The Causal Second Law
Abstract: I argue that if a special science satisfies certain key assumptions that are familiar from physicalist accounts of the special sciences and from physics, then its causal regularities have an associated notion of entropy, and that this causal entropy cannot decrease from a robust cause to its effect. Due to its analogy with the second laws of thermodynamics and statistical physics, I call the latter conclusion the causal second law. In this paper, I clarify the key assumptions, prove the causal second law, give sufficient conditions for causal entropy increase, relate the causal second law to statistical mechanics and thermodynamics, and argue that the reversibility objection does not threaten it. In addition, I claim that the causal second law is compatible with a non-metaphysical understanding of supervenience and the open systems view, argue that it does not imply a causal time arrow, reflect on relaxing the robustness condition, question whether it is necessary to invoke thermodynamics to show that special sciences' time arrows exist, and discuss a transition-relative-frequency-based, special-science-internal characterization of causal regularities.
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What is this paper about?
This paper introduces a simple idea called the “causal second law.” It says: in many sciences (like biology, psychology, economics), when a cause regularly leads to an effect, there is a way to measure how “spread out” or “numerous” the physical ways that cause and effect can happen. That measure is called causal entropy. The main claim is that causal entropy cannot go down from a robust cause to its effect. This is similar to the famous second law of thermodynamics, which says entropy doesn’t decrease.
What questions does the paper try to answer?
- Can we define an “entropy” for everyday and special-science causes and effects, not just for heat and gases in physics?
- Under what conditions does this causal entropy never decrease from cause to effect?
- When does it strictly increase?
- How does this idea relate to the usual second law in thermodynamics and statistical mechanics?
- Does the fact that microscopic physics can be time-reversal-invariant (you can imagine running a movie backward) cause problems for this causal second law?
- Can we relax the assumptions and still keep the main result useful for real-world sciences?
How does the paper approach the problem?
The paper uses a dynamical systems viewpoint—the idea that physical states evolve over time according to rules.
Here are the two key assumptions, explained simply:
- State-supervenience: Every description you use in a special science (like “the house has a burning match and a lot of flammable material”) sits on top of many detailed physical arrangements of particles and fields that make that description true. Think of a special-science description as a label, and behind it there’s a big set of fine-grained physical “microstates” that instantiate it.
- Measure-preservation: As physical states evolve over time, the “size” (think of it like area or volume on a giant map of all possible microstates) of any set of microstates doesn’t change. This is a well-known property in many physical theories (like classical mechanics) and is often called “phase-volume preservation.” Imagine ink blobs flowing across a surface: they can stretch and twist, but their total area stays the same.
Using these, the paper defines:
- Causal entropy: For a cause or an effect, causal entropy is how big the set of physical microstates is that make that description true. Bigger set = higher causal entropy.
- Robust causal regularity: A cause is robust if, almost every time it happens in the underlying physical world, it leads to its effect within a typical time. “Almost” here means exceptions are extremely rare in a precise, math-based way.
Analogy for the main proof: Think of the cause as a cluster of dots on a map (the microstates). The laws of physics move each dot forward in time, but the total area of dots stays the same. If “almost all” dots from the cause end up inside the effect’s region, then the effect’s region must be at least as big as the cause’s region. So causal entropy cannot decrease from a robust cause to its effect.
What did the paper find, and why is it important?
- The causal second law: If a special science uses robust causal regularities and the two key assumptions hold, then the causal entropy from cause to effect cannot decrease. That means effects are at least as “physically plentiful” (or more) than their causes.
- When does causal entropy strictly increase?
- Multiple possible causes: If one effect can be brought about by several distinct robust causes (e.g., a burned-down house can come from a lit match, a lit lighter, an electrical spark, or a flamethrower), then the effect’s microstate-set is bigger than any single cause’s set. So causal entropy strictly increases.
- Mismatch in descriptions: Special sciences describe the world in big, coarse patches; physics describes it in super-fine detail. Often, the set of all microstates that lead to an effect can’t be cleanly captured by a single special-science description. Because the science can only describe bigger, coarser regions, the actual cause we can name is smaller than the full “pull-back” of the effect, and entropy strictly increases.
- Link to thermodynamics: The paper connects this idea to a classic reasoning by E. T. Jaynes. Read in reverse, Jaynes’ argument shows that thermodynamic entropy (like the Boltzmann or Gibbs entropy you learn about for gases) can be understood as this same causal entropy when the “special science” is thermodynamics. This helps justify using the word “entropy” outside of thermodynamics.
- Reversibility objection answered: Some worry that if microscopic physics is time-reversal-invariant, entropy can go down as easily as it goes up. The paper shows this does not threaten the causal second law:
- The law talks about robust cause-to-effect patterns. Time-reversal typically won’t turn the effect into a robust cause of the original cause, because the effect’s physical set is larger. With measure-preservation, that reverse robust pattern can’t hold.
- Special-science descriptions are not necessarily “closed” under time reversal. Flipping velocities at the micro-level doesn’t guarantee a meaningful special-science description.
- The proof doesn’t need to assign entropy to each single microstate; it focuses on cause and effect descriptions.
What are the bigger takeaways?
- Entropy beyond physics: Many fields—medicine, psychology, economics, geology—use robust cause-effect patterns. Under familiar and often reasonable assumptions, each field can have its own version of “entropy,” tied to how many physical ways its causes and effects can be realized.
- Practical links: For thermodynamics, the paper shows a path from causal entropy to the standard, measurable thermodynamic entropy. That suggests similar practical links might be built for other sciences.
- Time arrows: The paper distinguishes “entropy from cause to effect” from “entropy in time.” It argues we might not need thermodynamics to explain why many special sciences show arrows of time in their own domains. Also, this causal second law does not force a claim like “causes must come before effects” in a deep metaphysical way—it’s about robust patterns and physical realizations, not about imposing a new time-direction rule on the world.
- Flexible assumptions: The strong metaphysical reading can be relaxed. You can understand “supervenience” as an empirical, theory-to-theory fit rather than a strict metaphysical dependency. The approach can handle multiple realizations and, in future work, even certain kinds of stochastic (random) dynamics if they preserve the right kind of measure.
A simple summary
- Define how many physical ways there are for a cause or an effect to happen (causal entropy).
- If a cause almost always leads to its effect, and the physical laws preserve the “size” of sets of these tiny physical states, then the effect can’t be based on fewer physical possibilities than the cause.
- Often, the effect is based on more possibilities, especially when many different causes can lead to it or when our special-science descriptions are coarse compared to physics.
- This mirrors the second law of thermodynamics and helps unify how we think about cause and effect across many sciences.
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