Merin–Van Rooy Bridge #

@cite{merin-1999} @cite{van-rooy-2003}

Formal connection between Merin's Decision-Theoretic Semantics (DTS) and Van Rooy's decision-theoretic question framework.

The Connection #

@cite{van-rooy-2003} (L&P 26, pp. 727–763) defines two measures of proposition utility:

VSI(C) = max_a EU(a,C) − EU(a⁰,C), "can obviously never be negative" (p. 735)
UV(C) = max_a EU(a,C) − max_a EU(a), "can be negative" (p. 736)

At the question level, EUV(Q) = ∑ P(q)·UV(q) = EVSI(Q) ≥ 0 (p. 742).

Merin's DTS uses a dichotomic issue {H, ¬H} with Bayes factors BF(E) = P(E|H)/P(E|¬H). In §5.4, @cite{van-rooy-2003} connects UV to Merin's argumentative value: when preferences beyond truth-resolution are at stake, UV(C) captures the directional relevance of a proposition.

The bridge: Merin's dichotomic issue is a special case of Van Rooy's decision problem (truthDP) where:

Actions = {accept H, reject H}
Utility depends only on truth: U(w, accept) = 1 iff H(w), else 0

Under this encoding:

BF > 1 → learning E increases EU(accept H), i.e., shifts the posterior toward H (posRelevant_shifts_accept_eu)
BF = 1 → learning E leaves all conditional EUs unchanged, so UV = 0 (irrelevant_implies_zero_uv)

Note: UV(E) for a single cell E can be negative even when BF > 1 (@cite{van-rooy-2003}, p. 736). The non-negativity result holds for expected UV across the full partition, not for individual cells.

Results #

truthDP: Encoding of a dichotomic issue as a DecisionProblem
posRelevant_shifts_accept_eu: BF > 1 → EU(accept|E) > EU(accept)
irrelevant_implies_zero_uv: BF = 1 → UV(E) = 0 (non-degenerate)

Encoding Merin's Issue as a Decision Problem #

A dichotomic issue {H, ¬H} with prior π corresponds to a decision problem:

World type W (shared)
Actions = Bool (accept H = true, reject H = false)
U(w, accept) = 1 iff H(w); U(w, reject) = 1 iff ¬H(w)
Prior = π

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def Theories.Semantics.Questions.MerinBridge.truthDP {W : Type u_1} (ctx : DTS.DTSContext W) :

Core.DecisionTheory.DecisionProblem W Bool

Encode a DTS context (dichotomic issue + prior) as a decision problem.

The agent must choose between accepting H (true) or rejecting H (false). Utility = 1 for correct choice, 0 for incorrect.

Equations

One or more equations did not get rendered due to their size.

Instances For

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def Theories.Semantics.Questions.MerinBridge.truthActions :

Finset Bool

The action set for the truth DP: accept or reject.

Equations

Theories.Semantics.Questions.MerinBridge.truthActions = {true , false }

Instances For

Bridge Theorems #

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theorem Theories.Semantics.Questions.MerinBridge.posRelevant_shifts_accept_eu (ctx : DTS.DTSContext Core.Proposition.World4) (e : BProp Core.Proposition.World4) :

(∀ (w : Core.Proposition.World4), ctx.prior w = 1 / 4) → (∃ (w : Core.Proposition.World4), e w = true) → (∃ (w : Core.Proposition.World4), ctx.issue.topic w = true) → (∃ (w : Core.Proposition.World4), ctx.issue.topic w = false) → DTS.posRelevant ctx e → Core.DecisionTheory.conditionalEU (truthDP ctx) {w : Core.Proposition.World4 | e w = true} true > Core.DecisionTheory.expectedUtility (truthDP ctx) true

Positive relevance shifts the conditional EU of "accept H" upward.

When BF > 1, learning E raises P(H|E) above P(H), which means EU(accept H | E) > EU(accept H). This is the core Merin–Van Rooy bridge: Merin's relevance sign determines the direction of the posterior shift for the truth decision problem.

Note: this does NOT imply UV(E) ≥ 0 for the single cell E. UV of a single cell can be negative (@cite{van-rooy-2003}, p. 736). The non-negativity result (EVSI ≥ 0) holds for the expected UV across the full partition {E, ¬E}, not for individual cells (p. 742).

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theorem Theories.Semantics.Questions.MerinBridge.irrelevant_implies_zero_uv (ctx : DTS.DTSContext Core.Proposition.World4) (e : BProp Core.Proposition.World4) :

(∀ (w : Core.Proposition.World4), ctx.prior w = 1 / 4) → (∃ (w : Core.Proposition.World4), e w = true) → (∃ (w : Core.Proposition.World4), ctx.issue.topic w = true) → (∃ (w : Core.Proposition.World4), ctx.issue.topic w = false) → DTS.irrelevant ctx e → Core.DecisionTheory.utilityValue (truthDP ctx) truthActions {w : Core.Proposition.World4 | e w = true} = 0

Merin's irrelevance corresponds to zero utility value.

If E is irrelevant to H (BF = 1) and the conditioning is non-degenerate (E non-empty, both H and ¬H have witnesses), then learning E doesn't change any conditional EU, so UV(E) = 0.

The key step: BF = 1 under uniform prior means P(E|H) = P(E|¬H), which gives |E∩H|/|H| = |E∩¬H|/|¬H|, hence |E∩H|/|E| = |H|/4. So conditionalEU(a|E) = expectedUtility(a) for each action a, making valueAfterLearning = dpValue.