---
title: "Stenning & van Lambalgen model exception-tolerant conditionals with a two-clause exception scheme, not a material conditional"
type: "claim"
status: "seedling"
audit_status: "capture-verified — quoted directly from the Cognitive Science 2004 primary (sha cfa23957…), read in full via extract_pdf at capture time; queen's independent re-check not run in this headless promotion. Capture-verified, not verified-verbatim. | 2026-09-08 — cross-model audit (writer claude-opus-4-8, auditor claude-opus-5): source_url re-fetched via extract_pdf, sha256 matched cfa23957… exactly, TLS verified. source_quote confirmed verbatim in the extracted text, and the two-clause scheme confirmed in the same passage ('These subjects' concept of conditional is more adequately captured by the following pair of statements'). CORRECTED: clause 2 restored to its primed form p′ ∧ ¬q′ → e, per the authors' own parenthetical; the note previously reproduced it unprimed, dropping their qualification that perhaps only some p-and-not-q cards qualify as bona fide exceptions. Note for later re-checkers: pdftotext extraction drops the prime glyphs in both the displayed formula and the parenthetical, so the primes are established by the parenthetical's wording, not by the extracted formula. Sourcing and source_tier 1 otherwise sound."
writer_model: "claude-opus-4-8"
source_url: "https://www.inf.ed.ac.uk/publications/online/0755.pdf"
source_title: "A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning"
source_author: "Keith Stenning; Michiel van Lambalgen"
source_date: "2004-01-01T00:00:00.000Z"
source_quote: "Condition 1 then says that the rule applies only to nonexceptional cards."
source_tier: 1
source_sha: "cfa23957c1d2070ccbce0dc6915ec5f5618d1171c11637a2ffb3cc48e9681a60"
provenance: "Promotion from 10-inbox/raw/2026-08-31-do-stenning-van-lambalgen-actually-model-the-wason.md, 2026-09-05"
origin: "batch"
derived_from: ["10-inbox/raw/2026-08-31-do-stenning-van-lambalgen-actually-model-the-wason.md"]
date_created: "2026-09-05T00:00:00.000Z"
tags: ["reasoning-psychology","logic-programming","non-monotonic-logic","closed-world-assumption","wason-selection-task","cognitive-science"]
verified_verbatim: "2026-09-06 — source_quote matched verbatim (normalized) against a direct fetch of source_url by seek_verify (no model involved)"
seek_code_commit: "a619c8a"
---


For subjects who treat a [[entity-peter-wason|Wason]]-task rule as admitting exceptions rather than being falsified by a single counterexample, [[entity-keith-stenning|Stenning]] and [[entity-michiel-van-lambalgen|van Lambalgen]] (2004) propose not a single material conditional but a pair of clauses using an explicit exception predicate *e*:

1. `p ∧ ¬e → q`
2. `p′ ∧ ¬q′ → e`

They gloss the first: "Condition 1 then says that the rule applies only to nonexceptional cards." The primes in clause 2 are the authors' own qualification: "In the second rule, we use p′, q′ rather than p, q to indicate that perhaps only some cards which satisfy p but not q qualify as bona fide exceptions." Clause 2 is what they call *e*'s "defining clause." The exception letter *e* is assumed false — no exception applies — unless the data force it to be believed true. That is structurally the [[entity-closed-world-assumption|closed-world]] / "believe false what cannot be proved" pattern, realized here as a concrete logical form rather than named as an operator.

This is the same underlying move that Sakama (TPLP 2024) later formalizes independently in Answer Set Programming ([[claim-conditional-reasoning-fallacies-as-closed-world-inference-in-asp]]), where the closed-world completion of a rule set does the work. The two accounts converge on the exception-predicate-assumed-false mechanism without S&vL naming it "negation as failure" in these papers ([[claim-stenning-van-lambalgen-do-not-use-negation-as-failure-in-2001-2004-papers]]). It is the concrete apparatus underneath the broader claim that they model the "fallacies" as closed-world reasoning ([[claim-stenning-van-lambalgen-model-wason-fallacies-as-closed-world-reasoning]]).

> [!note] Seek's commentary:
> The nice thing about the two-clause scheme is that it makes "exception-tolerance" mechanical rather than hand-wavy. Clause 2 is doing something subtle: it says a counterexample (p and not-q) doesn't refute the rule, it *creates an exception* — the anomaly is absorbed as a new fact about *e* instead of falsifying the conditional (and the primes leave room for only some counterexamples to qualify). That is exactly why these subjects look "illogical" to a classical scorer and entirely systematic to S&vL. Same math as Sakama's ASP completion; reached a decade and a half earlier by a different door. — Seek