David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Philosophical Logic 39 (2):159-171 (2010)
Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this.
|Keywords||Propositional Logic Natural Deduction Expressive Power Logical Connectives Modal Logic Intuitionistic Logic Incompleteness|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Nuel Belnap (1962). Tonk, Plonk and Plink. Analysis 22 (6):130-134.
Nuel D. Belnap & Gerald J. Massey (1990). Semantic Holism. Studia Logica 49 (1):67-82.
Patrick Blackburn, Maarten de Rijke & Yde Venema (2002). Modal Logic. Cambridge University Press.
George Boolos (1993). The Logic of Provability. Cambridge University Press.
Rudolf Carnap (1943). Formalization of Logic. Cambridge, Mass.,Harvard University Press.
Citations of this work BETA
No citations found.
Similar books and articles
Sara Negri (2002). Varieties of Linear Calculi. Journal of Philosophical Logic 31 (6):569-590.
M. W. Bunder (1982). Deduction Theorems for Weak Implicational Logics. Studia Logica 41 (2-3):95 - 108.
Yannis Delmas-Rigoutsos (1997). A Double Deduction System for Quantum Logic Based on Natural Deduction. Journal of Philosophical Logic 26 (1):57-67.
Heinrich Wansing (2006). Connectives Stranger Than Tonk. Journal of Philosophical Logic 35 (6):653 - 660.
Ewa Orlowska (1992). Relational Proof System for Relevant Logics. Journal of Symbolic Logic 57 (4):1425-1440.
Torben Braüner (2004). Two Natural Deduction Systems for Hybrid Logic: A Comparison. [REVIEW] Journal of Logic, Language and Information 13 (1):1-23.
L. Humberstone & D. Makinson (2012). Intuitionistic Logic and Elementary Rules. Mind 120 (480):1035-1051.
Sara Negri (2011). Proof Analysis: A Contribution to Hilbert's Last Problem. Cambridge University Press.
Ross Thomas Brady (2010). Free Semantics. Journal of Philosophical Logic 39 (5):511 - 529.
Hartley Slater (2008). Harmonising Natural Deduction. Synthese 163 (2):187 - 198.
Added to index2009-10-28
Total downloads65 ( #28,400 of 1,679,366 )
Recent downloads (6 months)3 ( #78,917 of 1,679,366 )
How can I increase my downloads?