UC Berkeley

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.