Subjunctive Dependence in Counterfactuals & FDT

• Subjunctive dependence is a formal relationship where one variable’s outcome covaries with hypothetical variations in another through logical, mathematical, or computational links.
• In decision theory, models like FDT incorporate subjunctive edges in graphical frameworks, allowing for interventions that capture logical correlations beyond pure causality.
• This concept underpins analyses in counterfactual conditionals, morphological learning, and cognitive modeling, enhancing predictions and decision-making strategies.

Subjunctive dependence is a formal concept central to the analysis of counterfactuals, advanced decision theories, and cognitive modeling. It refers to a specific kind of dependency structure: one in which the value of one variable or outcome systematically covaries with the hypothetical, counter-logical, or modal variation of another—not due solely to causal influence, but through logical, mathematical, or computational relations. Subjunctive dependence underlies the semantics of counterfactual conditionals, the distinctiveness of functional decision theory (FDT), select paradigms in morphological learning, and the distinction between material and counterfactual conditionals.

1. Formal Semantics of Subjunctive Dependence

Subjunctive dependence is formalized in both logical and graphical terms. In modal semantics, the semantics for counterfactuals (à la Stalnaker–Lewis) defines the truth of a subjunctive conditional "If $A$ were the case, $B$ would be the case" as holding at a world $w$ if and only if, in all $A$-worlds $w'$ closest to $w$, $B$ holds. This focuses on counterfactual dependence: $B$ varies with $A$ when $A$ is fixed by hypothetical intervention, up to minimal changes elsewhere (Gheorghiu, 13 Mar 2026).

Graphically, as formalized in functional decision theory, subjunctive dependence is encoded by adding "subjunctive edges" to a probabilistic graphical model. Let $B$0 be a graph representing random variables, with edges of two types:

• Causal edges (as in Pearlian DAGs), tracking physically possible interventions;
• Subjunctive edges, encoding logical/mathematical/computational correlations, such as those induced by perfect predictors, common source code, or algorithmic identity.

Formally, letting $B$1 be a probability distribution and $B$2 a graph with nodes $B$3 having causal parents $B$4 and subjunctive parents $B$5, then $B$6 subjunctively depends on $B$7 iff $B$8. Intervening on a subjunctive parent $B$9 (denoted $w$0) overwrites $w$1 and propagates the change only along subjunctive (and possibly causal) edges, updating only those nodes downstream of the change in the subjunctive structure (Yudkowsky et al., 2017).

2. Subjunctive vs. Causal and Material Dependencies

Subjunctive dependence is distinct from both causal dependence and the weaker correlations tracked by the material conditional.

• Causal dependence: Modeled by the “do-operator” in Pearlian frameworks, which tracks how the outcome of an intervention on $w$2 propagates strictly along causal arrows, ignoring logical or computational links. This captures "what would change if I physically forced X to be $w$3?" (Yudkowsky et al., 2017).
• Material conditional (indicative dependence): Defined truth-functionally ($w$4 is true unless $w$5 is true and $w$6 false), it tracks co-variation or patterns under a background theory, but not causation or subjunctive/counterfactual causality (Gheorghiu, 13 Mar 2026).
• Subjunctive dependence: Cuts across both, capturing “what would change if my decision algorithm produced output $w$7?”, preserving all logical, computational, and decision-theoretic linkages as relevant for counterfactual reasoning.

3. Subjunctive Dependence in Functional Decision Theory

Functional decision theory (FDT) explicitly elevates subjunctive dependence to the foreground of decision-making. In FDT, the agent's decision function is treated as a fixed mathematical object, and hypotheticals are evaluated by considering worlds where “the output of this function, given the agent’s inputs, is $w$8.” Precisely, FDT chooses an action by

$w$9

where $A$0 is the utility node and $A$1 is the subjunctive do-operator.

This intervention holds fixed all causal and subjunctive dependencies, intervening only on the decision-function output node. It respects both causal and logical pathways, allowing FDT agents to outperform causal decision theorists (CDT) and evidential decision theorists (EDT) on canonical problems wherever predictors or logical correlations are present (Yudkowsky et al., 2017).

Key examples from FDT:

Scenario	Subjunctive Dependence Role	Outcome for FDT versus CDT/EDT
Newcomb’s Problem	Predictor’s prediction subjunctively depends on agent’s output	FDT one-boxes, outperforms CDT/EDT
Smoking Lesion Problem	No subjunctive link from agent’s output to cancer	FDT smokes, like CDT
Parfit’s Hitchhiker	Driver’s rescue decision subjunctively depends on payment choice	FDT pays, ensuring rescue

CDT fails to cooperate with predictors/twins; EDT mismanages correlations. FDT, via subjunctive dependence, achieves the optimal outcome in all "fair" decision problems (Yudkowsky et al., 2017).

4. Subjunctive Dependence in Logic and Counterfactual Conditionals

Subjunctive dependence underlies the semantics of counterfactuals and "subjunctive conditionals." The classic possible-worlds semantics formalizes “If $A$2 were true, $A$3 would be true” as true in $A$4 if $A$5 holds in all $A$6-worlds closest to $A$7. This framework models counterfactual (subjunctive) conditionals, distinguishing them from the material conditional, which does not track counterfactual or causal dependence.

Structural-equation models sharpen this as follows: Subjunctive dependence holds when $A$8 and, for some $A$9, an intervention $w'$0 (the do-operator) yields a changed value of $w'$1. This captures “what-if” reasoning essential to science, causality, and advanced decision theory (Gheorghiu, 13 Mar 2026).

In ordinary language or scientific context, confusion often arises: material conditionals only register correlation, not subjunctive dependence, leading to paradoxical or "absurd" (but technically correct) inferences when speakers unconsciously shift between indicative and subjunctive interpretations (Gheorghiu, 13 Mar 2026).

5. Subjunctive Dependence in Morphological Learning

In cognitive modeling, especially in morphological learning, "subjunctive dependence" can be interpreted (editor's term) as the systematic co-variation of paradigm cells not justified by phonological, semantic, or syntactic motivation but by an abstract, possibly morphological-level, structural regularity.

For example, in the Spanish L-shaped morphome, the stem for the first-person singular indicative and all subjunctive forms covary, forming a class $w'$2. The empirical pattern:

$w'$3

reflects a higher-order dependency. Human generalization tends to privilege the dependence of 1SG IND on the subjunctive, while transformer models typically default to mood-based (indicative/subjunctive) generalization, indicating they can capture correlation but not the full abstraction of human morphological subjunctive dependence (Ramarao et al., 15 Feb 2026).

6. Philosophical Rationale and Broader Implications

Subjunctive dependence is justified as the appropriate expansion of the dependency concept for rational reasoning and action beyond strictly causal frameworks. Its main philosophical advantages:

• Unification: Combines the statistical strengths of evidential reasoning (EDT) with prediction-aware aspects of causal decision theory (CDT), avoiding each’s major failures in classic dilemmas (Yudkowsky et al., 2017).
• Reflective stability: Subjecting policies to subjunctive hypotheticals preserves ideal precommitment; FDT agents act as though precommitted without ad-hoc ratification steps.
• Universality and fairness: In “fair” dilemmas, subjunctive dependence ensures agents do not leave utility on the table or suffer for respecting mere statistical “news.”
• Multi-agent and sequential rationality: Subjunctive dependence enables robust reasoning in the presence of predictors, clones, and coalition game-theoretic structures, as it captures chains of mutual prediction and logical correlation (Yudkowsky et al., 2017).

A plausible implication is that formalizing subjunctive dependence—not just causal or evidential links—extends the expressive power of scientific, economic, and AI reasoning frameworks, aligning models more closely with the structural dependencies relevant for prediction, cooperation, and abstraction.

7. Cognitive and Computational Consequences

In human cognition, robust morphological generalization and the interpretation of counterfactuals suggest a sophisticated capacity for tracking subjunctive dependencies beyond patterns explicable by frequency or simple correlation. In neural network modeling, failure to reproduce human patterns of abstraction (as evidenced in transformer experiments on the L-shaped morphome) can often be traced to the lack of mechanisms for encoding or inferring subjunctive dependencies—contrasted with humans’ ability to induce schema-level connections even for rare patterns (Ramarao et al., 15 Feb 2026).

Future research directions include the formalization of morphological inductive biases, architectural enhancements for detecting and abstracting schematic subjunctive dependencies, and the integration of subjunctive structures in cognitive and AI systems (Ramarao et al., 15 Feb 2026).