David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Synthese 167 (2):271 - 315 (2009)
Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.
|Keywords||Boolean logic Tractability Semantic information Analytical reasoning Natural deduction|
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References found in this work BETA
Michael A. E. Dummett (1991). The Logical Basis of Metaphysics. Harvard University Press.
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Claude E. Shannon & Warren Weaver (1949). The Mathematical Theory of Communication. University of Illinois Press.
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Citations of this work BETA
Luciano Floridi (2014). Information Closure and the Sceptical Objection. Synthese 191 (6):1037-1050.
Luciano Floridi (2010). Information, Possible Worlds and the Cooptation of Scepticism. Synthese 175 (1):63 - 88.
Marie Duží (2010). The Paradox of Inference and the Non-Triviality of Analytic Information. Journal of Philosophical Logic 39 (5):473 - 510.
Jonas Clausen Mork (2015). Information Gain and Approaching True Belief. Erkenntnis 80 (1):77-96.
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