Marcello D'Agostino

A generalization of analytic deduction via labelled deductive systems. Part I: Basic substructural logics

In this series of papers we set out to generalize the notion of classical analytic deduction (i.e., deduction via elimination rules) by combining the methodology of labelled deductive systems (LDS) with the classical systemKE. LDS is a unifying framework for the study of logics and of their interactions. In the LDS approach the basic units of logical derivation are not just formulae butlabelled formulae, where the labels belong to a given “labelling algebra”. The derivation rules act on the labels as well as on the formulae, according to certain fixed rules of propagation. By virtue of the extra power of the labelling algebras, standard (classical or intuitionistic) proof systems can be extended to cover a much wider territory without modifying their structure. The systemKE is a new tree method for classical analytic deduction based on “analytic cut”.KE is a refutation system, like analytic tableaux and resolution, but it is essentially more efficient than tableaux and, unlike resolution, does not require any reduction to normal form.We start our investigation with the family of substructural logics. These are logical systems (such as Lambeks calculus, Anderson and Belnaps relevance logic, and Girards linear logic) which arise from disallowing some or all of the usual structural properties of the notion of logical consequence. This extension of traditional logic yields a subtle analysis of the logical operators which is more in tune with the needs of applications. In this paper we generalize the classicalKE system via the LDS methodology to provide a uniform refutation system for the family of substructural logics.The main features of this generalized method are the following: (a) each logic in the family is associated with a “labelling algebra”; (b) the tree-expansion rules (for labelled formulae) are the same for all the logics in the family; (c) the difference between one logic and the other is captured by the conditions under which a branch is declared closed; (d) such conditions depend only on the labelling algebra associated with each logic; and (e) classical and intuitionistic negations are characterized uniformly, by means of the same tree-expansion rules, and their difference is reduced to a difference in the labelling algebra used in closing a branch. In this first part we lay the theoretical foundations of our method. In the second part we shall continue our investigation of substructural logics and discuss the algorithmic aspects of our approach.

Are tableaux an improvement on truth-tables?

We show that Smullyans 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 Gentzens 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.

The Measurement of Rank Mobility

In this paper we investigate the problem of measuring social mobility when the social status of individuals is given by their rank. In order to sensibly represent the rank mobility of subgroups within a given society, we address the problem in terms of partial permutation matrices which include standard (“global”) matrices as a special case. We first provide a characterization of a partial ordering on partial matrices which, in the standard case of global matrices, coincides with the well-known “concordance” ordering. We then provide a characterization of an index of rank mobility based on partial matrices and show that, in the standard case of comparing two global matrices, it is equivalent to Spearman’s index.

Zsyntax: A formal language for molecular biology with projected applications in text mining and biological prediction

We propose a formal language that allows for transposing biological information precisely and rigorously into machine-readable information. This language, which we call Zsyntax (where Z stands for the Greek word ζωή, life), is grounded on a particular type of non-classical logic, and it can be used to write algorithms and computer programs. We present it as a first step towards a comprehensive formal language for molecular biology in which any biological process can be written and analyzed as a sort of logical “deduction”. Moreover, we illustrate the potential value of this language, both in the field of text mining and in that of biological prediction.

Semantics and proof-theory of depth bounded Boolean logics

We present a unifying semantical and proof-theoretical framework for investigating depth-bounded approximations to Boolean Logic, namely approximations in which the number of nested applications of a single structural rule, representing the classical Principle of Bivalence, is bounded above by a fixed natural number. These approximations provide a hierarchy of tractable logical systems that indefinitely converge to classical propositional logic. The framework we present here brings to light a general approach to logical inference that is quite different from the standard Gentzen-style approaches, while preserving some of their nice proof-theoretical properties, and is common to several proof systems and algorithms, such as KE, KI and Stalmarck?s method.

Cut-Based Abduction

In this paper we explore a generalization of traditional abduction which can simultaneously perform two differ- ent tasks: (i) given an unprovable sequent � � G, find a sentence H such that �, HG is provable (hypothesis generation); (ii) given a provable sequent � � G, find a sentence H such that � � H and the proof of �, HG is simpler than the proof of � � G (lemma generation). 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 Gentzens 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.

Semantic Information and the Trivialization of Logic: Floridi on the Scandal of Deduction

In this paper we discuss Floridi’s views concerning semantic information in the light of a recent contribution (in collaboration with the present author) [1] that defies the traditional view of deductive reasoning as “analytic” or “tautological” and construes it as an informative, albeit non-empirical, activity. We argue that this conception paves the way for a more realistic notion of semantic information where

Fibred Tableaux for Multi-Implication Logic

We investigate the notion of fibred tableaux which naturally arises from the idea of fibred semantics. Different implication operators peacefully cohabit and co-operate within the same labelled tableau method.

An informational view of classical logic

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 [21] 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.

A logical calculus for controlled monotonicity

Abstract In this paper we introduce a new deductive framework for analyzing processes displaying a kind of controlled monotonicity. In particular, we prove the cut-elimination theorem for a calculus involving series-parallel structures over partial orders which is built up from multi-level sequents, an interesting variant of Gentzen-style sequents. More broadly, our purpose is to provide a general, syntactical tool for grasping the combinatorics of non-monotonic processes.