We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach (...) laws and Switching laws. (shrink)

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).

This work is devoted to the relations between Lambek’s Syntactic Calculus and noncommutative variants of Girard’s Linear Logic; in particular the paper will consider: the geometrical representation of the laws of LC by means of proof-nets; the discovery - due to such a geometrical representation - of some laws of LC not yet considered; the discussion of possible linguistic uses of these new laws.

Hilbert's conference lectures during the year 1922, Neuebegründung der Mathematik. Erste Mitteilung and Die logischen Grundlagen der Mathematik (both are published in (Hilbert [1935] 1965) pp. 157-195), contain his first public presentation of an axiom system for propositional logic, or at least for a fragment of propositional logic, which is largely influenced by the study on logical woks of Frege and Russell during the previous years.The year 1922 is at the beginning of Hilbert's foundational program in its definitive form. The (...) final public presentation by Hilbert of an axiom system for propositional logic is contained in the 1934 book Grundlagen der Mathematik, I (Hilbert and Bernays [1934] 1968). Sequent .. (shrink)

We study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with "+ 1" instead of "- 1"). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator (...) F the point where the increasing F-sequence terminates. We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id) (ω) -sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in ID 1 . For a general presentation of the results stated in this paper, see [1]. We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25-31]. (shrink)

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach (...) laws and Switching laws. (shrink)

There is currently much interest in bringing together the tradition of categorial grammar, and especially the Lambek calculus, with the recent paradigm of linear logic to which it has strong ties. One active research area is designing non-commutative versions of linear logic (Abrusci, 1995; Retoré, 1993) which can be sensitive to word order while retaining the hypothetical reasoning capabilities of standard (commutative) linear logic (Dalrymple et al., 1995). Some connections between the Lambek calculus and computations in groups have long (...) been known (van Benthem, 1986) but no serious attempt has been made to base a theory of linguistic processing solely on group structure. This paper presents such a model, and demonstrates the connection between linguistic processing and the classical algebraic notions of non-commutative free group, conjugacy, and group presentations. A grammar in this model, or G-grammar is a collection of lexical expressions which are products of logical forms, phonological forms, and inverses of those. Phrasal descriptions are obtained by forming products of lexical expressions and by cancelling contiguous elements which are inverses of each other. A G-grammar provides a symmetrical specification of the relation between a logical form and a phonological string that is neutral between parsing and generation modes. We show how the G-grammar can be oriented for each of the modes by reformulating the lexical expressions as rewriting rules adapted to parsing or generation, which then have strong decidability properties (inherent reversibility). We give examples showing the value of conjugacy for handling long-distance movement and quantifier scoping both in parsing and generation. The paper argues that by moving from the free monoid over a vocabulary V (standard in formal language theory) to the free group over V, deep affinities between linguistic phenomena and classical algebra come to the surface, and that the consequences of tapping the mathematical connections thus established can be considerable. (shrink)

This set reprints volumes that were orginally published by Routledge and Kegan Paul Ltd. in 1953. Landmark volumes at the time of their original publication, these titles do not merely expound the theoretical constructions of Russian philosophers, but also relate these constructions to the general conditions of Russian life. Volume One examines the historical conditions of the development of philosophy in Russia and explores the general features of Russian philosophy. It also surveys the principal works on the history of Russian (...) philosophy. Volume Two includes biographies and examination of the themes of the following philosophers; Vladimir Solovyov, V.D. Kudryavtsev, N.F. Fyodorov and Later Hegelians such as Chicherin, Debolski and Bakunin. This volume also provides analysis of various schools of thought in Russian philosophy in the twentieth century, for example; Neo-Leibzianism, Modern Positivism and New-Marxism to name but a few. (shrink)

When William James spoke about belief to the philosophy clubs of Yale and Brown in 1896, he forewarned his audience of the nature of his comments by describing them as a “sermon on justification by faith” (James 13), titling the talk “The Will to Believe.” Although there is disagreement about the substance of James’s remarks, it is fairly innocuous to assert that James thought they were appropriate because of the prevalence of the “logical spirit” of many of those who practiced (...) academic philosophy that led them to the conclusion that religious faith was untenable. Aware of his audience, James presents his view on the permissibility of religious faith on the terms and grounds familiar to professional philosophers. .. (shrink)

In this paper, I argue that scholars such as John Braithwaite and Lode Walgrave rely on fictions when presenting their utopian vision of restorative justice. Three claims in particular are shown to be fictitious. Proponents of restorative justice maintain, first, that the offender and the victim voluntarily attend the restorative conference. Second, that the restorative conference enables the offender and the victim to take on active responsibility. Third, that the reparatory tasks on which the parties agree should not be understood (...) in terms of the intentional infliction of harm. These fictions, so I argue, are not merely a mistake, but instead serve an important function: the various parties need to believe that they adequately capture the reality of the restorative conference as they are more likely to acquiesce if they believe the fictions to be true. I conclude that the fictions are the driving force within the restorative endeavour. (shrink)

t. 1. Dialog s V. V. Davydovym -- t. 2. Nereshennye problemy razvivai︠u︡shchego obrazovanii︠a︡.

Монография предназначена для специалистов в области социальной философии, социологии и истории, а также для широкого круга читателей.

Substantive scientific letter writing is a cost-effective mode of complementing observational and experimental research. The value of such philosophically uncommitted and unsponsored well-balanced scientific activity has been relegated. Critical letter writing entails the abilities to: maintain rational scepticism; refuse to conform in order to explain data; persist in keeping common sense centre-stage; exercise logic to evaluate the biological significance of mathematical figures, including statistics, and the ability to sustain the will to share insights regarding disease mechanisms on an ostensibly lower (...) research platform. During peer review, innovative letter writing may share the occasionally unfortunate fate of innovative research. Rejected scientific letters do not automatically lose copyright. Periodicals with high letter loads will see some valuable contributions wasted, but that is the price for maintaining autonomy in scientific publication. The scientific community is an integrated whole that must respect the rights of authors at all levels. Unauthorised forwarding of rejected letters sets the dangerous precedent of justifying unjust means. (shrink)