Preference relations are at the heart of many fundamental concepts in artificial intelligence, ranging from utility comparisons, to defeat among strategies and relative plausibility among states, just to mention a few. Reasoning about such relations has been the object of extensive research and a wealth of formalisms exist to express and reason about them. One such formalism is conditional logic, which focuses on reasoning about the “best” alternatives according to a given preference relation. A “best” alternative is normally interpreted as an alternative that is either maximal (no other alternative is preferred to it) or optimal (it is at least as preferred as all other alternatives). And the preference relation is normally assumed to satisfy strong requirements (typically transitivity and some kind of well-foundedness assumption).

Here, we generalize this existing literature in two ways. Firstly, in addition to maximality and optimality, we consider two other interpretations of “best”, which we call unmatchedness and acceptability. Secondly, we do not inherently require the preference relation to satisfy any constraints. Instead, we allow the relation to satisfy any combination of transitivity, totality and anti-symmetry.

This allows us to model a wide range of situations, including cases where the lack of constraints stems from a modeled agent being irrational (for example, an agent might have preferences that are neither transitive nor total nor anti-symmetric) or from the interaction of perfectly rational agents (for example, a defeat relation among strategies in a game might be anti-symmetric but not total or transitive).

For each interpretation of “best” (maximal, optimal, unmatched or acceptable) and each combination of constraints (transitivity, totality and/or anti-symmetry), we study the sets of valid inferences. Specifically, in all but one case we introduce a sound and strongly complete axiomatization, and in the one remaining case we show that no such axiomatization exists.

偏好关系是人工智能中许多基本概念的核心，从效用比较到策略之间的失败以及国家之间的相对合理性，仅举几例。对这种关系进行推理一直是广泛研究的对象，并且存在大量的形式来表达和推理它们。一种这样的形式主义是条件逻辑，它侧重于根据给定的偏好关系推理“最佳”替代方案。“最佳”备选方案通常被解释为最大的备选方案（没有其他备选方案比它更受欢迎）或最佳备选方案（它至少与所有其他备选方案一样受欢迎）。并且偏好关系通常被假设为满足强烈的要求（通常是传递性和某种有根据的假设）。

在这里，我们以两种方式概括现有的文献。首先，除了最大化和最优之外，我们还考虑了对“最佳”的另外两种解释，我们称之为不匹配性和可接受性。其次，我们本质上并不要求偏好关系满足任何约束。相反，我们允许该关系满足传递性、整体性和反对称性的任意组合。

这使我们能够对各种情况进行建模，包括缺乏约束的情况是由于建模的代理是非理性的（例如，代理可能具有既不是传递的也不是完全的也不是反对称的偏好）或来自交互的情况。完全理性的代理人（例如，博弈中策略之间的失败关系可能是反对称的，但不是完全的或传递的）。

对于“最佳”的每种解释（最大、最优、不匹配或可接受）和约束的每种组合（传递性、整体性和/或反对称），我们研究了有效推论的集合。具体来说，除一种情况外，我们都引入了合理且强烈的完全公理化，而在剩下的一种情况下，我们证明不存在这样的公理化。