We analyze the expressive resources of \(\mathrm {IF}\) logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular \(\mathrm {IF}\) sentences, this amounts to the study of the fragment of \(\mathrm {IF}\) logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.

我们分析了不源于 Henkin（部分有序）量化的\(\mathrm {IF}\)逻辑的表达资源。当人们将注意力限制在常规\(\mathrm {IF}\)句子上时，这相当于对\(\mathrm {IF}\)片段的研究由动作回忆 (AR) 的博弈论属性个性化的逻辑。我们证明 prenex AR 句子的片段可以表达所有存在的二阶属性。然后我们表明，通过使用“分离信号”而不是 Henkin 或信号模式，可以在 AR 的非 prenex 片段中实现相同的效果。我们还研究了不规则 IF 逻辑（允许重新量化变量）并分析其与规则 IF 逻辑的对应关系。通过使用新方法，我们证明了知识记忆的博弈论特性也是不规则句子的一阶句法约束，并且我们识别了另一个新的一阶片段。最后，我们发现不规则前缀在有限和无限模型中的表现截然不同。特别是，我们表明，在无限结构上，每个不规则前缀都相当于一个规则前缀；我们提出了一个不规则前缀，它在有限模型上是二阶的，但在无限模型上折叠为一阶前缀。