Causality and Counterfactuals
Cause as branch structure
Alice flips a switch, and a moment later the lamp turns on. Everyone agrees the flip caused the light. Almost no one can say what that claim means.
The oldest answer, Hume’s, is regularity: flips of this kind are constantly conjoined with lightings of that kind. But regularity is too weak — night regularly follows day without causing it — and every attempt to patch it smuggles in the causal notion it was supposed to explain. The modern answer, due mainly to David Lewis, is counterfactual: the flip caused the light because if Alice had not flipped the switch, the lamp would not have turned on. That gets the logic right. Causal claims concern what would have happened otherwise; that is why we test drugs against control groups and assign blame by asking what the defendant could have done differently.
The trouble is what the counterfactual is supposed to be about. Lewis evaluated “if a had occurred, b would have occurred” using nearby possible worlds, which requires a similarity ordering. Modern causal models instead represent interventions explicitly, but they still require judgments about variables, structural equations, and which background conditions remain fixed. Counterfactual analysis gets the logic right; no ontology eliminates the modeling choices.
The Quantum Branching Universe (QBU) adds a physical interpretation of alternative outcomes and their weights — branches in which \(a\) occurs and branches in which it does not, physically actual and weighted rather than imagined. It does not automatically identify the correct contrast or nearest comparison history; the graph supplies ancestry and Measure once a coarse-graining is specified, and a causal model must still say what counts as Alice’s action, what is held fixed, and which differences are irrelevant.
The Definition
Given a Vantage \(V\), a causal model \(M\), an event \(a\), and an outcome \(b\), compare the conditional Measure of \(b\) where \(a\) occurs against its Measure under an appropriate matched contrast:
\[ \Delta_a b = \mu_V(b\mid a, C, M) - \mu_V(b\mid \neg a, C, M). \]
Here \(C\) is the contrast specification: the matched branch-sets, with the background conditions that isolate \(a\) held fixed. It is what an ordinary causal model builds with the \(do(\cdot)\) operator; in the QBU it also fixes which physically realized branches count as the comparison. \(C\) and \(M\) index the contrast and the causal model; they are not events conditioned on, and the bar here selects a comparison rather than recording an observation.
Then:
Event \(a\) is a cause of \(b\), relative to \(V\) and \(M\), when its occurrence changes the Measure of \(b\) against a matched contrast in which it does not occur, with the causal path respecting the QBU’s ancestry relation.
The deterministic case is the limit in which \(\mu_V(b\mid a,C,M)=1\) and \(\mu_V(b\mid \neg a,C,M)=0\). Most causes are not like that. They raise or lower an outcome’s Measure without guaranteeing it. The definition therefore has three explicit dependencies:
- Representation. The events and histories have been defined at a relevant coarse-graining.
- Contrast model. The contrast specifies what is changed and what is held fixed, isolating variation in \(a\) so the comparison is not mere observational conditioning.
- Measure difference. The occurrence of \(a\) changes the outcome’s conditional weight.
Ancestry rules out backward paths within the QBU representation; it does not by itself establish dependence. The matched contrast does that work.
Two clarifications keep the definition honest. First, the matched contrast is what separates causation from mere prediction. Umbrellas track rain, but the branches that differ in whether an umbrella is carried are not matched on the weather; a legitimate contrast holds the causally relevant background fixed and lets only \(a\) vary. What the QBU adds here is narrower than it may look: it makes the alternative continuations physically actual and weighted, so the counterfactual is a comparison among branches that exist rather than an imagined manipulation — which is why a supernova or the outbreak of a war, neither of which anyone can manipulate, still has a contrast. What branching does not supply is the isolation of \(a\) from its own causes. A shared ancestor gives common provenance, not the absence of common causes; the surgical severing that \(do(a)\) performs in an ordinary model is not something the branch structure does for free. Isolating \(a\) therefore remains a modeling requirement — discharged by structural assumptions, an admissible adjustment set, or another explicit no-confounding condition, and recorded in \(C\). Second, \(do(a)\) names one way of constructing that contrast — the interventional one — not the essence of causation; when \(a\) is an agent’s own policy, the contrast is one the agent produces, and the comparison becomes the intervention, and the control, of the previous chapters. Causation in general needs only a well-isolated matched contrast; agent control is its manipulable special case.
The events are picked out by Pattern Identifiers. When identity across histories matters, a Strong PI must include provenance from a declared ancestor; descriptive precision alone does not guarantee a common origin.
Run an ideal lamp through it. Event \(a\) is “Alice flips the switch at time \(t\)”; event \(b\) is “the lamp turns on at \(t+\Delta t\).” In a model that holds the power supply, bulb, and wiring fixed, suppose the flip makes the Measure of illumination nearly one while the matched no-flip branches leave it near zero. Then \(\Delta_a b\) is large and positive, so
\[ (\text{Alice flips switch}) \,\square\!\!\to\, (\text{Lamp turns on}) \]
and the everyday causal claim is represented as a model-relative contrast fact with physical weights.
The QBU contributes temporal ancestry and an Everettian interpretation of the weighted alternatives. The causal model contributes the contrast and comparison class. Neither can replace the other. Overdetermination, preemption, feedback, and coarse-graining remain hard cases; the definition is a framework for stating them, not a claim that ancestry alone has settled them.
A definition earns its keep by what it settles. Here are two questions it settles immediately.
Do Ideas Move Atoms?
In our brains, do ideas push around atoms, or do the atoms push the ideas? The question sounds like a koan, and centuries of dualist and epiphenomenalist hand-wringing have treated it as one. The inherited premise treats ideas as nonphysical while allowing only physical things to cause physical things. Ideas then become inert passengers unless something spooky is going on.
Apply the definition at the right grain. Ideas are higher-level patterns implemented by neural activity. If a matched contrast that preserves the relevant background while changing whether pattern A is instantiated systematically changes the Measure of later neural or behavioral outcome B, then A is causally relevant at that level. The contrast may be experimental, computational, or counterfactual; what matters is that the macrovariable supports stable difference-making rather than merely redescribing the outcome after the fact.
Ideas, as neural patterns, cause atomic events.
There is nothing spooky in this, and the software analogy shows why. A program is a higher-level description implemented by physical states, yet changing the program while holding hardware conditions appropriately fixed changes the machine’s behavior. Ideas can be causally explanatory in the same contrast-supporting sense.
The atomic and semantic descriptions need not compete for one causal slot. They track the same process at different grains and answer different contrasts. The higher-level claim is rigorous only where its variables remain stable under the comparison; not every story a mind tells about its motive will pass that test.
Probabilities for Things That Don’t Happen
The second question is a standing objection to the whole counterfactual program: if causation is predicated on counterfactuals, it cannot be probabilistic, because we cannot assign probabilities to things that don’t happen. A probability of an event that never occurs, the objection runs, is a number about nothing.
The objection confuses non-actual with measureless. Standard causal models assign probabilities under such contrasts without requiring that every alternative be a concrete world. The QBU offers a stronger ontological reading: alternative decoherent outcomes are components of the quantum state and carry Born Measure. On that reading, “Had Alice not flipped the switch, the lamp would probably still have been off” reports \(\mu_V(\text{off}\mid \neg\text{flip}, C, M)\).
Deterministic causation is the limiting case where the contrast drives the relevant Measure to zero or one; probabilistic causation changes it without fixing it. Everettian ontology is one way to interpret those weights physically, not a prerequisite for probabilistic causal reasoning.
That is where causation joins prediction. Knowledge, on my account, is pattern-encoded structure that reliably reduces an agent’s uncertainty about relevant futures. A correlation predicts under observation; a causal model predicts under intervention. Agents need the latter because control is evaluated against alternative policies. The resulting distributional difference can be measured in kybits only after this causal structure has supplied the baseline.