages like the simply typed lambda calculus. But as I argue, for the purposes of metaphysical theorizing, a different language would be better: the calculus of constructions. Since this language—like many pure type systems—allows for quantification over types, it is preferable to the languages currently being used in the philosophical literature. For the purposes of metaphysical theorizing, in other words, the calculus of constructions is the better language.

In this paper, I present several philosophical applications of pure type systems. As I explain, many pure type systems are more expressively powerful than the higher-order languages—like the simply typed lambda calculus—on which philosophers have focused. Consequently, pure type systems support the formulation of more general, and more unified, metaphysical theories. To illustrate this, I use pure type systems to formulate some extremely general principles of grounding: these principles describe ground-theoretic connections among items drawn from across the hierarchy of types. I also present both informal theories, and formally rigorous theories, of what pure type systems say about the world.

An introductory textbook, Logic for Justice covers, in full detail, the language and semantics of both propositional logic and first-order logic. It motivates the study of those logical systems by drawing on social and political issues. Basically, Logic for Justice frames propositional logic and first-order logic as two theories of the distinction between good arguments and bad arguments. And the book explains why, for the purposes of social justice and political reform, we need theories of that distinction. In addition, Logic for Justice is extremely lucid, thorough, and clear. It explains, and motivates, many different features of the formalism of propositional logic and first-order logic, always connecting those features back to real-world issues.

What sorts of aims, or goals, are constitutive of science? How does scientific evidence relate to the knowledge that science produces? And can the No Miracles Argument for scientific realism be defended against concerns about the explanatory capacity of truth? In Knowing Science, Bird engages with questions like these at length. In this paper, I engage with these questions too. I raise some concerns for the view that aiming at knowledge is constitutive of science. I provide three counterexamples to an epistemic principle which Bird uses to argue for the view that evidence is knowledge. And I formulate a version of the No Miracles Argument which, by basing realism on a particular approach to meaning and to the empirical science of linguistics, avoids any mention of truth whatsoever.

It is often assumed that theorizing in terms of natural properties is more objectively valuable than theorizing in terms of non-natural properties. But this assumption faces an explanatory challenge: explain the greater objective value of theorizing in terms of natural properties. In this paper, I answer that challenge by proposing and exploring three different accounts of the objective value of naturalness. Two appeal to constitutive natures: it is part of the constitutive nature of explanation, or of objective value, that theorizing in terms of natural properties is more objectively valuable. The third appeals to the theoretical role that naturalness plays.

In this chapter, we explore some ways in which reasoning based on mysticism can be rational, focusing on the episode “The Fortuneteller,” in which Aang, Katara, and Sokka save a village from a volcanic eruption. Throughout this episode, Sokka advocates a purely empirical approach to reasoning. The villagers, however, believe that no source of knowledge is more reliable than Aunt Wu, the local fortuneteller. At several points in the episode, Sokka claims that the villagers’ reliance on Aunt Wu is irrational. The villagers claim otherwise: since Aunt Wu has never led them astray, it is rational to rely on her. As we will see, the villagers are right. Their approach to reasoning – based on Aunt Wu’s fortunetelling – is more rational than it initially seems to be. Roughly put, despite their belief in the mystical, the villagers behave in accord with a standard philosophical theory of rational decision-making, a theory of what makes some decisions rational and other decisions not. So Sokka is wrong to claim that the villagers’ reliance on Aunt Wu is irrational. The villagers behave quite rationally.

If a proposition is typically true, given your evidence, then you should believe that proposition; or so I argue here. In particular, in this paper, I propose and defend a principle of rationality---call it the `Typical Principle'---which links rational belief to facts about what is typical. As I show, this principle avoids several problems that other, seemingly similar principles face. And as I show, in many cases, this principle implies the verdicts of the Principal Principle: so ultimately, the Typical Principle may be the more fundamental of the two.

According to the Best System Account of lawhood, laws of nature are theorems of the deductive systems that best balance simplicity and strength. In this paper, I advocate a different account of lawhood which is related, in spirit, to the BSA: according to my account, laws are theorems of deductive systems that best balance simplicity, strength, and also calculational tractability. I discuss two problems that the BSA faces, and I show that my account solves them. I also use my account to illuminate the nomological character of special science laws.

I formulate a theory of groups based on pluralities and counterparts: roughly put, a group is a plurality of entities at a time. This theory comes with counterpart-theoretic semantics for modal and temporal sentences about groups. So this theory of groups is akin to the stage theory of material objects: both take the items they analyze to exist at a single time, and both use counterparts to satisfy certain conditions relating to the modal properties, temporal properties, and coincidence properties of those items.

I explore the relationship between a prominent analysis of intrinsic properties, due to Langton and Lewis, and the phenomenon of quantum entanglement. As I argue, the analysis faces a puzzle. The full analysis classifies certain properties of entangled particles as intrinsic. But when combined with an extremely plausible assumption about duplication, the main part of the analysis classifies those properties as non-intrinsic instead. I conclude that much of Lewis’s metaphysics is in trouble: Lewis based many of his metaphysical views—his thesis of Humean supervenience, for instance, and his account of recombination—on an analysis of intrinsicality which does not sit well with quantum phenomena.

Philosophers have developed several detailed accounts of what makes some mathematical proofs explanatory. Significantly less attention has been paid, however, to what makes some proofs more explanatory than other proofs. That is problematic, since the reasons for thinking that some proofs explain are also reasons for thinking that some proofs are more explanatory than others. So in this paper, I develop an account of comparative explanation in mathematics. I propose a theory of the `at least as explanatory as' relation among mathematical proofs.

A popular method for comparing the structures of mathematical objects, which I call the ‘subset approach’, says that X has more structure than Y just in case X’s automorphisms form a proper subset of Y’s automorphisms. This approach is attractive, in part, because it seems to yield the right results in some comparisons of spacetime structure. But as I show, it yields the wrong results in a number of other cases. The problem is that the subset approach compares structure using automorphism sets. So a different approach is needed: structure should be compared using automorphism groups.

I show that centered propositions—also called de se propositions, and usually modeled as sets of centered worlds—pose a serious problem for various versions of Lewis's Principal Principle. The problem, put roughly, is that in scenarios like Elga's `Sleeping Beauty' case, those principles imply that rational agents ought to have obviously irrational credences. To solve the problem, I propose a centered version of the Principal Principle. My version allows centered propositions to be objectively chancy.