We make the following contributions to modal logics with propositional quantifiers and modal logics with comparative operators in this dissertation:

- We define a general notion of normal modal logics with propositional quantifiers. We call them normal Π-logics. Then, as was done by Scrogg's theorem on extensions of the modal logic S5, we study in general the normal Π-logics extending S5. We show that they are all complete with respect to their algebraic semantics based on complete simple monadic algebras. We also show that the lattice formed by these logics is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N with the natural order topology. Further, we show how to determine the computability of normal Pi-logics extending S5Π. A corollary is that they can be of arbitrarily high Turing-degree.

- Regarding the normal Π-logics extending the modal logic KD45, we identify two important axioms: Immod: □∀p(□p → p) and 4∀: ∀p□φ→□∀p□φ. We argue that when □ is interpreted as the belief operator, we should not accept Immod in the logic of belief while 4∀ is desirable in settings with full introspection. Then we provide algebraic semantics based on complete well-connected pseudo monadic algebras for Box and ∀p and show that with respect to these algebras, normal Π-logics extending KD45 and 4∀ with a finite list of formulas are complete. We also give a general completeness theorem for atomic complete well-connected pseudo monadic algebras and a sufficient condition for the decidability of logics obtained by classes of these algebras, atomic or not. A special case of these general theorems is that the normal Π-logic of serial, transitive, and Euclidean Kripke frames, that is, the Kripke frames validating KD45, is axiomatized by KD45, 4∀, Immod, and ∃p(p∧∀q(q→□(p→q))) together with the usual axioms and rules for propositional quantifiers, and this logic is decidable.

Other than completeness and decidability, we also show that 4∀ is not in the smallest normal Π-logic extending KD45, using a countermodel based on a possible-world frame with propositional contingency, and that 4∀ is valid in any complete Boolean algebra expansion validating KD45.

- For modal logics with comparative operators, we first provide an axiomatization of the logic of comparing the cardinality of sets, as defined by Cantor. The main technical contribution is the observation that in a purely comparative language, we can define finiteness well enough so that an axiomatization can be done by combining the logic of comparative cardinality for finite sets and the logic of comparative cardinality for infinite sets. Note that these two logics are very different: the former contains the axiom of qualitative additivity: |A| ≥ |B| iff |A \ B| ≥ |B \ A| but not the axiom of absorption: if |A| ≥ |B| and |A| ≥ |C| then |A| ≥ |B ∪ C|, while the latter does the opposite.

- Then we consider modal logics for comparative imprecise probability. In these logics, comparisons are made according to a set of probability measures and can be intuitively read as either "at least as likely as" (symbolized by ≥) or "more likely than" (symbolized by >). We first disambiguate two interpretations of "more likely than" based on a set of probability measures and show that the stronger interpretation is not definable from "at least as likely as" while the weaker sense is. Then, we go on to axiomatize the logic of imprecise probability in a sequence of languages obtained by adding to the language with just "at least as likely as", one by one, the comparative operator > for "more likely than" (in the stronger sense), a unary operator ♢ for "possibly", and a binary operator <φ>ψ for "possibly φ, and after learning the truth of φ, ψ''. We also comment on the expressivity of these languages and the decidability of the logics in these languages. In particular, we show that many distinctive features of the imprecise probability approach of representing the doxastic states of agents, such as the problem of dilation, are observable at this purely comparative level. Finally, we add a pair of operators Ipφ and Ip-φ, intuitively read, respectively, as "after learning the existence of an actually true new proposition, now named by p, φ", and "after learning the existence of an actually false new proposition, now named by p, φ". We show that this pair of operators allow us to formalize a common kind of information dynamics and will boost the expressivity of the language to a quantitative level.

This dissertation consists of four largely independent chapters. The first two chaptersconcern counterparts of classical theorems in modal logic in more general semantics: the Sahlqvist Correspondence Theorem inter alia for possibility semantics in Chapter 1 and the Goldblatt-Thomason Theorem and Fine’s Canonicity Theorem for neighborhood semantics in Chapter 2. Chapter 3 contains various results on Heyting algebras, among which is the topological-dynamical study of the automorphism group of the smallest existentially closed Heyting algebra. The last chapter establishes choice-free duality between the category of ortholattices and a category of certain spectral spaces.

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures.

This dissertation explores an alternative to such representations known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices.

The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-contructive dualities via Vietoris constructions.

The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint.

In the 1941 edition of Introduction to Logic and to the Methodology of the DeductiveSciences Alfred Tarski laments that

[T]he methodology of empirical sciences constitutes an important domainof scientific research. The knowledge of logic is of course valuable in the study of this methodology, as it is in the case of any other discipline. It must be admitted, however, that up to the present, logical concepts and methods have not found any specific or fertile applications in this domain.

This dissertation aims to partially realize Tarski’s project of applying mathematicallogic to questions beyond the scope of pure mathematics through an investigation of the relationship between a theory and the evidence that (partially) confirms or refutes it. It is constituted by three projects: On Falsification, On Rational Jurisprudence, and On the Sufficiency of First-Order Logic.

The first major chapter, On Falsification, is concerned with the question of when atheory is refutable with certainty on the basis of sequence of primitive observations. Beginning with the simple definition of falsifiability as the ability to be refuted by some finite collection of observations, I assess the literature on falsification and its descendants along the lines of its static and dynamic components. The static case is broadly concerned with the question of how much of a theory can be subjected to falsifying experiments. In much of the literature, this question is tied up with whether the theory in question is axiomatizable by a collection of universal first-order sentences. I argue that this is too narrow a conception of falsification by demonstrating that a natural class of theories of distinct model-theoretic interest—so-called NIP theories—are themselves highly falsifiable. The dynamic case, by contrast, is concerned with the question of how falsifiable a single proposition is in the short and long run. Formal Learning theorists such as Schulte and Juhl have argued that long-run falsifiability is characterized by the topological notion of nowhere density in a suitable topological space. I argue that the short-run falsifiability of a hypothesis is in turn characterized by the VC finiteness of the hypothesis. Crucially, VC finite hypotheses correspond precisely to definable sets in NIP structures, pointing to a robust interplay between the static and dynamic cases of falsification. Finally, I end the chapter by giving rigorous foundations for Mayo’s conception of severe testing by way of a combinatorial, non-probabilistic notion of surprise. VC finite hypotheses again appear as the hypotheses with guaranteed short-run surprise bounds. Therefore, NIP theories and VC finite hypotheses capture the notion of short-run falsifiability.

The next chapter, On Rational Jurisprudence, is concerned with the epistemic question of confirming a hypothesis—the guilt of a defendant—by way of testimony heardby a juror over the course of an American-style criminal trial. In it, I attempt to settle a dispute between two strands of the legal community over the issue of whether the methods of Bayesian rationality are incompatible with jurisprudential principles such as the Presumption of Innocence. To this end, I prove a representation theorem that shows that so long as a juror would not convict the defendant having heard no testimony (the Presumption of Innocence) but would convict upon hearing some collection of testimony (Willingness to Convict), then this juror’s disposition to convict the defendant is representable as the disposition of a Bayesian threshold juror in Posner’s sense. This result indicates that relevant notion of a Bayesian threshold juror is insufficiently specified to render this debate a substantive one.

Finally, On the Sufficiency of First-Order Logic is concerned with the limits of soundinference. The starting point of this chapter is a reflection on a principle that Barwise terms the First-Order Thesis, namely that “logic is first-order logic, so that anything that cannot be defined in first-order logic is outside the domain of logic.” Barwise was chiefly concerned with the relative inexpressiveness of First-Order Logic. Despite this, I argue that First-Order Logic—while not the most expressive abstract logic—is sufficient to represent and carry out any inference a finitistic agent might carry out. The main mathematical result here is the Σ1 completeness of the consequence relation |=FO of First-Order logic as a Turing functional. A consequence of Σ1 completeness is that any notion of logical consequence which is machine verifiable given an oracle naming the theory Γ is in fact able to be internalized into a First-Order proof system. While the translation function will generally not preserve semantics on the nose—after all, First-Order Logic is not particularly expressive—the inferential structure of such a system is captured by a first-order system by way of a computable translation function. I then consider a recent argument of Warren’s that agents like us in nearby possible worlds can implement the ω rule of inference, a sound but non-recursive pattern of inference. While I do not refute his premise directly, I do argue that there is a clear finitistic interpretation Warren’s purported example ω rule, saving the plausibility of the position that agents like ourselves are only capable of performing finitary inferences. I conclude the chapter by reflecting on Hilbert’s Thesis, a position constituted by two related theses:

1. Hilbert’s Expressibility Thesis (HET): All mathematical (extra-logical) assumptions may be expressed in first-order logic, and

2. Hilbert’s Provability Thesis (HPT): The informal notion of provable is madeprecise by the formal notion of provable in first-order logic.

Kripke has argued that(HET) + (HPT) ⇒ Church’s Thesis on the basis of G ̈odel’s Completeness Theorem. The results of this chapter indicate a partial converse. The Σ1 completeness of |=FO shows that Church’s Thesis + In-Principle Machine Verifiability of Proofs ⇒ (HPT). First-Order Logic is neither the only nor most expressive logic with Σ1 complete entailment relation, but the above argument shows that First-Order Logic is sufficiently expressive to simulate any machine-verifiable inferential system.

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power. In doing so, I show how these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language.

When we attempt to describe the world, we are trying to distinguish the way things are from all the many ways things could have been—in other words, we are trying to locate ourselves within a region of logical space. According to this picture, languages can be thought of as ways of carving logical space or, more formally, as maps from sentences to classes of models. For example, the language of first-order logic is just a mapping from first-order formulas to model-assignment pairs that satisfy those formulas. Almost all formal languages discussed in metaphysics and logic, as well as many of those discussed in natural language semantics, can be characterized in this way.

Using this picture of language, I analyze two different approaches to defining expressive power, each of which is motivated by different roles a language can play in a debate. One role a language can play is to divide and organize a shared conception of logical space. If two languages share the same conception of logical space (i.e., are defined over the same class of models), then one can compare the expressive power of these languages by comparing how finely they carve logical space. This is the approach commonly employed, for instance, in debates over tense and modality, such as the primitivism-reductionism debate.

But a second role languages can play in a debate is to advance a conception or theory of logical space itself. For example, consider the debate between perdurantism, which claims that objects persist through time by having temporal parts located throughout that time, and endurantism, which claims that objects persist through time by being wholly present at that time. A natural thought about this debate is that perdurantism and endurantism are simply alternative but equally good descriptions of the world rather than competing theories. Whenever the endurantist says, for instance, that an object is red at time t, the perdurantist can say that the object’s temporal part at t is red. On this view, one should conceive of perdurantism and endurantism not as theories picking out disjoint regions of logical space, but as theories offering alternative conceptions of logical space: one in which persistence through time is analogous to location in space and one in which it is not. A similar distinction applies to other metaphysical debates, such as the mereological debate between universalism and nihilism.

If two theories propose incommensurable conceptions of logical space, we can still compare their expressive power utilizing the notion of a translation, which acts as a correlation between points in logical space that preserves the language’s inferential connections. I build a formal theory of translation that explores different ways of making this notion precise. I then apply this theory to two metaphysical debates, viz., the debate over whether composite objects exist and the debate over how objects persist through time. This allows us to get a clearer picture of the sense in which these debates can be viewed as genuine.

Subjective Bayesianism and Humean decision theory are dominant as both prescriptive and descriptive accounts of reasoning and rational decision-making. A subsequent genre of work within these paradigms acknowledges that the basic theories suffer from certain limitations or unexplained paradoxes, then seeks to modify them so as to remedy the defect. I develop arguments both against the original, "pure" paradigms and against certain attempts to correct them, making a case for a pluralist account of knowledge and decision-making.