Deterministic Theories

TL;DR

The paper critiques current attempts to formalize determinism via possible-worlds and haecceitistic notions, arguing that a Carnapian explication grounded in formal model theory offers a clearer, physics-compatible account. By introducing formal criteria ($D1$–$D3$) based on duplications and isomorphisms, it shows how to assess determinism as a property of theories, not of metaphysical world-picking. The authors defend full determinism through precise formalism (notably $D3$) and analyze counterexamples (Belot, Hawthorne, Melia) as instances where the formal criteria depend on the chosen notion of duplications. They demonstrate that the hole argument and related debates in general relativity can be resolved within this framework, thereby fostering a rigorous dialogue between physics and metaphysics. The work emphasizes that interpretation is a formal, not purely semantic, matter and that a careful formal approach yields robust predictions about when theories are deterministic.

Abstract

Determinism is (roughly) the thesis that the past determines the future. But efforts to define it precisely have exposed deep methodological disagreements. Standard possible-worlds formulations of determinism presuppose an "agreement" relation between worlds,but this relation can be understood in multiple ways, none of which is particularly clear. We critically examine the proliferation of definitions of determinism in the recent literature, arguing that these definitions fail to deliver clear verdicts about actual scientific theories. We advocate a return to a formal approach, in the logical tradition of Carnap, that treats determinism as a property of scientific theories, rather than an elusive metaphysical doctrine. We highlight two key distinctions: (1) the difference between qualitative and "full" determinism, as emphasized in recent discussions of physics and metaphysics, and (2) the distinction between weak and strong formal conditions on the uniqueness of world extensions. We argue that defining determinism in terms of metaphysical notions such as haecceities is unhelpful, whereas rigorous formal criteria such as Belot's D1 and D3 offer a tractable and scientifically relevant account. By clarifying what it means for a theory to be deterministic, we set the stage for a fruitful interaction between physics and metaphysics.

Figures (4)

• Figure 1: Space with three places and two times, with two $\beta$ particles at the later time. Since symmetries are assumed to fix $\alpha$, the center blocks can be omitted from the model without changing the conclusions we draw.
• Figure 2: Theories that are supposed to be qualitatively, but not fully, deterministic.
• Figure 3: Melia's symmetric world
• Figure 4: The toy examples in the literature are D3-indeterministic, while GR is D3-deterministic. Belot's example can be interpreted in two ways, one deterministic and one indeterministic.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Example : toy Leibnizian spacetime
• Example : toy Newtonian spacetime