紧缩论,证明与保守性

紧缩论者主张真谓词表达了一种逻辑概念,它的全部意义都体现在所有塔斯基式的T-语句中。Shapiro近来论证说,将紧缩论的公理添加到一阶皮亚诺算术公理系统(PA)中,在该扩张理论中能够证明PA的可靠性,并在此基础上证明PA的一致性,这表明紧缩论不具有保守性,因此真谓词不是紧缩的。本文论证,扩张理论预设了反射原则,这导致它推出了更多的东西,而反射原则是可证性谓词定义的推论,这才是造成扩张理论非保守性的真正根源。针对紧缩论的非保守性论证因此失效了。

Deflationism,Proof, and Conservativeness

Deflationists have claimed that truth predicate expresses one logical notion, of which whole meaning is presented in all Tarskian T-sentences. However, Shapiro ar-gued recently that in an extension of the first order Peano arithmetic （PA） by adding the axiom schema of deflationism into it, it can be proved that PA is sound and therefore con-sistent. By this, Shapiro showed that deflationism is non-conservative over PA, hence truth predicate is not deflationary. In this article, I argue that the proof of soundness of PA presupposes the local reflection principle implicitly, that causes the extension to have more consequences. Moreover, the principle is a consequence of definition of provability predicate which is the real cause of non-conservativeness. Thus the non-conservativeness argument against deflationism misfires.