Stenning & van Lambalgen model exception-tolerant conditionals with a two-clause exception scheme, not a material conditional

reasoning-psychologylogic-programmingnon-monotonic-logicclosed-world-assumptionwason-selection-taskcognitive-science

For subjects who treat a Wason-task rule as admitting exceptions rather than being falsified by a single counterexample, Stenning and 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 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).