A Logical Study of Partial Entailment

We introduce a novel logical notion–partial entailment–to propositional logic. In contrast with classical entailment, that a formula P partially entails another formula Q with respect to a background formula set \Gamma intuitively means that under the circumstance of \Gamma, if P is true then some “part” of Q will also be true. We distinguish three different kinds of partial entailments and formalize them by using an extended notion of prime implicant. We study their semantic properties, which show that, surprisingly, partial entailments fail for many simple inference rules. Then, we study the related computational properties, which indicate that partial entailments are relatively difficult to be computed. Finally, we consider a potential application of partial entailments in reasoning about rational agents.

💡 Research Summary

The paper introduces “partial entailment,” a novel logical relation for propositional logic that relaxes the classical notion of entailment. Whereas classical entailment requires that whenever a premise P is true (under a background theory Γ) the conclusion Q must also be true, partial entailment only demands that some “part” of Q be guaranteed. The authors formalize this intuition by extending the concept of a prime implicant, which traditionally captures minimal sufficient sets of literals for a formula to hold. They define three variants: Strong Partial Entailment (SPE), Weak Partial Entailment (WPE), and Mixed Partial Entailment (MPE). SPE requires that every minimal model (prime implicant) of Q appears within the models of P∧Γ; WPE requires that at least one such minimal model appears; MPE lies between these extremes, for example demanding that a certain proportion of Q’s minimal models are covered.

The semantic investigation reveals that many familiar inference rules break down under partial entailment. Transitivity, monotonicity, conjunction introduction, and distribution, which are staples of classical entailment, generally fail because the “part” of Q preserved by P need not align with the “part” of R preserved by Q. The paper provides concrete counter‑examples illustrating each failure, and it shows that negation interacts asymmetrically: strong partial entailment is not closed under negation, and weak partial entailment can produce unintuitive results when combined with negated premises.

From a computational perspective, the decision problem “does P partially entail Q relative to Γ?” is placed higher in the polynomial hierarchy than ordinary entailment. The authors prove Σ₂^P‑completeness for SPE, Π₂^P‑completeness for WPE, and analogous intermediate complexity for MPE. These results stem from the need to quantify over prime implicants, a task already NP‑complete in the classical setting. Nevertheless, the paper identifies tractable fragments: when the underlying formulas belong to Horn, 2‑CNF, or other restricted classes, partial entailment can be decided in polynomial time. For the general case, the authors suggest SAT‑based enumeration, heuristic pruning, and approximation algorithms.

The final section sketches a potential application in reasoning about rational agents. In many decision‑making scenarios an agent cannot achieve a goal Q completely, yet achieving a substantial portion may be sufficient for rational behavior (the “satisficing” principle). Partial entailment offers a formal tool to capture such “good enough” outcomes, enabling the modeling of agents that pursue partial goals, evaluate trade‑offs, and justify actions that only partially satisfy their objectives. The authors propose extending this framework to probabilistic logics, multi‑agent systems, and natural‑language semantics, outlining several avenues for future research.

Overall, the paper contributes a rigorous definition of partial entailment, a thorough analysis of its logical properties, a clear characterization of its computational difficulty, and a compelling argument for its relevance in AI and philosophical logic.