Hintikka and Sandu originally proposed Independence Friendly Logic (IF) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the IF semantics exhibits some limitations. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, the truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, IF does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. In this paper, we investigate an extension of IF, called Alternating Dependence/Independence Friendly Logic (ADIF), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for IF. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants ADIF the full descriptive power of Second Order Logic. We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of ADIF, the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.

翻译：欣蒂卡与桑杜最初提出的独立友好逻辑（IF）是一种含不完美信息的一阶逻辑，用于描述自然语言语义背后的博弈论现象。该逻辑能够表达量化变量间的独立约束（与亨金量词类似），并具有基于不完美信息博弈的优美博弈论语义。然而，IF语义存在某些局限性：它在评估语句真值与假值时不对称地对待博弈双方，仅考虑其中一方具有不完美信息。此外，语句的真假判定等价于语义不完美信息博弈中某一方存在统一获胜策略，这使得IF允许出现既非真亦非假的未定语句，从而违背排中律。本文研究了一种名为交替依赖/独立友好逻辑（ADIF）的IF扩展，旨在克服上述局限。为此我们提出一种新型组合语义，该语义推广了霍奇斯为IF提出的基于王牌（trumps）的语义框架。新语义：(i)能够同时有意义地限制博弈双方，(ii)满足博弈论确定性，(iii)恢复语句的排中律，(iv)赋予ADIF与二阶逻辑相当的完整描述能力。我们还为ADIF的前束片段提供了等效的赫布兰德-斯科伦语义和博弈论语义，后者通过一种确定性的无限持续博弈定义，该博弈在有限结构上精确捕获了前两种语义。