Posts com a Tag ‘CARNIELLI Walter (Aut)’
Contradictions, from Consistency to Inconsistency – CARNIELLI; MALINOWSKI (M)
CARNIELLI, Walter; MALINOWSKI, Jacek. Contradictions, from Consistency to Inconsistency. Trends in Logic 47. Springer International Publishing, 2018. VI+322 pagesp. Resenha de: TESTA, Rafael R. Manuscrito, Campinas, v.42 n.1 Jan./Mar. 2019.
Trends in Logic is the conference series of the journal Studia Logica, covering contemporary formal logic and its relations to other disciplines. The works collected in this volume were initiated by the discussions that took place at the conference to commemorate the 40th Anniversary of the Centre for Logic, Epistemology and History of Science. The title of the event celebrates one of the three main areas of CLE – that has been called as the epicentre of a “Brazilian school of paraconsistency”. The reasons for that are the original works of da Costa, followed by his pupils and collaborators that are part of CLE’s history. Simply put their interest include the development of systems strong enough to encompass most of mathematics, while avoiding some well-known logical paradoxes.
Not surprisingly most of the works in this volume are developed around paraconsistent logics or, as it is explained by the editors in the introductory chapter, they are concerned about the subtle distinctions between consistency and non-contradiction, as well as among contradiction, inconsistency and triviality. There are many interesting problems discussed in the book, some of them are well-known among readers familiar with paraconsistency. Setting aside the introduction, the book itself does not intend to be a historical review on the main questions regarding the subject. Rather, the chapters help the reader to taste some information about where paraconsistency is now and where it is heading, as well as cast new lights on some old questions regarding the consistency of formal theories. In what follows I succinctly present the central elements of each chapter.
The introduction briefly presents some state-of-the-art discussion regarding the central questions that permeate the book. In the homonym chapter, Carnielli and Malinowski explain the title of the book and show the relevance of the subject in contemporary discussions in logic and philosophy of science – themes that are familiar to the authors. Walter Carnielli is full professor of Logic at the State University of Campinas (Unicamp) and served as the director of CLE, as well as editor and member of editorial boards of major journals. Some of his works encompass for instance combinations of logics, many-valued and paraconsistent logics – like the logics of formal inconsistency advanced by Carnielli and Marcos (2002) that systematises a large class of paraconsistent logics. Jacek Malinowski is the editor-in-chief of Studia Logica, Head of the Department of Logic and Cognitive Science at the Polish Academy of Science and Head of the Section of Logical Semiotics at Nicolaus Copernicus University in Torun, Poland. He has published works in several areas, for instance logical foundations of computer sciences, nonmonotonic and cognitive logic, just to name a few. The book reflects the multidisciplinary interest of the editors.
The second chapter (the first of 13 collaborative papers) brings Arenhart’s investigation on an overlooked argument advanced by da Costa (1997) to the effect that there may be true contradictions about the concrete world. The novelty of the chapter “The Price of True Contradictions About the World” is bridging da Costa’s argument to a well-known dialetheist understanding of paraconsistency. By advancing several objections to the argument, the daring conclusion drawn by the author is that the acceptance of true contradictions about the world comes with heavy prices to pay: for instance adopting an inconvenient conservative and pessimistic attitude towards change in science.
In “The Possibility and Fruitfulness of a Debate on the Principle of Non-contradiction”, Estrada-González and del Rosario Martínez-Ordaz go back to the Aristotelian arguments regarding the principle of non-contradiction (PNC) originally advanced in his Metaphysics. The aim is to show how they can be used for a better understanding of the different standpoints that are present in the contemporary debate. The authors advance five major stances regarding the debate on the PNC, namely: Detractors, Fierce supporters, Demonstrators, Methodologists and Calm supporters. They suggest how we can find elements of those instances in several authors in the literature, from Aristotle up to the present. Maybe the main claim of this chapter is that one can find all the elements of Calm supporters already in Aristotle’s works.
Friend and del Rosario Martínez-Ordaz explore a formal method to model the fact that sometimes mathematicians and scientists reason with inconsistent premises while denying that this is possible or makes any sense – a tooll called Chunck and Permeate (C&P) advanced by Bryson and Priest (2004). Roughly speaking, C&P divides a given proof with inconsistent premises into consistent subsets, called chunks, and allows only some information to permeate from one chunk to the next. In “Keeping Globally Inconsistent Scientific Theories Locally Consistent”, the authors extend C&P by adding a visual representation of chunks in the form of bundle diagrams. By extending it, they apply the method to analyse a case in physics and discuss the implications of inconsistency toleration in science, possibly opening up avenues for other discussions in the role of logic in science.
In “What is a Paraconsistent Logic?”, Barrio, Pailos and Szmuc recall some canonical definitions of paraconsistent logics (advanced for instance by Priest, Tanaka and Weber (2016); Carnielli and Coniglio (2016); and Ripley (2015)) in order to suggest a new one. By taking into account a meta-inferential notion of explosion, the authors bring into the light the fact that some logical systems might validate the Explosion Principle but invalidate a meta-inferential version of it. Relaying on some well-formulated logical and philosophical reasons, this chapter advances the novel thesis that a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. Being so, a number of systems in the literature turn out to be, in that sense, paraconsistent logics.
Gaytán, D’Ottaviano and Morado present a system motivated by the problems of modelling explanation from the point of view of Philosophy of Science. In the chapter “Provided You’re not Trivial: Adding Defaults and Paraconsistency to a Formal Model of Explanation”, the authors advance the so-called GMD framework. Within that formal system it is possible to make an analysis of the interaction between rules and a minimal conception of context – composed by a set of beliefs (a minimal idea of a theory) in interaction with an inferential engine (a logic). In order to illustrate this novel epistemic system, the authors adopt it to analyse the concept of explanation using Reiter’s default theories and a specific paraconsistent logic of da Costa.
In the chapter “Para-Disagreement Logics and Their Implementation Through Embedding in Coq and SMT”, Woltzenlogel Paleo advances a novel approach to para-disagreement logics. The basic language is the usual propositional one, extended with box and diamond operators from modal logics and the @ operator from hybrid logics. The semantics are very similar to possible worlds for modal logics with small differences regarding the representation of world reachability. This framework allows a fine-tuned approach regarding information source, so that conflicting information from different sources can be consistently combined. By suggesting some possible semantical embeddings in Coq and SMT, the author advocates the implementation of automated reasoning tools for these logics.
Džamonja and Panza, in “Asymptotic Quasi-completeness and ZFC”, put forward a thesis that the axioms ZFC of first order set theory is actually very powerful at some infinite cardinal, contrary to what it could be stated. Since ZFC axioms are subject to Gödel’s Incompleteness Theorems (cf. Gödel (1931)), if they are assumed to be consistent then they are necessarily incomplete – a fact that can be supported by various concrete statements, including the celebrated Continuum Hypothesis. In order to illustrate their thesis, it is explained that by looking at limits of uncountable cardinals, such as אω, and working with singular cardinals (which are necessarily limits, cf. Kojman (2011)), at such cardinals there is a very serious limit to independence. Furthermore, many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. The thesis then follows by the fact that the behaviour of the set-theoretic universe is asymptotically determined at singular cardinals by the behaviour that the universe assumes at the smaller regular cardinals. Being so, ZFC foundationally provides an asymptotically univocal image of the universe of sets around the singular cardinals.
“Interpretation and Truth in Set Theory” also presents an inquiry on some fundamental questions of set theory. In this chapter, Freire grasps concrete axiom systems in terms of a double-layer schema: respectively containing the conceptual and the deductive components of the system. The conceptual component is identified with a criterion given by directive principles, supposable bounding the subject matter of the system. After advancing two lists of directive principles for the set theory, the set-theoretic truth and the fixation of truth-values in each double-layer picture that emerged from these lists are then analysed. It is worth noticing that the general approach that is forwarded in this chapter can be applied to other mathematical theories with interesting results.
In the short but sturdy chapter “Coherence of the Product Law for Independent Continuous Events”, Mundici demonstrates a formal result regarding probability theory: the product law for logically independent events (for Boolean as well as for continuous MV-algebraic events) follows from de Finetti’s fundamental notion of a coherent set of betting odds, in the same sense that it was originally demonstrated for the additivity law by de Finetti’s 1932 Dutch Book theorem.
In the chapter “A Local-Global Principle for the Real Continuum”, Magossi and Rioul present a logical flow of proofs in the most influential undergraduate and graduate textbooks on Real Analysis in the U.S.A., France and Brazil in order to start a discussion regarding the local-global principle (LG) as a new efficient and enjoyable tool for proving the basic theorems of real analysis. Both, LG (any local and additive property is global ) and the related principle of global-limit (GL: any global and subtractive property has a limit point) could be used as basis for a new presentation of the integral, just as Cousin’s lemma was used to build the Kurzweil-Henstock integral – what the authors intend to advance in future works.
The chapter “Quantitative Logic Reasoning” by Finger brings an unifying approach on some logical systems, namely propositional Probabilistic Logic (classical propositional logic enhanced with probability assignments over formulas); first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic (a multi-valued logic for which there exists a well-founded probability theory). From the viewpoint of Quantitative Logic Reasoning, the author shows that analogous properties hold throughout that class of systems, and presents for each one a language, semantics and decision problem, followed by normal form presentation and satisfiability characterization. Furthermore, complexity results and decision algorithms are also advanced.
Carnielli, Mariano and Matulovic advance an algebraic method based on the polynomial representation of first-order sentences in order to introduce algebraic semantics for first-order logic, departing from modern forms of “algebraizing a Logic” tradition like presented by Blok and Pigozzi (1989). In “Reconciling First-Order Logic to Algebra” the authors employ the notion of M-rings, rings equipped with infinitary operations that can be naturally associated to the first-order structures and each first-order theories. It is shown that infinitary versions of the Boolean sums and products are able to express algebraically first-order logic from a new perspective. This chapter also suggests an unifying algebraical approach to logic by opening-up avenues for possible generalizations of the method to n-valued and other non-classical logics.
In the book’s last chapter, Marcelino, Caleiro and Rivieccio clarify the efficiency of some novel techniques in the study of Hilbert-style logics. In “Plug and Play Negations” the authors focus on the negation fragments of logics which result from different possible choices of well-known rules involving the connectives {→,⊥}, with a few variations – in fact negation is usually introduced as a derived connective by making ¬p:=p→⊥¬p:=p→⊥ (that is, using the material implication → and the falsum constant ⊥). In turn the degree of tolerance to inconsistencies of a logic (degree of paraconsistency) can be determined by the interaction among these three connectives. The techniques used allow for a modular analysis of the logics, providing complete semantics based on (non-deterministic) logical matrices and complexity upper bounds.
Albeit the great diversity of themes discussed in the chapters, all of them can be subsumed into a broadly understood study of consistency – the book perfectly demonstrates how issues surrounding that study go well beyond traditional inquiries on paraconsistent logics, taking novel perspectives that are not too far away from such inquiries. The selection helps the reader to perceive how those works intersect with core traditional mathematical and philosophical questions. Contradictions, from Consistency to Inconsistency nicely supplements the existing literature on the subject. This is a volume that it is well worth reading!
References
ARISTOTLE: Metaphysics. In The Complete Works of Aristotle, ed. J Barnes (vol. 2). Princeton N.J.: Princeton University Press, 1984. [ Links ]
BLOK, W.J. and D. PIGOZZI. “Algebraizable Logics”. Memoirs of the AMS. 396, American Mathematical Society, Providence, USA, 1989. [ Links ]
BROWN, B. and PRIEST, G. “Chunk and permeate, a paraconsistent inference strategy. part i: the infinitesimal calculus”. Journal of Philosophical Logic 33 (4): 379-388. August, 2004. [ Links ]
CARNIELLI, W. and CONIGLIO, M. Paraconsistent Logic: Consistency, Contradiction and Negation. Dordrecht: Springer, 2016. [ Links ]
_____ and MARCOS, J. “A taxonomy of C-systems”. Paraconsistency: The Logical Way to the Inconsistent. Lecture Notes in Pure and Applied Mathematics 228, 1-93, 2002. [ Links ]
DA COSTA, N.C.A. Logiques Classiques et Non Classiques. Essai sur les fondements de la logique. Paris: Masson, 1997. [ Links ]
DE FINETTI, B. Theory of Probability, A Critical Introductory Treatment, Translated by Antonio Machí and Adrian Smith. Chichester, UK: Wiley, 2017. [ Links ]
GÖDEL, K. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”. Monatshefte für Mathematik und Physik 38: 173-198, 1931. [ Links ]
KOJMAN, M. “Singular Cardinals: from Hausdorff’s gaps to Shelah’s pcf theory”. In Sets and Extensions in the Twentieth Century ed. by Dov M. Gabbay, Akihiro Kanamori, and John Woods, vol. 6 of Handbook of the History of Logic, pp. 509-558. Elsevier, 2011. [ Links ]
PRIEST, G., TANAKA, K. and WEBER, Z. “Paraconsistent logic”. In The Stanford Encyclopedia of Philosophy, Winter 2016 ed, ed. Edward Zalta. Stanford University, 2016. http://plato.stanford.edu/archives/win2016/entries/logic-paraconsistent/. [ Links ]
RIPLEY, D. “Paraconsistent logic”. Journal of Philosophical Logic 44 (6): 771-780, 2015. [ Links ]
Rafafel R. Testa – University of Campinas. Center for Logic and Epistemology. Campinas, SP. Brazil. E-mail: [email protected]
Modalities and Multimodalities – CARNIELLI (M)
CARNIELLI, Walter; PIZZI, Claudio. Modalities and Multimodalities. [?]: Springer, 2008. 320 p. Resenha de: COSTA-LEITE, Alexandre. Manuscrito, Campinas, v.36 n.1 Jan./June 2013.
Many philosophers are taking advantage of modal logic in order to approach their problems. This happens especially because the development of modal tools to formalize philosophical issues has been shown to be decisive to understand deep topics, for instance, in metaphysics, epistemology, ethics and their connections. This is evident when we read articles on philosophical logic. Modal tools play a role in building concepts, modelling theories and solving paradoxes, just to mention a few examples. In this sense, modal logic has been a necessary (but not sufficient) condition for philosophical research.
The book under review contains several of these modal tools. It has nine chapters dealing with a great plurality of aspects of modal logics. By the end of each chapter, there are exercises (but answers are not provided) and also a brief history of the main concepts introduced through the text. The book is accessible to anyone with a background in classical logic and it is of interest to all those studying logic. This is the English version of a book published years ago in Italian. Comparing this actual version with the old one, we see many improvements but no crucial changes.
The reader finds in chapter one an overview of classical propositional logic – called by the authors standard propositional calculus (PC)– because all modal logics studied in the book are extensions of this logic. The most important topic explored in this chapter is the constructive completeness proof of PC by Kalmár’s method. Moreover, other metalogical properties of PC are studied.
The syntactical aspects of modal logics besides their proof theory are introduced in chapter two. Beginning with Aristotle’s modal square of oppositions, this chapter goes up to the hierarchy of normal modal systems ranging from K to S5. Many properties of S5 are examined. In order to prove its consistency, they use the concept of translation of logics. The proof of the reduction theorem for modalities in S5 is presented in detail. In this chapter, the reader starts to understand the fundamental role of induction proofs in modal logics: this kind of proof is used to demonstrate the reduction theorem and the syntactical deduction theorem for modal logics. One very positive aspect of this book is that theorems are proved in detail and this is really helpful in order to precisely follow the whole proof.
After exploring syntactical aspects in chapter two, chapter three deals with semantics, and the famous Dugundji’s theorem which banned modal logic from the many-valued scenario is proved in detail. This theorem, although very important in modal logic, is almost never mentioned in modal logic textbooks. Afterwards, the fundamental distinction between Carnapian models (without accessibility relations) and Kripke models (with accessibility relations) is addressed. This lead us to correspondence theory (a branch of modal logic which investigates the connections between modal and first-order logic) and more fashionable concepts such as bissimulations, p-morphisms, and the Goldblatt-Thomason theorem. This dense chapter ends with tableaux for several modal systems. The authors show how to get different kinds of tableaux depending on the kind of accessibility of a given system.
The essential notion of (modal) completeness is analyzed in chapter four. It contains a detailed completeness proof of the general modal system K+G∞ which has as instantiations a great variety of modal logics. In this sense, proving completeness for this general system allows immediate proofs for all systems of the hierarchy from K to S5. This completeness proof is realized by what is called Henkin’s method. Thus, results such as the Lindenbaum theorem and the fundamental theorem of canonical models are presented. The chapter ends with a study of the logic of provability (linking modal logic with arithmetic), because this logic does not fall under the K+G∞ schema and, therefore, another strategy is required to prove completeness.
Following the same line, and developing metalogical properties of modal systems, chapter five shows how some modal logics are, indeed, incomplete. Notions like modal algebra and finite model property are presented. The latter is connected with the problem of proving decidability of modal logics.
From now on, the book, despite still examining a plurality of technical results, touches modal notions which are richer from the philosophical viewpoint. The theme of chapter six is that of temporal/tense logics. Temporal logics are studied from the syntactical but also from the semantical viewpoint. Tableaux are used as the main proof-theoretical strategy and completeness and incompleteness proofs of these temporal logics are also based on this technique. Tense logics are very useful in order to formalize models of time such as, for instance: branching, linear and circular time. Each temporal logic reflects some properties of these readings of time. The reader finds here a broad discussion on these subjects and a discussion of other interesting systems as, for example, hybrid logics.
Still going into modal notions with an intense philosophical flavour, the theme of chapter seven is that of epistemic logics. These are responsible for the mathematics of notions like knowledge and belief which are extremely important in computer science (and, needless to say, epistemology). Single and multi-agent epistemic logics are also studied from the proof and semantical theoretical viewpoints. There is a special session on doxastic logics as well as on common and distributed knowledge. Standard discussions on epistemic logic like positive and negative introspection are also examined.
It is rare to find a book dealing with multimodalities. These are exactly the topic of chapter eight. Indeed, it is important to note that central philosophical problems are formulated in sentences containing interactive concepts like, for instance, combined notions from metaphysics and epistemology. Thus, in order to reason about philosophy, multimodalities are the rule. This part of the book shows how to treat these multimodalities. Some multimodal systems are discussed: epistemic doxastic logics, deontic temporal logics, epistemic temporal logics etc. There is also a completeness proof of a very general multimodal system.
The last chapter (that is, chapter nine) enters into the landscape of complicated quantified modal logics which have a strong expressive power. Here we find a discussion on necessary and contingent identities, on rigid designation, and on the wild topic of quantification and multimodalities.
Despite the fact that the word “multimodalities” appears in the title, there is only one chapter dedicated to the theme. It is difficult to write a perfect book. Authors should prepare an errata correcting typos and other problems. There are no big errors, only minor and peripheral mistakes. However, some general negative aspects should be pointed out: 1) There is almost no mention of non-classical modal logics which are also pretty much investigated nowadays (this is surprising given that one the authors – W. A. Carnielli – works on non-classical logic); 2) Multimodal logics are introduced without appeal to methods like fusions and products (this is also surprising given that W. A. Carnielli does intensive research on combining logics); 3) Many-dimensional modal logics are not touched; 4) There is almost nothing about deontic logic; 5) Complexity of modal logics is not studied. This is a negative point, given that exploring concepts from the realm of computational complexity such as NP-completeness would make the book more attractive to the computer science community. Of course, some of these broad negative aspects would be, in a certain sense, too much for the purposes of this introductory guide to modal logic. This is the reason why I hope authors write a second volume covering all or some of these topics with the same elegance and simplicity.
I have used Modalities and Multimodalities to teach at the undergraduate level and the experience has been very good: after studying it, undergraduate students were able to develop creative research, going from non-abstract and primitive reasoning about the actual world to the richness and beauty of limitless modal inference.