Rules and derivations in an elementary logic course

When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them, and are related to nothing they already know, unlike, for instance, the notion of model, that can be seen as a generalization of the notion of algebraic structure. In this note, we defend the idea that one strategy to introduce these notions is to start with the notion of inductive definition [1]. Then, the notion of derivation comes naturally. We also defend the idea that derivations are pervasive in logic and that defining precisely this notion at an early stage is a good investment to later define other notions in proof theory, computability theory, automata theory, ... Finally, we defend the idea that to define the notion of derivation precisely, we need to distinguish two notions of derivation: labeled with elements and labeled with rule names. This approach has been taken in [2]

Principles and Implementation of Deductive Parsing

We present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as to implement the corresponding parser. The method generalizes easily to parsers for augmented phrase structure formalisms, such as definite-clause grammars and other logic grammar formalisms, and has been used for rapid prototyping of parsing algorithms for a variety of formalisms including variants of tree-adjoining grammars, categorial grammars, and lexicalized context-free grammars.Comment: 69 pages, includes full Prolog cod

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

An Abstract Approach to Stratification in Linear Logic

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic

Data-Oriented Language Processing. An Overview

During the last few years, a new approach to language processing has started to emerge, which has become known under various labels such as "data-oriented parsing", "corpus-based interpretation", and "tree-bank grammar" (cf. van den Berg et al. 1994; Bod 1992-96; Bod et al. 1996a/b; Bonnema 1996; Charniak 1996a/b; Goodman 1996; Kaplan 1996; Rajman 1995a/b; Scha 1990-92; Sekine & Grishman 1995; Sima'an et al. 1994; Sima'an 1995-96; Tugwell 1995). This approach, which we will call "data-oriented processing" or "DOP", embodies the assumption that human language perception and production works with representations of concrete past language experiences, rather than with abstract linguistic rules. The models that instantiate this approach therefore maintain large corpora of linguistic representations of previously occurring utterances. When processing a new input utterance, analyses of this utterance are constructed by combining fragments from the corpus; the occurrence-frequencies of the fragments are used to estimate which analysis is the most probable one. In this paper we give an in-depth discussion of a data-oriented processing model which employs a corpus of labelled phrase-structure trees. Then we review some other models that instantiate the DOP approach. Many of these models also employ labelled phrase-structure trees, but use different criteria for extracting fragments from the corpus or employ different disambiguation strategies (Bod 1996b; Charniak 1996a/b; Goodman 1996; Rajman 1995a/b; Sekine & Grishman 1995; Sima'an 1995-96); other models use richer formalisms for their corpus annotations (van den Berg et al. 1994; Bod et al., 1996a/b; Bonnema 1996; Kaplan 1996; Tugwell 1995).Comment: 34 pages, Postscrip

Informal proof, formal proof, formalism

Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened