Doing and Showing

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure

Reasoning about Knowledge in Linear Logic: Modalities and Complexity

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has no

A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Linear logic for constructive mathematics

We show that numerous distinctive concepts of constructive mathematics arise automatically from an interpretation of "linear higher-order logic" into intuitionistic higher-order logic via a Chu construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of classical concepts using the choice between multiplicative and additive linear connectives. Linear logic thus systematically "constructivizes" classical definitions and deals automatically with the resulting bookkeeping, and could potentially be used directly as a basis for constructive mathematics in place of intuitionistic logic.Comment: 39 page

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later

Gentzen-Prawitz Natural Deduction as a Teaching Tool

We report a four-years experiment in teaching reasoning to undergraduate students, ranging from weak to gifted, using Gentzen-Prawitz's style natural deduction. We argue that this pedagogical approach is a good alternative to the use of Boolean algebra for teaching reasoning, especially for computer scientists and formal methods practionners

Automatically detecting open academic review praise and criticism

This is an accepted manuscript of an article published by Emerald in Online Information Review on 15 June 2020. The accepted version of the publication may differ from the final published version, accessible at https://doi.org/10.1108/OIR-11-2019-0347.Purpose: Peer reviewer evaluations of academic papers are known to be variable in content and overall judgements but are important academic publishing safeguards. This article introduces a sentiment analysis program, PeerJudge, to detect praise and criticism in peer evaluations. It is designed to support editorial management decisions and reviewers in the scholarly publishing process and for grant funding decision workflows. The initial version of PeerJudge is tailored for reviews from F1000Research’s open peer review publishing platform. Design/methodology/approach: PeerJudge uses a lexical sentiment analysis approach with a human-coded initial sentiment lexicon and machine learning adjustments and additions. It was built with an F1000Research development corpus and evaluated on a different F1000Research test corpus using reviewer ratings. Findings: PeerJudge can predict F1000Research judgements from negative evaluations in reviewers’ comments more accurately than baseline approaches, although not from positive reviewer comments, which seem to be largely unrelated to reviewer decisions. Within the F1000Research mode of post-publication peer review, the absence of any detected negative comments is a reliable indicator that an article will be ‘approved’, but the presence of moderately negative comments could lead to either an approved or approved with reservations decision. Originality/value: PeerJudge is the first transparent AI approach to peer review sentiment detection. It may be used to identify anomalous reviews with text potentially not matching judgements for individual checks or systematic bias assessments