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Towards a canonical classical natural deduction system

2013
*
Annals of Pure and Applied Logic
*

This paper studies

doi:10.1016/j.apal.2012.05.008
fatcat:wseksboskbbdzknydmqtu4eznq
*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. ...##
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Towards a Canonical Classical Natural Deduction System
[chapter]

2010
*
Lecture Notes in Computer Science
*

This paper studies

doi:10.1007/978-3-642-15205-4_24
fatcat:bsgzp3s2mrgqnkkfvfgobmtpbq
*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. ...##
###
The Deduction Theorem for Strong Propositional Proof Systems
[chapter]

2007
*
Lecture Notes in Computer Science
*

In particular, this yields

doi:10.1007/978-3-540-77050-3_20
fatcat:2b7dlniphnfv3ofvduq62i3tce
*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? ...##
###
The Deduction Theorem for Strong Propositional Proof Systems

2008
*
Theory of Computing Systems
*

In particular, this yields

doi:10.1007/s00224-008-9146-6
fatcat:tn76ejrx25fp3j6wg4i725v7c4
*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? ...##
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Hybrid Deduction–Refutation Systems

2019
*
Axioms
*

I then illustrate the concept on

doi:10.3390/axioms8040118
fatcat:ar4xhq35yjcsvjxa7hnz5fqvim
*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 ...##
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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). ...##
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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*...##
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Hybrid Deduction-Refutation Systems for FDE-Based Logics

2021
*
The Australasian Journal of Logic
*

The proofs of completeness are presented in

doi:10.26686/ajl.v18i6.7080
fatcat:r4h2lnb2vrct3bhietnbtqw2u4
*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*...##
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Logic or reason?

2013
*
Logic and Logical Philosophy
*

It argues against the wide spread assumptions that logic is:

doi:10.12775/llp.2012.008
fatcat:xaodvn7wejfrvko75lemwhj4pm
*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*). ...##
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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. ...##
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The Fundamental Problem of General Proof Theory

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

doi:10.1007/s11225-018-9785-9
fatcat:nkzfmdsqmjcobptej2ytgp4yue
*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*. ...##
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A Rewrite System for Proof Constructivization

2016
*
Proceedings of the Eleventh Workshop on Logical Frameworks and Meta-Languages: Theory and Practice - LFMTP '16
*

We use the ability of rewrite

doi:10.1145/2966268.2966270
dblp:conf/lfmtp/Cauderlier16
fatcat:dcaqfyvsrjc5rcylifejfkzd4q
*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*. ...##
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Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural Logicism

2014
*
Philosophia Mathematica
*

This is Part II of

doi:10.1093/philmat/nku009
fatcat:lr7aadqo5zgltlhzmicgmfapwi
*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 ...##
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Logical Frameworks
[chapter]

2001
*
Handbook of Automated Reasoning
*

For the definition of

doi:10.1016/b978-044450813-3/50019-9
fatcat:h5gede7ne5eidp5d5aqyviuo3q
*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*. . . ...##
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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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