A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

Some arguments use ‘generic’, or ‘typical’, objects. An explanation for this idea in terms of ‘almost all’ is suggested. The intuition of ‘almost all’ as ‘but for a few exceptions’ is rendered precise by means of ultrafilters. A logical system, with generalized quantifiers for ‘almost all’, is proposed as a basis for generic reasoning. This logic is monotonic, has a simple sound and complete deductive calculus, and is a conservative extension of classical first-order logic, with which it shares several properties. (...) For generic reasoning, generic individuals are introduced and internalized as generic constants, thereby producing conservative extensions where one can reason about generic objects as intended. A many-sorted version of this logic is introduced to handle distinct notions of ‘large’ subsets. Other possible applications for this logic are indicated. (shrink)

We discuss some basic issues underlying the natural numbers: induction and recursion. We examine recursive formulations and their use in establishing universal and particular properties.

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert’s influential monograph Foundations of Geometry. Particular attention is bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt’s postulate. On the one hand, we argue that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly (...) relevant for a better understanding of the respective geometrical practices. On the other hand, we claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting. (shrink)

Vague notions, such as ‘generally’, ‘rarely’, ‘often’, ‘almost always’, ‘a meaningful subset of a whole’, ‘most’, etc., occur often in ordinary language and in some branches of science. We introduce modal logical systems, with generalized operators, for the precise treatment of assertions involving some versions of such vague notions. We examine modal logics, constructed in a modular fashion, with generalized operators corresponding to some versions of ‘generally’ and ‘rarely’.

A lógica não contém teoremas puramente existenciais: as únicas sentenças existenciaisválidas são aquelas com análogas universais válidas. Aqui, mostramos que istorealmente é assim quando corretamente interpretado: toda validade ex- istencial possuiuma análoga universal simples, que também é válida. Também caracterizamos validadesuniversais e existenciais em termos de tautologias.

A new, simpler proof, based on internalisation of interpretations, of the Modularisation Theorem for logical specifications is presented. This result is a basic tool for composing implementations and specialisation by parameter instantiation.

Logics for ‘generally’ are intended to express some vague notions, such as ‘generally’, ‘several’, ‘many’, ‘most’, etc., by means of the new generalized quantifier ∇ and to reason about assertions with ‘generally’ . We introduce the idea of functional interpretation for ‘generally’ and show that representative functions enable elimination of ∇ and reduce consequence to classical theories. Thus, one can use proof procedures and theorem provers for classical first-order logic to reason about assertions involving ‘generally’.

Logics for generally were introduced for handling assertions with vague notions,such as generally, most, several, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable elimination of the generalized quantifier. These devices permit using methods for classical first-order logic to reason about consequence in ultrafilter logic.

One of the main problems in computer science is to ensure that programs are implemented in such a way that they satisfy a given specification. There are many studies about methods to prove correctness of programs. This work presents a method, belonging to the constructive synthesis or proofs-as-programs paradigm, that comes from the Curry-Howard isomorphism and extracts the computational contents of intuitionist proofs. The synthesis process proposed produces a program in an imperative language from a proof in many-sorted intuitionist logic, (...) where the axioms define the abstract data types, using Natural Deduction as deductive system. It is proved the correctness, as well as the completeness of the method regarding the Heyting arithmetic with ω-rule . A discussion about the use of the finitary induction instead of computational ω-rule concludes the work. (shrink)

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed.

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational (...) axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares. (shrink)

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational (...) axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is ᵎ x ᵎ for some set ᵎ) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares. (shrink)

Proper fork algebras are algebras of binary relations over a structured set. The underlying set has changed from a set of pairs to a set closed under an injective function. In this paper we present a representation theorem for their abstract counterpart, that entails that proper fork algebras — whose underlying set is closed under an injective function — constitute a finitely based variety.1.

Implementation of formal specifications is very important in formal software development and can be described in terms of simple logical concepts. Formal specifications are presentations of theories in many-sorted first-order logic, and an implementation of a formal specification on another formal specification amounts to an interpretation of the former into a conservative extension of the latter. Here we present and analyse some sort introducing constructs akin to those found in many programming languages. This is of importance because it occurs often (...) in implementing formal specifications, when new sorts are ‘constructed’ from the concrete ones. We specify these constructs and show that these specifications provide extensions with behaviour close to that of the familiar extensions by definitions. (shrink)

Logics for ‘generally’ were introduced for handling assertions with vague notions , which occur often in ordinary language and in science. LG’s provide a framework for distinct notions of ‘generally’: one builds a specific logic for the notion one has in mind. We introduce deductive systems, in natural deduction style, for LG’s and show that these systems are normalizable.

We consider a paradigm of applications of Logic Engineering to illustrate the information interchange among different areas of knowledge, through the formal approach to some aspects of computing. We apply the paradigm to the area of distributed systems, taking the demand for specification formalisms, treated in three areas of knowledge: modal logics, first-order logic and algebra. In doing so, we obtain transfer of intuitions and results, establishing that, as far as input/output representation is concerned, these three formalisms are equivalent.

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.

We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...) correspondence language, so both can express the same input–output behavior of processes. (shrink)

We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...) correspondence language, so both can express the same input-output behavior of processes. (shrink)

Logics for ‘generally’ were introduced for handling assertions with vague notions, by non-standard generalized quantifiers, and to reason qualitatively about them . Filter logic is intended to address ‘most’. Here, we show that filter logic can be faithfully embedded into a classical first-order theory of certain predicates, called compatible. We also use representative predicates to enable elimination of the generalized quantifier. These devices permit using classical first-order methods to reason about consequence in filter logic and help clarifying the role of (...) such extended logics for ‘generally’. (shrink)

Veloso-Pereira-Haeusler_On-what-there-must-be.