Semantics and proof-theory of depth bounded Boolean logics

An informational view of classical logic

“…Quine, but ignored by the logical community. Within the general framework presented in [21] we provide an in-depth discussion of this informational semantics and a detailed analysis of a specific infinite hierarchy of tractable approximations to classical propositional logic that is based on it. This hierarchy can be used to model the inferential power of resource-bounded agents and admits of a uniform proof-theoretical characterization that is half-way between a classical version of Natural Deduction and the method of semantic tableaux.…”

“…Within the general framework presented in [21] we provide an in-depth discussion of this informational semantics and a detailed analysis of a specific infinite hierarchy of tractable approximations to classical propositional logic that is based on it. This hierarchy can be used to model the inferential power of resource-bounded agents and admits of a uniform proof-theoretical characterization that is half-way between a classical version of Natural Deduction and the method of semantic tableaux.Keywords: Classical Propositional Logic, Informational Semantics, Non-deterministic matrices, Computational Complexity, Natural Deduction sep Semantic Tableaux.$ This paper elaborates on ideas and results arising from my collaboration with Marcelo Finger, Luciano Floridi and Dov Gabbay [22,21] and is based on an invited talk given at the 8th Workshop on Logical and Semantic Frameworks with Applications (LSFA) held in São Paulo on 2-3 September 2013. A preliminary and partial version was published in the proceedings of the workshop [20].…”

“…$ This paper elaborates on ideas and results arising from my collaboration with Marcelo Finger, Luciano Floridi and Dov Gabbay [22,21] and is based on an invited talk given at the 8th Workshop on Logical and Semantic Frameworks with Applications (LSFA) held in São Paulo on 2-3 September 2013. A preliminary and partial version was published in the proceedings of the workshop [20].…”

Semantic Information and the Trivialization of Logic: Floridi on the Scandal of Deduction

“…The answer is yes. The details can be found in [1], [38] and [40], where it is shown that (i) the deducibility relation is a logic in Tarski's sense, (ii) it satisfies the subformula property, and (ii) whether or not Γ ϕ can be decided in polynomial (quadratic) time.…”

“…Such a tree of arguments involving virtual information is a proof of ϕ from Γ whenever each subargument either ends with an explicit inconsistency or with the conclusion T a ϕ. This is only the intuitive idea, the formal details are in [1,[38][39][40]. For each n ∈ N, k is the consequence relation that allows for bounded applications of this proof rule up to a given fixed depth k. It can then be shown that classical logic is the limit of the sequence of these bounded consequence relations k as k tends to infinity.…”