Decision-making assisted by algorithms developed by machine learning is increasingly determining our lives. Unfortunately, full opacity about the process is the norm. Would transparency contribute to restoring accountability for such systems as is often maintained? Several objections to full transparency are examined: the loss of privacy when datasets become public, the perverse effects of disclosure of the very algorithms themselves, the potential loss of companies’ competitive edge, and the limited gains in answerability to be expected since sophisticated algorithms usually are (...) inherently opaque. It is concluded that, at least presently, full transparency for oversight bodies alone is the only feasible option; extending it to the public at large is normally not advisable. Moreover, it is argued that algorithmic decisions preferably should become more understandable; to that effect, the models of machine learning to be employed should either be interpreted ex post or be interpretable by design ex ante. (shrink)
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. (shrink)
Recent years have been blessed with an abundance of logical systems, arising from a multitude of applications. A logic can be characterised in many different ways. Traditionally, a logic is presented via the following three components: 1. an intuitive non-formal motivation, perhaps tie it in to some application area 2. a semantical interpretation 3. a proof theoretical formulation. There are several types of proof theoretical methodologies, Hilbert style, Gentzen style, goal directed style, labelled deductive system style, and so on. The (...) tableau methodology, invented in the 1950s by Beth and Hintikka and later per fected by Smullyan and Fitting, is today one of the most popular, since it appears to bring together the proof-theoretical and the semantical approaches to the pre of a logical system and is also very intuitive. In many universities it is sentation the style first taught to students. Recently interest in tableaux has become more widespread and a community crystallised around the subject. An annual tableaux conference is being held and proceedings are published. The present volume is a Handbook a/Tableaux pre senting to the community a wide coverage of tableaux systems for a variety of logics. It is written by active members of the community and brings the reader up to frontline research. It will be of interest to any formal logician from any area. (shrink)
We show that Smullyan's analytic tableaux cannot p-simulate the truth-tables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cut-free proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableau-like method without affecting its analytic nature.
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  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. (shrink)
We investigate the semantics of the logical systems obtained by introducing the modalities and into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in (...) a previous paper. (shrink)
In this paper we explore a generalization of traditional abduction which can simultaneously perform two different tasks: given an unprovable sequent Γ ⊢ G, find a sentence H such that Γ, H ⊢ G is provable ; given a provable sequent Γ ⊢ G, find a sentence H such that Γ ⊢ H and the proof of Γ, H ⊢ G is simpler than the proof of Γ ⊢ G . We argue that the two tasks should not be distinguished, (...) and present a general procedure for finding suitable hypotheses or lemmas. When the original sequent is provable, the abduced formula can be seen as a cut formula with respect to Gentzen's sequent calculus, so the abduction method is cut-based. Our method is based on the tableau-like system KE and we argue for its advantages over existing abduction methods based on traditional Smullyan-style Tableaux. (shrink)
The papers collected in this volume are based on the best contributions to the conference of the Italian Society for Logic and Philosophy of Science (SILFS) that took place in Milan on 8-10 October 2007. The aim of the Society, since its foundation in 1952, has always been that of bringing together scholars - working in the broad areas of Logic, Philosophy of Science and History of Science - who share an open-minded approach to their disciplines and regard them as (...) essentially requiring continuous confrontation and bridge-building to avoid the danger of over-specialism. In this perspective, logicians and philosophers of science should not indulge in inventing and cherishing their own "internal problems" - although these may occasionally be an opportunity for conceptual clarification - but should primarily look at the challenging conceptual and methodological questions that arise in any genuine attempt to extend our objective knowledge. As Ludovico Geymonat used to put it: " good] philosophy should be sought in the folds of science itself." Contributions are distributed into six sections, five of which - "Logic and Computing," "Physics and Mathematics," "Life Sciences," "Economics and Social Sciences," "Neuroscience and Philosophy of Mind" - are devoted to the discussion of cutting-edge problems that arise from current-day scientific research, while the remaining section on "General Philosophy of Science" is focused on foundational and methodological questions that are common to all areas. (shrink)