Paradox without Self-Reference
Paradox without Self-Reference
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没有自我参照的悖论
作者：
S. Yablo
该解释背后的基本观点是，我们需要区分一组假设的直接不一致性或自相矛盾与悖论性。两者都涉及荒谬性的证明（1）。但是与直接矛盾相关的荒谬性证明是可归约的，而与悖论相关的证明则不是。当证明可以通过有限次应用归约程序而化为范式时，它们就是可归约的。这些归约程序旨在消除不必要的冗长。最重要的是，通过应用一个逻辑算子的引入规则，然后立即应用相应的消除规则，就可能产生这种冗长。其结果是在证明中出现一个句子，它作为引入规则应用的结论以及相应消除规则应用的主要前提。归约消除了这种“最大”句子的出现，它们在证明中就像不需要的“关节”。逻辑算子的归约程序旨在消除证明中这种不必要的迂回。1
The basic idea behind that account is that we need to distinguish straightforward inconsistency, or self-contradiction, of a set of assumptions, from paradoxicality. Both involve proofs of absurdity (1). But the proofs of absurdity in connection with straightforward contradictions are normalizable, whereas those in connection with paradoxes are not. Proofs are normalizable when they can be brought into normal form by a finite sequence of applications of reduction procedures. These reduction procedures are designed to get rid of unnecessary prolixity. Such prolixity can arise, most importantly, by applying an introduction rule for a logical operator and then immediately applying the corresponding elimination rule. The result is a sentence occurrence within the proof standing as the conclusion of an application of the introduction rule and as the major premiss of an application of the corresponding elimination rule. Reductions get rid of such 'maximal' sentence occurrences, which stand as unwanted 'knuckles' in the proof. The reduction procedures for the logical operators are designed to eliminate such unnecessary detours within proofs.1