Probabilistic Conjunction Error (Logical)

Also Known As: Linda Problem Conjunction Rule Violation

Formal Fallacy ID: conjunction_fallacy_logic

Definition

The logical form of the conjunction fallacy: concluding that a conjunction of events is more probable than one of its conjuncts alone. This violates a basic axiom of probability theory and formal logic concerning subset relationships.

Examples

Linda is more likely to be a bank teller AND a feminist activist than just a bank teller.

Mark is a retired army officer who frequently attends gun shows and has written letters to his local newspaper opposing federal regulations. People consistently rate it as more probable that Mark is a veteran AND a member of the NRA than that Mark is simply a veteran — despite the conjunction always being less probable than either component alone.

A weather app user is told there is a 70% chance of rain tomorrow. They reason that it is more likely to rain AND be cold than it is to simply rain, because rain and cold 'go together' — violating the basic probability rule that a conjunction cannot exceed the probability of its least likely conjunct.

Formal Logic Pattern

FOL Pattern

The First-Order Logic formula representing this reasoning pattern's logical structure.

FOL (First-Order Logic) uses quantifiers (∀ = for all, ∃ = there exists), connectives (∧ = and, ∨ = or, ⇒ = implies, ¬ = not), and predicates to capture the essential form of a reasoning pattern. For example, the Ad Hominem fallacy: Person(x) ∧ HasFlaw(x) ⇒ Invalid(Claim(x)). These patterns allow automated verification of logical validity.

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P(A ∧ B) ≤ P(A) violated

Formal Verification: Formally invalid

Verification Steps

Verification Steps

Binary yes/no questions that an AI must answer to detect a reasoning pattern in a text.

Each of the 452 aspects has verification steps — simple yes/no questions designed to systematically detect whether a pattern appears in a text. For ad hominem: "Does the argument attack a person rather than their claim?" For false dichotomy: "Are only two options presented when more exist?" This ensures consistent, reproducible analysis.

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Binary (yes/no) questions an LLM must answer to identify this aspect:

1

Does the argument claim that a specific, detailed scenario is more likely than a more general one?
Type: binary
2

Is the specific scenario a subset of the general scenario (i.e., the specific scenario includes all conditions of the general one plus more)?
Type: binary
3

Does the reasoning violate the rule that P(A and B) cannot exceed P(A)?
Type: binary