Semantics and proof-theory of depth bounded Boolean logics

We present a unifying semantical and proof-theoretical framework for investigating depth-bounded approximations to Boolean Logic, namely approximations in which the number of nested applications of a single structural rule, representing the classical Principle of Bivalence, is bounded above by a fixed natural number. These approximations provide a hierarchy of tractable logical systems that indefinitely converge to classical propositional logic. The framework we present here brings to light a general approach to logical inference that is quite different from the standard Gentzen-style approaches, while preserving some of their nice proof-theoretical properties, and is common to several proof systems and algorithms, such as KE, KI and Stalmarck?s method.

[1]  Stephen Cook,et al.  Corrections for "On the lengths of proofs in the propositional calculus preliminary version" , 1974, SIGA.

[2]  Dov M. Gabbay,et al.  Cut and Pay , 2006, J. Log. Lang. Inf..

[3]  Dov M. Gabbay,et al.  The New Logic , 2001, Log. J. IGPL.

[4]  Krysia Broda,et al.  A Solution To A Problem Of Popper , 1995 .

[5]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[6]  Arnon Avron,et al.  Natural 3-valued logics—characterization and proof theory , 1991, Journal of Symbolic Logic.

[7]  Magnus Björk,et al.  First Order Stålmarck , 2008, Journal of Automated Reasoning.

[8]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[9]  Magnus Björk.  A First Order Extension of Stålmarck's Method , 2005, LPAR.

[10]  Luciano Floridi,et al.  The enduring scandal of deduction Is propositional logic really uninformative ? , 2007 .

[11]  J. Hintikka.  Logic, language-games and information : Kantian themes in the philosophy of logic , 1973 .

[12]  Mukesh Dalal.  Anytime Families of Tractable Propositional Reasoners Introduction , .

[13]  Marcelo Finger.  Towards Polynomial Approximations of Full Propositional Logic , 2004, SBIA.

[14]  M. Dummett.  Elements of Intuitionism , 2000 .

[15]  Reiner Hähnle,et al.  Tableaux and Related Methods , 2001, Handbook of Automated Reasoning.

[16]  Anna Zamansky,et al.  Cut-Elimination and Quantification in Canonical Systems , 2006, Stud Logica.

[17]  Arnon Avron,et al.  Canonical Propositional Gentzen-Type Systems , 2001, IJCAR.

[18]  Marcello D'Agostino,et al.  The enduring scandal of deduction , 2009, Synthese.

[19]  Marco Mondadori.  Efficient Inverse Tableaux , 1995, Log. J. IGPL.

[20]  Nuel D. Belnap,et al.  How a Computer Should Think , 2019, New Essays on Belnap-­Dunn Logic.

[21]  Arnon Avron,et al.  Non-deterministic Matrices and Modular Semantics of Rules , 2005 .

[22]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[23]  Marcelo Finger,et al.  The universe of propositional approximations , 2006, Theor. Comput. Sci..

[24]  James M. Crawford,et al.  A Non-Deterministic Semantics for Tractable Inference , 1998, AAAI/IAAI.

[25]  Marcello D'Agostino.  Analytic Inference And The Informational Meaning Of The Logical Operators , 2014 .

[26]  Arnon Avron,et al.  A Non-deterministic View on Non-classical Negations , 2005, Stud Logica.

[27]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[28]  G. Gentzen.  Untersuchungen über das logische Schließen. II , 1935 .

[29]  Marcello D'Agostino,et al.  The Taming of the Cut. Classical Refutations with Analytic Cut , 1994, J. Log. Comput..

[30]  Marcello D'Agostino,et al.  Are tableaux an improvement on truth-tables? , 1992, J. Log. Lang. Inf..

[31]  Marcelo Finger.  Polynomial Approximations of Full Propositional Logic via Limited Bivalence , 2004, JELIA.

[32]  Marcello D'Agostino,et al.  Tableau Methods for Classical Propositional Logic , 1999 .

[33]  Marcelo Finger,et al.  Approximate and Limited Reasoning: Semantics, Proof Theory, Expressivity and Control , 2004, J. Log. Comput..

[34]  Sebastian Sequoiah Grayson.  The Scandal of Deduction. Hintikka on the Information Yield of Deductive Inferences , 2008 .

[35]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[36]  Sebastian Sequoiah-Grayson.  The Scandal of Deduction , 2008, J. Philos. Log..

[37]  Marco Schaerf,et al.  Approximate Reasoning and Non-Omniscient Agents , 1992, TARK.

[38]  Andrzej Indrzejczak,et al.  Natural Deduction, Hybrid Systems and Modal Logics , 2010 .

[39]  Mukesh Dalal.  Anytime clausal reasoning , 2004, Annals of Mathematics and Artificial Intelligence.

[40]  Mary Sheeran,et al.  A Tutorial on Stålmarck's Proof Procedure for Propositional Logic , 2000, Formal Methods Syst. Des..

[41]  Enrico Moriconi,et al.  On Inversion Principles , 2008 .

[42]  Hykel Hosni,et al.  Tractable Depth-Bounded Logics and the Problem of Logical Omniscience , 2010 .

[43]  P. Dangerfield.  Logic , 1996, Aristotle and the Stoics.

[44]  Marcello D'Agostino,et al.  Classical Natural Deduction , 2005, We Will Show Them!.

[45]  Richard L. Epstein,et al.  The Semantic Foundations of Logic Volume 1: Propositional Logics , 1990 .

[46]  Marcello D'Agostino,et al.  Investigations into the complexity of some propositional calculi , 1990 .

[47]  Andrzej Indrzejczak,et al.  Extended Natural Deduction , 2010 .