Seminars of Philosophy of Perception, Mind, and Language

Marcello D'Agostino

February 22 MON — 16.00-18.00 [PLEASE NOTE UNUSUAL TIME!!]

Sala "Enzo Paci" — Direzione del Dipartimento (Via Festa del Perdono 7, Milano)

Informational semantics and the problem of logical omniscience.

Abstract: Logic is informationally trivial and, yet, computationally hard. This is one of the most baffling paradoxes arising from the traditional account of logical consequence. Triviality stems from the widely accepted characterization of deductive inference as non-ampliative: the information carried by the conclusion is (in some sense) contained in the information carried by the premises. Computational hardness stems from the well-known results showing that most interesting logics are undecidable or, even when decidable, very likely to be intractable. This situation leads to the so-called ``scandal of deduction'' and to the related ``problem of logical omniscience'': if logic is informationally trivial, then a rational agent should always be informed that a certain sentence is a logical consequence of the data. However, this is clearly not the case when the logic in question lacks a feasible decision procedure. A typical response to this paradox is that logical systems are idealizations and, as such, not intended to faithfully describe/prescribe the actual deductive behaviour of real agents. To address this problem, we present a unifying semantic and proof-theoretical framework for investigating depth-bounded approximations to Boolean Logic. These approximations provide a hierarchy of tractable logical systems that indefinitely converge to classical propositional logic and can be usefully employed to model the inferential activity of real-life, resource-bounded agents. The operational rules, which are shared by all approximation systems, are justified by an ``informational semantics'' whereby the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent. We show that the informational meaning of the logical operators that arises from this semantics is consistent with a ``strong manifestability requirement'': any agent who grasps the (informational) meaning of the logical operators should be able to tell, in practice and not only in principle, whether or not s(he) holds the information that a given complex sentence is true, or the information that it is false, or neither of the two. This informational semantics also allows us to draw a sharp demarcation between ``analytic'' (uninformative, tautological) and ``synthetic'' (informative, non-tautological) inferences in propositional logic, which defies the empiricist dogma that all logical inferences should belong to the first class. It also provides the means for defining degrees of syntheticity of logical inferences that may be related, on the one hand, to the ``cognitive effort'' required by an agent to recognize their validity and, on the other, to the computational resources that need to be consumed for this task.