Ruy J. G. B. de Queiroz

The Functional Interpretation of Modal Necessity

Since the early days of Kripke-style possible-worlds semantics for modalities, there has been a significant amount of research into the development of mechanisms for handling and characterising modal logics by means of ‘naming’ possible worlds, either directly by introducing identifiers, or indirectly by some other means (such as, e.g., using formulas to identify the possible world). To list a few, we have [Gabbay, forthcomingl’s ‘labelling’ formulas with names for worlds in the framework of Labelled Deductive Systems; [Fitch, 1966b]’s tree-proof deduction procedures for modal logics; [Thomason and Stalnaker, 1968]’s device of predicate abstraction introduced to handle Skolemisation across worlds and characterise non-rigid designators; [Fitting, 1972, 1975]’s e-calculus based axiom systems for modal logics, as well as his tableaux systems for modal logics with predicate abstraction [Fitting, 1981, 1989]; the irreflexivity rule of [Gabbay and Hodkison, 1990]’s axiomatic systems of temporal logic; the explicit reference to possible worlds in the deterministic modal logics of [Farinas del Cerro and Herzig, 1990]; [Ohlbach, 1991]’s semantics-based translation methods for modal logics and its functional representation of possible worlds structures. In first-order predicate logics the individuals over which one quantifies are naturally assumed to have names. The main connectives of modal logics are such that they quantify over (higher-order) objects which are not usually given names, in some cases for methodological reasons.

Labelled Natural Deduction

The functional interpretation of logical connectives is concerned with a certain harmony between, on the one hand, a indexfunctional! calculus functional calculus on the expressions built up from the recording of the deduction steps (the labels), and, on the other hand, a logical calculus on the formulae. It has been associated with Curry’s early discovery of the correspondence between the axioms of intuitionistic implicational logic and the type schemes of the so-called ‘combinators’ of Combinatory Logic [12], and has been referred to as the formulae-as-types interpretation. Howard’s [80] extension of the formulae-as-types paradigm to full intuitionistic first-order predicate logic meant that the interpretation has since been referred to as the ‘Curry-Howard’ functional interpretation. Although Heyting’s [75, 76] intuitionistic logic did fit well into the formulae-as-types paradigm, it seems fair to say that, since Tait’s [117, 118] intensional interpretations of Godel’s [69] Dialectica system of functionals of finite type, there has been enough indication that the framework would also be applicable to logics beyond the realm of intuitionism. Ultimately, the foundations of a functional approach to formal logic are to be found in Prege’s [47, 50, 51] system of ‘concept writing’, not in Curry, or Howard or, indeed, Heyting.

Intuitionistic N-Graphs

The geometric system of deduction called N-Graphs was introduced by De Oliveira in 2001. The proofs in this system are represented by means of digraphs and, while its derivations are mostly based on Gentzen’s sequent calculus, the system gets its inspiration from geometrically based systems, such as the Kneales’ tables of development, Statman’s proofs-as-graphs, Buss’ logical flow graphs and Girard’s proof-nets. Given that all these geometric systems appeal to the classical symmetry between premises and conclusions, providing an intuitionistic version of any of these is an interesting exercise in extending the range of applicability of the geometric system in question. In this paper we produce an intuitionistic version of N-Graphs, based on Maehara’s LJ’ system, as described by Takeuti. Recall that LJ’ has multiple conclusions in all but the essential intuitionistic rules e.g. implication right and negation right. We show soundness and completeness of our intuitionistic N-Graphs with respect to LJ’. We also discuss how we expect to extend this work to a version of N-Graphs corresponding to the intuitionistic logic system FIL (Full Intuitionistic Logic) of de Paiva and Pereira and sketch future developments.

Proof-graphs: a Thorough Cycle Treatment, Normalization and Subformula Property

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure, where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof. Moreover, the subformula and separation properties are determined.

Transformations via Geometric Perspective Techniques Augmented with Cycles Normalization

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof.

Natural Deduction for Equality: The Missing Entity

The conception of the very first decision procedures for first-order sentences brought about the need for giving citizenship to function symbols (e.g. Skolem functions). We argue that a closer look at proof procedures for first-order sentences with equality brings about the need for introducing (function) symbols for rewrites. This is appropriately done via the framework of labelled natural deduction which allows to formulate a proof theory for the “logical connective” of propositional equality. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Lof’s equality types (both intensional and extensional). What we end up with is a formulation of what appears to be a middle ground solution to the ‘intensional’ versus ‘extensional’ dichotomy which permeates most of the work on characterizing propositional equality in natural deduction style. (Part of this material was presented at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions.)

Meaning, Function, Purpose, Usefulness, Consequences – Interconnected Concepts

Further to the connections between meaning and use, it seems useful to consider the (explanation of the immediate) consequences one is allowed to draw from a proposition as something directly related to its meaning/use. And indeed, Wittgenstein’s references to the connections between meaning and the consequences, as well as between use and consequences are sometimes as explicit as his celebrated ‘definition’ of meaning as use given in the Investigations. Here we attempt to collect some of these references, discussing how an intuitive basis for the construction of a more convincing proof-theoretic semantics (than, say, assertability conditions semantics) for the mathematical language can arise out of this connection meaning/use/(explanation of the immediate) consequences.1

On the Identity Type as the Type of Computational Paths

We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is easy to understand and use. We make this point clear constructing terms of some relevant types using our proposed elimination rule. We also show that the identity type, as defined by our approach, induces a groupoid structure. This result is on par with the fact that the traditional identity type induces a groupoid, as exposed by Hofmann \& Streicher (1994).

Linear Time Proof Verification on N-Graphs: A Graph Theoretic Approach

This paper presents a linear time algorithm for proof verification on N-Graphs. This system, introduced by de Oliveira, incorporates the geometrical techniques from the theory of proof-nets to present a multiple-conclusion calculus for classical propositional logic. The soundness criterion is based on the one given by Danos and Regnier for Linear Logic. We use a DFS-like search to check the validity of the cycles in a proof graph, and some properties from trees to check the connectivity of every switching a concept similar to D-R graph. Since the soundness criterion in proof graphs is analogous to Danos-Regniers procedure, the algorithm can also be extended to check proofs in the multiplicative linear logic without units MLL- with linear time complexity.

Breach of internet privacy through the use of cookies

This paper describes the issue of invasion of privacy on the Internet using the techniques of cookies, as subtle means to commit such a crime. The issue of data privacy and information of millions of internet users becomes increasingly critical in terms of maintaining social order, for large corporations that dominate the World Wide Web is one of the main beneficiaries of these illegal practices that are happening daily.