We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can be safely expanded with a transparent truth predicate. Nevertheless, CM cannot recapture every classically valid meta-metainference. We will afterwards develop a hierarchy of consequence relations CMn for metainferences of level n (for 1 ≤ n < ω). Each CMn recovers every metainference of level n or less, and can be nontrivially expanded with a transparent truth predicate, but cannot recapture every classically valid metainferences of higher levels. Finally, we will present a logic CMω, based on the hierarchy of logics CMn, that is fully classical, in the sense that every classically valid metainference of any level is valid in it. Moreover, CMω can be nontrivially expanded with a transparent truth predicate.

中文翻译：

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以非结构手段为特征的完全经典的真理理论

我们将提出一个三值后果关系元推论, 称为厘米，通过定义英石和TS, 两个众所周知的底层结构推论的后果关系。尽管英石恢复每个经典有效的推理，它使一些经典有效的元推理无效。尽管厘米作为英石在推理层面，它还恢复了每一个经典有效的元推理。而且，厘米可以使用透明的真值谓词安全地扩展。尽管如此，厘米无法重新捕获每一个经典有效的元元推理。之后我们将开发后果关系的层次结构厘米n水平参考n（对于 1 ≤n<ω）。每个厘米n恢复水平的每一个metaferencen或更少，并且可以用透明的真值谓词进行非平凡的扩展，但不能重新捕获更高级别的每个经典有效元推论。最后，我们将提出一个逻辑厘米ω, 基于逻辑层次厘米n， 那是完全古典的，从某种意义上说，任何级别的每一个经典有效的元推理在其中都是有效的。而且，厘米ω可以用透明的真值谓词进行非平凡的扩展。