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Towards a canonical classical natural deduction system

José Espírito Santo
2013 Annals of Pure and Applied Logic  
This paper studies a new classical natural deduction system, presented as a typed calculus named λµlet.  ...  This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of "read-back" of calculi in  ...  So, nothing like a canonical system is obtained through Gentzen-Prawitz approach to classical natural deduction. Moreover, the design difficulties are accompanied by technical problems.  ... 
doi:10.1016/j.apal.2012.05.008 fatcat:wseksboskbbdzknydmqtu4eznq

Towards a Canonical Classical Natural Deduction System [chapter]

José Espírito Santo
2010 Lecture Notes in Computer Science  
This paper studies a new classical natural deduction system, presented as a typed calculus named λµlet.  ...  This calculus and the mentioned isomorphism Θ offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of "read-back" of calculi in  ...  So, nothing like a canonical system is obtained through Gentzen-Prawitz approach to classical natural deduction. Moreover, the design difficulties are accompanied by technical problems.  ... 
doi:10.1007/978-3-642-15205-4_24 fatcat:bsgzp3s2mrgqnkkfvfgobmtpbq

The Deduction Theorem for Strong Propositional Proof Systems [chapter]

Olaf Beyersdorff
2007 Lecture Notes in Computer Science  
In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NP-pairs.  ...  We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and  ...  Are there natural strong proof systems besides Frege systems that satisfy the weak deduction property?  ... 
doi:10.1007/978-3-540-77050-3_20 fatcat:2b7dlniphnfv3ofvduq62i3tce

The Deduction Theorem for Strong Propositional Proof Systems

Olaf Beyersdorff
2008 Theory of Computing Systems  
In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NP-pairs.  ...  We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and  ...  Are there natural strong proof systems besides Frege systems that satisfy the weak deduction property?  ... 
doi:10.1007/s00224-008-9146-6 fatcat:tn76ejrx25fp3j6wg4i725v7c4

Hybrid Deduction–Refutation Systems

Valentin Goranko
2019 Axioms  
I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which I show soundness and completeness for both deductions and refutations.  ...  Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability  ...  Hybrid Extensions of the System of Natural Deduction for PL I will illustrate here the concept of canonical hybrid extension, applied to the system of Natural Deduction for the classical propositional  ... 
doi:10.3390/axioms8040118 fatcat:ar4xhq35yjcsvjxa7hnz5fqvim

Page 3769 of Mathematical Reviews Vol. , Issue 86h [page]

1986 Mathematical Reviews  
The classic results for definite clause logic programs are extended in a simple and natural manner.  ...  theories for which there exists a canonical system of rewriting rules (in a theory with a complete set of reductions).  ... 

Page 134 of Ratio Vol. 2, Issue 1 [page]

1959 Ratio  
What has no place in the Humian system is his account of the apodictic nature of deductive sciences.  ...  There is no doubt that Hume had a deep-rooted antagonism toward Aristotelian logic which he associates with the philosophy of the schoolmen; and, I think, in this he shares the view of other classical  ... 

Hybrid Deduction-Refutation Systems for FDE-Based Logics

Eoin Moore
2021 The Australasian Journal of Logic  
The proofs of completeness are presented in a uniform way. The paper builds on work by Goranko, who presented a deductively and refutationally sound and complete hybrid system for classical logic.  ...  Hybrid deduction-refuation systems are presented for four first-degree entailment based logics. The hybrid systems are shown to be deductively and refutationally sound with respect to their logics.  ...  Hybrid deduction systems for four FDE based logics Defining the hybrid deductive systems The goal of this paper is to work out a uniform treatment for FDE based systems, so let us specify a natural deduction  ... 
doi:10.26686/ajl.v18i6.7080 fatcat:r4h2lnb2vrct3bhietnbtqw2u4

Logic or reason?

P. Rush
2013 Logic and Logical Philosophy  
It argues against the wide spread assumptions that logic is: a model of reason; a model of correct reason; the laws of thought, or indeed is related to reason at all such that the essential nature of the  ...  I note that due to such assumptions, our current understanding of the nature of logic itself is thoroughly entangled with the nature of reason.  ...  S is a deductive system).  ... 
doi:10.12775/llp.2012.008 fatcat:xaodvn7wejfrvko75lemwhj4pm

Page 8925 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
This paper gives a direct proof of the Church-Rosser property of the cut-elimination procedure for full first-order classical natural deduction.  ...  Between the submission (1999) and publication (2002) of this article, a system of natural deduction has been published [J. von Plato, Arch. Math.  ... 

The Fundamental Problem of General Proof Theory

Dag Prawitz
2018 Studia Logica: An International Journal for Symbolic Logic  
It has been surprisingly neglected in logic and philosophy of mathematics with two exceptions: Gentzen's remarks about what justifies the rules of his system of natural deduction and proposals in the intuitionistic  ...  tradition about what a proof is.  ...  The Idea of Valid Argument This was how I understood and developed the remarks that Gentzen made after having defined his system of natural deduction.  ... 
doi:10.1007/s11225-018-9785-9 fatcat:nkzfmdsqmjcobptej2ytgp4yue

A Rewrite System for Proof Constructivization

Raphaël Cauderlier
2016 Proceedings of the Eleventh Workshop on Logical Frameworks and Meta-Languages: Theory and Practice - LFMTP '16  
We use the ability of rewrite systems to represent partial functions to implement heuristics for proof constructivization in Dedukti, a logical framework based on rewriting in which proofs are first-class  ...  Proof constructivization is the problem of automatically extracting constructive proofs out of classical proofs.  ...  Among these 1369 proofs, 1258 (91.9%) were translated to classical natural deduction.  ... 
doi:10.1145/2966268.2966270 dblp:conf/lfmtp/Cauderlier16 fatcat:dcaqfyvsrjc5rcylifejfkzd4q

Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural Logicism

N. Tennant
2014 Philosophia Mathematica  
This is Part II of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic.  ...  Belnap, Jr. [1975] . 12 The system of minimal logic is due to Johansson [1936] . ural nesting of sets of natural-deduction rules in, for example, the canonical presentation of the three systems C, I  ...  '. 4 See Tennant [1978] , at p. 74, where in discussing a detailed proof of soundness of the system of natural deduction for classical first-order logic, we gave a clear indication of rule-circularity  ... 
doi:10.1093/philmat/nku009 fatcat:lr7aadqo5zgltlhzmicgmfapwi

Logical Frameworks [chapter]

Frank Pfenning
2001 Handbook of Automated Reasoning  
For the definition of canonical forms as a deductive system we need two mutually recursive judgments: canonical and atomic forms.  ...  One can still formulate some reductions for classical derivations, but natural deduction is at heart an intuitionistic calculus.  ...  . . . . . . . . . . . . . . . . . . . . 55 C canonical form . . . . . . . . . . . . . . . . . . . . . 59, 65 D deduction theorem . . . . . . . . . . . . . . . . . . . . 53 deductive system . . .  ... 
doi:10.1016/b978-044450813-3/50019-9 fatcat:h5gede7ne5eidp5d5aqyviuo3q

Page 1368 of Mathematical Reviews Vol. , Issue 98C [page]

1998 Mathematical Reviews  
This result is an important step toward a new the- ory of reduction based on expansion rules, and gives a natural interpretation to the notion of second-order y-long normal forms used in higher-order resolution  ...  The authors then modify the translation by introducing a variable denoting “a set of pos- sible valuations” to establish equivalence between general modal deduction in Ks and Q-deduction.  ... 
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