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Quasi-conjunction of trivalent conditionals

Quasi-conjunction combines trivalent conditional objects by conjoining their material implications under the union of their conditioning events.

Version
v1 · 2026-10-31 · History
Domain-specific #
14561
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Mathematical Logic, Probability Theory → Mathematics

Core Idea

A conditional a|b is true when a and b hold, false when b holds without a, and inapplicable when b fails. Over a Boolean event algebra, quasi-conjunction combines two such objects as (a|b) ∧Q (c|d) = ((b→a)∧(d→c))|(b∨d), where → is material implication. The operation defines a conditional object without requiring a probability assignment. Probability-coherent entailment is a further, qualified use.[1][2]

How would you explain it like I'm…

Two Promises Together

Think of a promise like 'If it's sunny, we go to the park.' If it's sunny and we go, the promise is kept. If it's sunny and we don't go, it's broken. If it isn't sunny, the promise just doesn't count today. Quasi-conjunction puts two promises together: the pair doesn't count only when neither 'if' happens, and it's broken if any promise that counts gets broken.

Joining If-Then Rules

An 'if' statement like 'if b, then a' can be true, false, or not apply at all: it is true when b and a both happen, false when b happens without a, and it doesn't count when b doesn't happen. Quasi-conjunction joins two of these 'if' statements into one. The joined statement counts whenever at least one of the two 'if' parts happens. When it counts, it is true only if neither statement is broken, meaning no applicable 'if' part happened without its matching result. It is just a rule for combining statements, not a way of computing chances.

Combining Three-Valued Conditionals

A conditional a|b ('a given b') has three possible values: true when a and b both hold, false when b holds but a does not, and inapplicable when b fails. Quasi-conjunction combines two such conditionals into one: (a|b) ∧Q (c|d) = ((b→a) ∧ (d→c)) | (b∨d), where → is ordinary 'if-then' (material implication). So the combined conditional applies whenever b or d holds. When it applies, it is false exactly when one of the original conditionals is false; a conditional that does not apply in that case counts as harmless because its material implication is true. This defines an operation on conditionals; it does not assign probabilities, and using it to reason about probability needs extra assumptions.

 

In trivalent (three-valued) treatments of conditionals, a conditional event a|b over a Boolean algebra of events takes the value true on a∧b, false on ¬a∧b, and void (inapplicable) on ¬b. Quasi-conjunction is the binary operation (a|b) ∧Q (c|d) = ((b→a) ∧ (d→c)) | (b∨d), where → is material implication in the underlying event algebra. The result's conditioning event is the union b∨d, so it is applicable whenever at least one component is. Within that region, the consequent (b→a)∧(d→c) is false exactly when some applicable component is false, because material implication is vacuously true where its antecedent fails. This defines an algebraic operation on conditional objects only; it assigns no probabilities, and using it in probabilistic or coherence-based entailment requires further assumptions.

Examples

Derived four-outcome calculation. In P({1,2,3,4}), the Q of ({1}|{1,2}) and ({2}|{2,3}) has raw implication intersection {1,4} and context {1,2,3}. It is therefore ({1}|{1,2,3}) after context restriction. This is a calculation derived here from the published Q formula, not a source-reported or empirical case.[1]

Published formal knowledge base. In Gilio and Sanfilippo's Example 2, logically independent party propositions L, S, G, N underlie a stipulated p-consistent family F. Selected rules S|L and ¬N|(L∧S) have quasi-conjunction (S∧¬N)|L. Their conclusion that F p-entails ¬N|L additionally uses coherence, p-consistency and conditional inclusion. The party scenario is illustrative, not a deployed system.[2]

Boundaries

Q differs from interval conjunction on the same trivalent conditional family. Boolean Conditional-Event Algebra Constructions instead uses a Boolean constructed carrier, while Default Logic uses extension semantics. The historical associations in the 2020 comparison should be read as that paper's account: the cited Schay, Calabrese and Goodman–Nguyen originals were not directly examined for this article.[1][3]

Neighborhood in Abstraction Space

Quasi-conjunction of trivalent conditionals sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Models & Logical Foundations (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

References

[1] Tommaso Flaminio, Lluís Godo and Hykel Hosni, “Boolean algebras of conditionals, probability and logic,” Artificial Intelligence 286 (2020), 103347, §8.1. https://doi.org/10.1016/j.artint.2020.103347; author full text. registry ↩a ↩b ↩c

[2] Angelo Gilio and Giuseppe Sanfilippo, “Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects,” Information Sciences 245 (2013), 146–167. https://doi.org/10.1016/j.ins.2013.03.019; Definitions 3–5, Corollary 1 and Example 2. Author PDF. registry ↩a ↩b

[3] Didier Dubois and Henri Prade, “Measure-free conditioning, probability and non-monotonic reasoning,” Proceedings of IJCAI 1989, §§2, 4. Conference PDF. registry ↩