相干逻辑关系语义的推理解释
|
摘要
关系语义是相干逻辑中最为重要的语义学之一,但是关系语义一开始就以"纯粹"的形式语义的面貌出现,其中三元关系R的直观意思是什么并不清楚,于是出现了关于关系语义的多种解释。我们认为,R所代表的是推理规则集、前提集和结论集三者之间的关系,据此提出了推理语义。推理语义以推理的形式结构为背景,有明确的直观意义。本文以相干逻辑系统R+为例,证明出推理语义是与关系语义等价的语义。从推理语义与关系语义的这个等价关系上看,推理语义完全可以作为对于关系语义直观解释的一个中间环节。由此不仅使得关系语义有了推理结构的解释,同时说明了相干逻辑是一种关于推理的逻辑。这与相干逻辑产生的历史也完全吻合。
Relational semantics is one of the most popular semantics for relevance logic. However, because this semantics is a kind of "purely" formal semantics, the ternary relation R lacks intuition, and that is why there are various interpretations for relational semantics. As to our opinion, R is a representation of the relation among rules, premises and conclusions. Based on this opinion, inference semantics is presented. The background of inference semantics is the formal structure of inference and this feature makes inference semantics intuitive for our understanding. In this paper, we will prove that relational semantics and inference semantics are equivalent with respect to the relevance logical system R+. From this equivalence relation, inference semantics could be viewed as an intermediate link for the intuitive interpretation of relational semantics. By inference semantics, relational semantics could got an interpretation from the structure of inference. Relevance logic is a logic of inference, so inference semantics is coincident with the history of relevance logic.
Relational semantics is one of the most popular semantics for relevance logic. However, because this semantics is a kind of "purely" formal semantics, the ternary relation R lacks intuition, and that is why there are various interpretations for relational semantics. As to our opinion, R is a representation of the relation among rules, premises and conclusions. Based on this opinion, inference semantics is presented. The background of inference semantics is the formal structure of inference and this feature makes inference semantics intuitive for our understanding. In this paper, we will prove that relational semantics and inference semantics are equivalent with respect to the relevance logical system R+. From this equivalence relation, inference semantics could be viewed as an intermediate link for the intuitive interpretation of relational semantics. By inference semantics, relational semantics could got an interpretation from the structure of inference. Relevance logic is a logic of inference, so inference semantics is coincident with the history of relevance logic.
引文
[1]A.R.Anderson and N.Belnap,1975,Entailment:The Logic of Relevance and Necessity,volume I,Princeton:Princeton University Press.
[2]A.R.Anderson,N.Belnap and J.M.Dunn,1992,Entailment:The Logic of Relevance and Necessity,volume II,Princeton:Princeton University Press.
[3]J.Barwise,1993,“Constraint,channels and the flow of information”,in P.Aczel,D.Israel,Y.Katagiri and S.Peters(eds.),Situation Theory and Its Application,3,pp.3–27,Stanford:CSLI Publication.
[4]D.D.Israel and J.Perry,1990,“What is information?”,in P.Hanson(ed.),Information,Language and Cognition,6,pp.1–19,Vancouver:University of British Columbia Press.
[5]J.Dunn,1986,“Relevance logic and entailment”,in D.Gabbay and F.Guenthner(eds.),Handbook of Philosophical Logic,3,pp.117–124,Dordrecht:Riedel Publishing Company.
[6]J.Dunn and G.Restall,2002,“Relevance logic”,in D.Gabbay and F.Guenthner(eds.),Handbook of Philosophical Logic 2nd,6,pp.86–98,Netherlands:Kluwer Academic Publishers.
[7]E.Mares,1997,“Relevant logic and the theory of information”,Synthese,109:354–360.
[8]E.Mares,2004,Relevant Logic:A Philosophical Interpretation,Cambridge:Cambridge University Press.
[9]G.Restall,1996,“Information flow and relevant logics”,in D.W.J.Seligman(ed.),Logic,Language and Computation,1,pp.463–478,Stanford:CSLI Publication.
[10]R.Routley and R.Meyer,1972,“The semantics of entailment,II”,Journal of Philosophical Logic,(1):53–73.
[11]R.Routley and R.Meyer,1972,“The semantics of entailment,III”,Journal of Philosophical Logic,(1):192–208.
[12]R.Routley and R.Meyer,1973,“The semantics of entailment”,in H.Leblanc(ed.),Truth,Syntax and Modality,pp.199–243,Amsterdam:North-Holland.
[13]冯棉,相干逻辑研究,2010年,华东师范大学出版社。
[14]胡光远,“相干逻辑的情境论解释”,逻辑学研究,2013年第6卷第4期,第105–115页。
[15]周北海,“衍推系统Efde的推理模型”,自然辩证法研究(增刊),1996年,第10–12页。
[16]周北海,“推理、推理模型和推理语义”,逻辑今探——中国逻辑学会第五次代表大会暨学术讨论会论文集(1996),1999年,中国社会科学文献出版社,第3–15页。
1 这些条件因为依赖关系与原文的排序有所不同(详情见命题2)。
2 R2(ab)cd=x(Rabx;Rxcd),R2df a(bc)d=df x(Raxd Rbcx)。
3这是Rabc的严格表述。确切的说,[ab]是由规则集a和前提集b得到的结论集。Rabc即[ab]包含于c。c是包含[ab]的集合,为表述方便也简称结论集。
4这个现象大概与推理语义产生的过程有关。Dunn&Restall([6])指出这一语义是由Routly和Urquhart在二十世纪六十年代和七十年代所分别提出的操作语义(operational semantics)的发展。操作语义中的框架是一个三元组,其中K是信息片段(pieces of information)的集合,o是K上信息片段之间的复合运算,0表示的是“空的信息片段”。
5在本文的写作过程中,我们询问了N.Belanp关于关系语义的三元关系直观意义的问题。Belnap在回复中说:“我花了很多年来考虑相干逻辑中三元关系的问题。最后,关于三元关系的直观意义,我所能说的也就是Anderson&Belnap([2])中三处非常短的引述。这三处引述是由R.Meyer、A.Urquhart和K.Fine发现的。关系语义虽然形式上非常漂亮,但是我却不知道它到底在说什么。因为这三条引述非常短,所以我认为没什么人能在关系语义的直观意义问题上说些更多的东西。……但是在我看来,这始终是‘眼中的沙子’”。
[2]A.R.Anderson,N.Belnap and J.M.Dunn,1992,Entailment:The Logic of Relevance and Necessity,volume II,Princeton:Princeton University Press.
[3]J.Barwise,1993,“Constraint,channels and the flow of information”,in P.Aczel,D.Israel,Y.Katagiri and S.Peters(eds.),Situation Theory and Its Application,3,pp.3–27,Stanford:CSLI Publication.
[4]D.D.Israel and J.Perry,1990,“What is information?”,in P.Hanson(ed.),Information,Language and Cognition,6,pp.1–19,Vancouver:University of British Columbia Press.
[5]J.Dunn,1986,“Relevance logic and entailment”,in D.Gabbay and F.Guenthner(eds.),Handbook of Philosophical Logic,3,pp.117–124,Dordrecht:Riedel Publishing Company.
[6]J.Dunn and G.Restall,2002,“Relevance logic”,in D.Gabbay and F.Guenthner(eds.),Handbook of Philosophical Logic 2nd,6,pp.86–98,Netherlands:Kluwer Academic Publishers.
[7]E.Mares,1997,“Relevant logic and the theory of information”,Synthese,109:354–360.
[8]E.Mares,2004,Relevant Logic:A Philosophical Interpretation,Cambridge:Cambridge University Press.
[9]G.Restall,1996,“Information flow and relevant logics”,in D.W.J.Seligman(ed.),Logic,Language and Computation,1,pp.463–478,Stanford:CSLI Publication.
[10]R.Routley and R.Meyer,1972,“The semantics of entailment,II”,Journal of Philosophical Logic,(1):53–73.
[11]R.Routley and R.Meyer,1972,“The semantics of entailment,III”,Journal of Philosophical Logic,(1):192–208.
[12]R.Routley and R.Meyer,1973,“The semantics of entailment”,in H.Leblanc(ed.),Truth,Syntax and Modality,pp.199–243,Amsterdam:North-Holland.
[13]冯棉,相干逻辑研究,2010年,华东师范大学出版社。
[14]胡光远,“相干逻辑的情境论解释”,逻辑学研究,2013年第6卷第4期,第105–115页。
[15]周北海,“衍推系统Efde的推理模型”,自然辩证法研究(增刊),1996年,第10–12页。
[16]周北海,“推理、推理模型和推理语义”,逻辑今探——中国逻辑学会第五次代表大会暨学术讨论会论文集(1996),1999年,中国社会科学文献出版社,第3–15页。
1 这些条件因为依赖关系与原文的排序有所不同(详情见命题2)。
2 R2(ab)cd=x(Rabx;Rxcd),R2df a(bc)d=df x(Raxd Rbcx)。
3这是Rabc的严格表述。确切的说,[ab]是由规则集a和前提集b得到的结论集。Rabc即[ab]包含于c。c是包含[ab]的集合,为表述方便也简称结论集。
4这个现象大概与推理语义产生的过程有关。Dunn&Restall([6])指出这一语义是由Routly和Urquhart在二十世纪六十年代和七十年代所分别提出的操作语义(operational semantics)的发展。操作语义中的框架是一个三元组
5在本文的写作过程中,我们询问了N.Belanp关于关系语义的三元关系直观意义的问题。Belnap在回复中说:“我花了很多年来考虑相干逻辑中三元关系的问题。最后,关于三元关系的直观意义,我所能说的也就是Anderson&Belnap([2])中三处非常短的引述。这三处引述是由R.Meyer、A.Urquhart和K.Fine发现的。关系语义虽然形式上非常漂亮,但是我却不知道它到底在说什么。因为这三条引述非常短,所以我认为没什么人能在关系语义的直观意义问题上说些更多的东西。……但是在我看来,这始终是‘眼中的沙子’”。