Article

One-and-a-halfth-order logic

Authors:

Murdoch J. Gabbay,

Aad MathijssenAuthors Info & Claims

PPDP '06: Proceedings of the 8th ACM SIGPLAN international conference on Principles and practice of declarative programming

Pages 189 - 200

https://doi.org/10.1145/1140335.1140359

Published: 10 July 2006 Publication History

Abstract

The practice of first-order logic is replete with meta-level concepts. Most notably there are the meta-variables themselves (ranging over predicates, variables, and terms), assumptions about freshness of variables with respect to these meta-variables, alpha-equivalence and capture-avoiding substitution. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both algebraic and sequent specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within itWe discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.

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Cited By

Gabbay M(2016)Semantics Out of ContextJournal of the ACM10.1145/270081963:3(1-66)Online publication date: 28-Jun-2016
https://dl.acm.org/doi/10.1145/2700819
Gabbay M(2013)Nominal Terms and Nominal Logics: From Foundations to Meta-mathematicsHandbook of Philosophical Logic10.1007/978-94-007-6600-6_2(79-178)Online publication date: 9-Apr-2013
https://doi.org/10.1007/978-94-007-6600-6_2
Gabbay MMulligan D(2011)Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal setsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.71.571(58-75)Online publication date: 31-Oct-2011
https://doi.org/10.4204/EPTCS.71.5
Show More Cited By

Index Terms

One-and-a-halfth-order logic
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Theorem proving algorithms
2. Theory of computation
  1. Logic
    1. Proof theory

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One-and-a-halfth-order Logic

The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions ...

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cover image ACM Conferences

PPDP '06: Proceedings of the 8th ACM SIGPLAN international conference on Principles and practice of declarative programming

July 2006

280 pages

ISBN:1595933883

DOI:10.1145/1140335

General Chair:
Annalisa Bossi
Università Ca' Foscari di Venezia, Italy
,
Program Chair:
Michael Maher
National ICT Australia & University of New South Wales, Australia

Copyright © 2006 ACM.

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Publication History

Published: 10 July 2006

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PPDP06: Principles and Practice of Declarative Programming

July 10 - 12, 2006

Venice, Italy

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Overall Acceptance Rate 230 of 486 submissions, 47%

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Cited By

Gabbay M(2016)Semantics Out of ContextJournal of the ACM10.1145/270081963:3(1-66)Online publication date: 28-Jun-2016
https://dl.acm.org/doi/10.1145/2700819
Gabbay M(2013)Nominal Terms and Nominal Logics: From Foundations to Meta-mathematicsHandbook of Philosophical Logic10.1007/978-94-007-6600-6_2(79-178)Online publication date: 9-Apr-2013
https://doi.org/10.1007/978-94-007-6600-6_2
Gabbay MMulligan D(2011)Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal setsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.71.571(58-75)Online publication date: 31-Oct-2011
https://doi.org/10.4204/EPTCS.71.5
Clouston R(2011)Nominal Logic with Equations OnlyElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.71.471(44-57)Online publication date: 31-Oct-2011
https://doi.org/10.4204/EPTCS.71.4
Gabbay M(2009)A study of substitution, using nominal techniques and Fraenkel-Mostowksi setsTheoretical Computer Science10.1016/j.tcs.2008.11.013410:12-13(1159-1189)Online publication date: 20-Mar-2009
https://dl.acm.org/doi/10.1016/j.tcs.2008.11.013
Gabbay MMulligan D(2009)Two-level Lambda-calculusElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2009.07.018246(107-129)Online publication date: 1-Aug-2009
https://dl.acm.org/doi/10.1016/j.entcs.2009.07.018
Gabbay MMathijssen A(2008)Capture-avoiding substitution as a nominal algebraFormal Aspects of Computing10.1007/s00165-007-0056-120:4-5(451-479)Online publication date: 15-Jan-2008
https://doi.org/10.1007/s00165-007-0056-1
Gabbay MMulligan D(2008)One-and-a-Halfth Order Terms: Curry-Howard and Incomplete DerivationsLogic, Language, Information and Computation10.1007/978-3-540-69937-8_16(179-193)Online publication date: 2008
https://doi.org/10.1007/978-3-540-69937-8_16
Gabbay MMathijssen A(2007)A formal calculus for informal equality with bindingProceedings of the 14th international conference on Logic, language, information and computation10.5555/1770176.1770188(162-176)Online publication date: 2-Jul-2007
https://dl.acm.org/doi/10.5555/1770176.1770188
Clouston RPitts A(2007)Nominal Equational LogicElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2007.02.009172(223-257)Online publication date: 1-Apr-2007
https://dl.acm.org/doi/10.1016/j.entcs.2007.02.009
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