Categorical Harmony and Paradoxes in Proof-Theoretic Semantics
Categorical Harmony and Paradoxes in Proof-Theoretic Semantics
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证明理论语义学中的范畴和谐与悖论
DOI：
10.1007/978-3-319-22686-6_6
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Y. Maruyama
中科院分区：
文献类型：
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作者：
K. Abe;Y. Giga;K. Shade;T. Suzuki;K. Abe;Y. Maruyama;K.Abe;Y. Maruyama;阿部健;Y. Maruyama
意义理论有两大阵营：以戴维森为代表的指称主义阵营和以达米特和布兰德为代表的推理主义阵营。证明论语义学（Prooftheoretical Semantics）是一项语义学事业，在根岑、普拉维茨和马丁-洛夫的证明论传统中，阐明了逻辑常数和推论的意义的推理主义解释，用另一条路径“从证明到意义”取代戴维森的路径“从真理到意义”。本文旨在为范畴证明论语义学的发展做出贡献，提出范畴和谐原则，从而对Prior的“tonk”和相关的悖论逻辑常数提供结构性的解释。范畴和谐建立在Lawvere的概念，逻辑常数作为伴随函子，这相当于双线规则的某种形式的推理条款。在概念上，范畴和谐支持逻辑的迭代概念。根据范畴和谐，逻辑常数的悖论性存在内涵程度;根据内涵区分，Russell型悖论常数悖论性最强，Tonk悖论性较弱。对tonk问题的分类诊断是tonk混淆了二元真与假常数，将真与假等同起来;因此普赖尔的tonk悖论是由模棱两可引起的，而罗素悖论不是。这告诉我们，普赖尔的唐克型悖论可以通过消歧来解决，而罗素型悖论则不能。因此，范畴和谐使我们能够在唐克型的伪悖论和罗素型的真正悖论之间划定一条界线。最后，我认为，基于范畴逻辑方法的范畴语义学甚至可能为调和和统一这两个阵营铺平道路。
There are two camps in the theory of meaning: the referentialist one including Davidson, and the inferentialist one including Dummett and Brandom. Prooftheoretic semantics is a semantic enterprise to articulate an inferentialist account of the meaning of logical constants and inferences within the proof-theoretic tradition of Gentzen, Prawitz, and Martin-Löf, replacing Davidson’s path “from truth to meaning” by another path “from proof to meaning”. The present paper aims at contributing to developments of categorical proof-theoretic semantics, proposing the principle of categorical harmony, and thereby shedding structural light on Prior’s “tonk” and related paradoxical logical constants. Categorical harmony builds upon Lawvere’s conception of logical constants as adjoint functors, which amount to double-line rules of certain form in inferential terms. Conceptually, categorical harmony supports the iterative conception of logic. According to categorical harmony, there are intensional degrees of paradoxicality of logical constants; in the light of the intensional distinction, Russell-type paradoxical constants are maximally paradoxical, and tonk is less paradoxical. The categorical diagnosis of the tonk problem is that tonk mixes up the binary truth and falsity constants, equating truth with falsity; hence Prior’s tonk paradox is caused by equivocation, whereas Russell’s paradox is not. This tells us Prior’s tonk-type paradoxes can be resolved via disambiguation while Russell-type paradoxes cannot. Categorical harmony thus allows us to demarcate a border between tonk-type pseudo-paradoxes and Russell-type genuine paradoxes. I finally argue that categorical semantics based on the methods of categorical logic might even pave the way for reconciling and uniting the two camps.