Ser (algo) ou não ser: existência e predicação na lógica e na metafísica de Aristóteles

This paper offers a new interpretation of Aristotle’s logical system that allows us to break away from a presumably inescapable dilemma inherent to any attempt of justification from a strictly formal standpoint; namely, either the system stands apart from any standard system of mathematical logic — propositional logic, class theory, quantification theory, etc. — or it is lacking in consistency. While some other attempts have been made that seem to avoid this dilemma, they have always resulted in an intolerably strong restriction of the Semantics underpinning the Aristotelian system. These are, therefore, unsatisfactory solutions that undermine some of the key principles of Aristotelian metaphysics. One by-product of the solution presented here is a reappraisal of some elements in Aristotle’s theory of Being that breaks down several contemporary misconceptions on the subject. Key words: logic, Aristotle, justification, mathematical logic.Este artigo oferece uma nova interpretação do sistema lógico de Aristóteles que nos permite tomar distância de um dilema supostamente sem escapatória inerente a qualquer tentativa de justificação desde um ponto de partida estritamente formal; mais especificamente, ou o sistema se identifica com algum sistema lógico-matemático – lógica proposicional, teoria de classes, teoria da quantificação, etc. – ou é considerado inconsistente. Embora tenham sido feitas outras tentativas que parecem evitar este dilema, sempre resultaram numa restrição intoleravelmente forte à Semântica que dá sustentação ao sistema aristotélico. Estas são, consequentemente, soluções insatisfatórias que minam alguns princípios chaves da metafísica aristotélica. Uma das soluções propostas aqui é uma revalorização de alguns elementos da teoria aristotélica do Ser que derruba alguns mal-entendidos contemporâneos sobre o tema. Palavras-chave: lógica, Aristóteles, justificação, lógica matemática

Tarski’s Convention T: condition beta

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus

Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

Reuse and integration of specification logics: the hybridisation perspective

Hybridisation is a systematic process along which the characteristic features of hybrid logic, both at the syntactic and the semantic levels, are developed on top of an arbitrary logic framed as an institution. It also captures the construction of first-order encodings of such hybridised institutions into theories in first-order logic. The method was originally developed to build suitable logics for the specification of reconfigurable software systems on top of whatever logic is used to describe local requirements of each system’s configuration. Hybridisation has, however, a broader scope, providing a fresh example of yet another development in combining and reusing logics driven by a problem from Computer Science. This paper offers an overview of this method, proposes some new extensions, namely the introduction of full quantification leading to the specification of dynamic modalities, and exemplifies its potential through a didactical application. It is discussed how hybridisation can be successfully used in a formal specification course in which students progress from equational to hybrid specifications in a uniform setting, integrating paradigms, combining data and behaviour, and dealing appropriately with systems evolution and reconfiguration.This work is financed by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation—COMPETE 2020 Programme, and by National Funds through the FCT (Portuguese Foundation for Science and Technology) within project POCI-01-0145-FEDER-006961. M. Martins was further supported by project UID/MAT/04106/2013. A. Madeira and R. Neves research was carried out in the context of a post-doc and a Ph.D. grant with references SFRH/BPD/103004/2014 and SFRH/BD/52234/2013, respectively. L.S. Barbosa is also supported by SFRH/BSAB/ 113890/2015

The Ancient Master Argument and Some Examples of Tense Logic

Abstract The Master Argument of Diodorus Cronus has been long debated by logicians and philosophers. During the Hellenistic period it was so famous that doxographers and commentators took for granted its notoriety and none of them gave us a detailed report. The first section presents a philosophical account of the ancient Master Argument, by trying to retrace its meaning, originated from the Megarian context, and so halfway between ancient logic and metaphysics. The second section introduces a logical analysis of the Master Argument against the backdrop of the Jarmużek-Pietruszczak semantics for the tense logic K t 4P; but the main aim of the section is to deal with one of the most fascinating attempts to peruse the Master Argument, i.e. A. Prior's reconstruction. Prior stays true to the Diodorean philosophical stance even if he uses modern logical tools. The significance of the work by Prior marks the beginning of tense logic. The third section expounds an argument by Øhrstrøm-Hasle. Danish logicians do not consider additional premises for the Master Argument. They give, in primis, a sentential example for the third premise, proving its inconsistency with the first two. The deterministic conclusion is the implicit result of this stratagem. Finally, in the fourth section, we compare the strategies by Prior and Øhrstrøm-Hasle

The genealogy of ‘∨’

The use of the symbol ∨ for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or”, vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established