An informational view of classical logic

Abstract

We present an informational view of classical propositional logic that stems from a kind of informational semantics whereby the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent. In this view the inferential power of logical agents is naturally bounded by their limited capability of manipulating "virtual information", namely information that is not implicitly contained in the data. Although this informational semantics cannot be expressed by any finitely-valued matrix, it can be expressed by a non-deterministic 3-valued matrix that was first introduced by W.V.O. 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.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]. I wish to thank Maribel Fernandez and Marcelo Finger for inviting me to the workshop and all the participants for an interesting discussion. I also wish to thanks two anonymous referees for valuable comments and suggestions.

Decidability and complexity of fibred logics without shared connectives

“…Last but not least, refining the idea above, a particularly appealing application of our results seems to be in approximate reasoning in the sense used also in [23,15]. In the scenario above, even when for each Γ and ϕ there is a finite A Γ,ϕ ⊆ Ax such that Γ, Ax 12 ϕ if and only if Γ, A Γ,ϕ 12 ϕ, it will often be the case that this set A Γ,ϕ is too big, or not effectively/efficiently calculable.…”

Depth-Bounded Approximations of Probability

“…Before that, let us finish this section recalling two important properties of the DB logics, already mentioned in the introduction, and shown e.g. in [3,4]. First, DB logics provide a hierarchy of consequence relations approximating the classical one, that is, k ⊆ k+1 and lim k→∞ k = , where stands for classical derivability.…”

“…A related issue has been investigated in logic, where the family of DB logics [3,4] relies on the idea of separating two kinds of (classically valid) inferences: the inferences which only serve the purpose to make explicit the information that agents already possess, i.e. those using only their actual information on the one hand, and those which make use of virtual information on the other.…”

Classical logic, argument and dialectic

“…Natural deduction proofs apply intuitive introduction and elimination rules (or "intelim rules" for short) that are akin to natural modes of human reasoning (in contrast to other proof theories e.g., axiomatic systems, Gentzen-style sequent calculi or resolution), and are thus particularly appropriate if one is to simulate human understanding and reasoning. 23 We use a non-standard version of classical natural deduction [20,22,23], that we call "C-intelim" (for "classical intelim"). As discussed in [20,24], this version is more faithful to the intuitive classical meaning of the logical operators and naturally suggests a simple measure of the "depth" of an argument.…”

“…For example, in the leftmost branch of the first tree in Figure 4, the formula ¬q is obtained from the actual assumption p → ¬q at the top of the tree and from the virtual assumption p by means of an application of the rule → E1; and r is obtained from the derived formula ¬q and the actual assumption q ∨ r by means of an application of ∨E1. [22,23] argue that the minimum number of nested applications of RB required to develop a deductive argument, provides a natural and plausible measure of the 'difficulty' involved in constructing it.…”