The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by W´ojcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued semantics can be reduced to a two-valued one. So, why should one even consider using logics with more (...) than two values? Because, we argue, one has to decide how to deal with bivalence and settle down the tradeoff between logical 2-valuedness and truth-functionality, from a pragmatical standpoint. -/- This paper will illustrate the ups and downs of a two-valued reduction of logic. Suszko’s reductive result is quite non-constructive.We will exhibit here a way of effectively constructing the two-valued semantics of any logic that has a truth-functional finite-valued semantics and a sufficiently expressive language. From there, as we will indicate, one can easily go on to provide those logics with adequate canonical systems of sequents or tableaux. The algorithmic methods developed here can be generalized so as to apply to many non-finitely valued logics as well —or at least to those that admit of computable quasi tabular two-valued semantics, the so-called dyadic semantics. (shrink)

We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL toward providing a meaningful algebraic counterpart also to logics with a many-sorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new (...) algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way toward a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach. (shrink)

The standard notion of formal theory, in logic, is in general biased exclusively towards assertion: it commonly refers only to collections of assertions that any agent who accepts the generating axioms of the theory should also be committed to accept. In reviewing the main abstract approaches to the study of logical consequence, we point out why this notion of theory is unsatisfactory at multiple levels, and introduce a novel notion of theory that attacks the shortcomings of the received notion by (...) allowing one to take both assertions and denials on a par. This novel notion of theory is based on a bilateralist approach to consequence operators, which we hereby introduce, and whose main properties we investigate in the present paper. (shrink)

. The semantic collapse problem is perhaps the main difficulty associated to the very powerful mechanism for combining logics known as fibring. In this paper we propose cryptofibred semantics as a generalization of fibred semantics, and show that it provides a solution to the collapsing problem. In particular, given that the collapsing problem is a special case of failure of conservativeness, we formulate and prove a sufficient condition for cryptofibring to yield a conservative extension of the logics being combined. For (...) illustration, we revisit the example of combining intuitionistic and classical propositional logics. (shrink)

Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, one-step derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction (...) systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequences systems. (shrink)

We discuss the axiomatization of generalized consequence relations determined by non-deterministic matrices. We show that, under reasonable expressiveness requirements, simple axiomatizations can always be obtained, using inference rules which can have more than one conclusion. Further, when the non-deterministic matrices are finite we obtain finite axiomatizations with a suitable generalized subformula property.

The modern view of logical reasoning as modeled by a consequence operator has allowed for huge developments in the study of logic as an abstract discipline. Still, it is unable to explain why it is often the case that the same designation is used, in an ambiguous way, to describe several distinct modes of reasoning over the same logical language. A paradigmatic example of such a situation is ‘modal logic’, a terminology which can encompass reasoning over Kripke frames, but also (...) over Kripke models, and in any case either locally or globally . Herein, we adopt a novel abstract notion of logic presented as a lattice-structured hierarchy of consequence operators, and explore some common proof-theoretic and model-theoretic ways of presenting such hierarchies through a collection of meaningful examples. In order to illustrate the usefulness of the notion of hierarchical consequence operators we address a few questions in the theory of combined logics, where a suitable abstract presentation of the logics being combined is absolutely essential. We show how to define the operation, in general, and we achieve a number of interesting preservation results for fibring within the context of 2-hierarchies. (shrink)

It is well-known that da Costa's C-systems of paraconsistent logic do not admit a Blok-Pigozzi algebraization. Still, an algebraic flavored semantics for them has been proposed in the literature, namely using the class of so-called da Costa algebras. However, the precise connection between these semantic structures and the C-systems was never established at the light of the theory of algebraizable logics. In this paper we propose to study the C-systems from an algebraic point of view, and to fill in this (...) gap by using the tools and techniques of the newly developed behavioral approach to abstract algebraic logic. As a by-product of the approach, we also rediscover the bivaluation semantics of the logics. (shrink)

We define and study abstract valuation semantics for logics, an algebraically well-behaved version of valuation semantics. Then, in the context of the behavioral approach to the algebraization of logics, we show, by means of meaningful bridge theorems and application examples, that abstract valuations are suited to play a role similar to the one played by logical matrices in the traditional approach to algebraization.

The Fregean-inspired Principle of Compositionality of Meaning for formal languages asserts that the meaning of a compound expression is analysable in terms of the meaning of its constituents, taking into account the mode in which these constituents are combined so as to form the compound expression. From a logical point of view, this amounts to prescribing a constraint—that may or may not be respected—on the internal mechanisms that build and give meaning to a given formal system. Within the domain of (...) formal semantics and of the structure of logical derivations, the PoC is often directly reflected by metaproperties such as truth-functionality and analyticity, characteristic of computationally well-behaved logical systems. (shrink)

We study the general problem of strengthening the logic of a given matrix with a set of axioms, using the idea of rexpansion. We obtain two characterization methods: a very general but not very effective one, and then an effective method which only applies under certain restrictions on the given semantics and the shape of the axioms. We show that this second method covers a myriad of examples in the literature. Finally, we illustrate how to obtain analytic multiple-conclusion calculi for (...) the resulting logics. (shrink)

In previous work, we introduced a version of distributed temporal logic that is well-suited both for verifying security protocols and as a metalogic for reasoning about, and relating, different security protocol models. In this paper, we formally investigate the relationship between our approach and strand spaces, which is one of the most successful and widespread formalisms for analyzing security protocols. We define translations between models in our logic and strand-space models of security protocols, and we compare the results obtained with (...) respect to the level of abstraction that is inherent in each of the formalisms. This allows us to clarify different aspects of strand spaces that are often left implicit, as well as pave the way to transfer results, techniques and tools across the two approaches. (shrink)

We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations—nowadays more commonly called multiple-conclusion logics—to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a congruence); the (...) problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras. (shrink)

Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$. On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok (...) and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as,, or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$, with,, and. (shrink)

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...) for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer (...) result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established. (shrink)

This article aims to clarify the sharp contrast between the “short path” of Heideggerian ontology and the “long path” adopted by Ricoeur as a representation of the many necessary mediations in the constitution of his philosophical anthropology. Heidegger breaks into contemporary thought with the publication of his treatise “Being and Time” (1927) as a kind of direct settlement in the field of ontology. In contrast, Ricoeur is seen - and sees himself - as a thinker of conceptual mediations in his (...) long itinerary, since adherence to Husserlian phenomenology and subsequent hermeneutic inflection, passing through confrontations with Freudian psychoanalysis, structuralism and analytical philosophy, to the creation of the book “Oneself as Another” (1990). However, the two philosophers built their works on the common horizon of hermeneutics; therefore, are there possible points of convergence between them? And how could we interpret the relevance of this possible interlocution? These questions cannot be answered without a thorough comparative reading. This text does not intend to answer them, but only to suggest elements for further investigations. (shrink)

La 4e de couverture indique : "Dès la fin du XVIIIe siècle, la référence à Rousseau devient incontournable pour la pensée juridique, comme si toute réflexion sur le droit, et pas uniquement dans le domaine du droit public, devrait régler ses comptes avec le Genevois. Bien entendu, cette référence changera de signification politique avec le temps et l’évolution des institutions étatiques européennes. Mais l’extension du suffrage universel et la consolidation de la démocratie au XXe siècle sont loin de signer la (...) fin des querelles autour de l’œuvre. Au contraire, la question Rousseau revient dans toute grande théorie du droit public. Et l’histoire de ce cheminement nous livre un ensemble d’indices dont l’intérêt va au-delà du droit. Les différents chapitres de l’ouvrage se proposent d’explorer, à partir de l’étude de quelques grands juristes français et allemands, la portée que la philosophie de Rousseau a pu avoir pour la pensée juridique.". (shrink)

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

Many of the computing systems programmed using Machine Learning are opaque: it is difficult to know why they do what they do or how they work. Explainable Artificial Intelligence aims to develop analytic techniques that render opaque computing systems transparent, but lacks a normative framework with which to evaluate these techniques’ explanatory successes. The aim of the present discussion is to develop such a framework, paying particular attention to different stakeholders’ distinct explanatory requirements. Building on an analysis of “opacity” from (...) philosophy of science, this framework is modeled after accounts of explanation in cognitive science. The framework distinguishes between the explanation-seeking questions that are likely to be asked by different stakeholders, and specifies the general ways in which these questions should be answered so as to allow these stakeholders to perform their roles in the Machine Learning ecosystem. By applying the normative framework to recently developed techniques such as input heatmapping, feature-detector visualization, and diagnostic classification, it is possible to determine whether and to what extent techniques from Explainable Artificial Intelligence can be used to render opaque computing systems transparent and, thus, whether they can be used to solve the Black Box Problem. (shrink)

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of (...) its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate "partial meet contraction functions", which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are "relational" and "transitively relational", are studied in detail, and their connections with certain "supplementary postulates" of Gardenfors investigated, with a further representation theorem established. (shrink)

v. 1. Ensino secundário e médio -- v. 2. Estudos sobre o ensino primário -- Planejamento educacional e contribuições à educação do futuro.

The received view of dynamical explanation is that dynamical cognitive science seeks to provide covering law explanations of cognitive phenomena. By analyzing three prominent examples of dynamicist research, I show that the received view is misleading: some dynamical explanations are mechanistic explanations, and in this way resemble computational and connectionist explanations. Interestingly, these dynamical explanations invoke the mathematical framework of dynamical systems theory to describe mechanisms far more complex and distributed than the ones typically considered by philosophers. Therefore, contemporary dynamicist (...) research reveals the need for a more sophisticated account of mechanistic explanation. (shrink)

Models developed using machine learning are increasingly prevalent in scientific research. At the same time, these models are notoriously opaque. Explainable AI aims to mitigate the impact of opacity by rendering opaque models transparent. More than being just the solution to a problem, however, Explainable AI can also play an invaluable role in scientific exploration. This paper describes how post-hoc analytic techniques from Explainable AI can be used to refine target phenomena in medical science, to identify starting points for future (...) investigations of causal relationships, and to generate possible explanations of target phenomena in cognitive science. In this way, this paper describes how Explainable AI—over and above machine learning itself—contributes to the efficiency and scope of data-driven scientific research. (shrink)

In a context of pervasive digitalization of the social world, both before and during the COVID-19 pandemic, the field of education has undergone major changes with the development of digital practices and settings. However, the physical presence of the subjects and the body remain something primordial and irreplaceable in traditional educational processes. Thus, it is often assumed that virtuality is opposed to the corporeal reality of the subjects involved in teaching, learning and studying. In this paper we aim to critically (...) challenge this assumption by addressing the phenomenon of virtuality in a more original sense: as a fundamental dimension of corporeality itself. Drawing on Merleau-Ponty’s phenomenological insights, we argue that the experience of the virtual is inherently embodied and fully real. We understand corporeality and virtuality as entangled and multidimensional phenomena. At the same time, this leads us to rethink the digital as only one possible medium in which virtuality can appear as inherent to embodied experience. We develop our phenomenological approach in three instances. First, we question and bracket two common assumptions about corporeality. From our critical perspective, assumptions about a disembodied digitality and a devirtualized corporeality represent extremes within which educational discourses currently move. Second, we address the inherent virtuality of embodied experience. We outline the heuristic concept of the virtual body to describe corporeality and virtuality as entangled and multidimensional phenomena that encompass extension, intercorporeality and intermediality. Finally, we make some remarks on the relationships of corporeality and virtuality considering the challenges of digital education. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)

Bayesian reverse-engineering is a research strategy for developing three-level explanations of behavior and cognition. Starting from a computational-level analysis of behavior and cognition as optimal probabilistic inference, Bayesian reverse-engineers apply numerous tweaks and heuristics to formulate testable hypotheses at the algorithmic and implementational levels. In so doing, they exploit recent technological advances in Bayesian artificial intelligence, machine learning, and statistics, but also consider established principles from cognitive psychology and neuroscience. Although these tweaks and heuristics are highly pragmatic in character and (...) are often deployed unsystematically, Bayesian reverse-engineering avoids several important worries that have been raised about the explanatory credentials of Bayesian cognitive science: the worry that the lower levels of analysis are being ignored altogether; the challenge that the mathematical models being developed are unfalsifiable; and the charge that the terms ‘optimal’ and ‘rational’ have lost their customary normative force. But while Bayesian reverse-engineering is therefore a viable and productive research strategy, it is also no fool-proof recipe for explanatory success. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

The philosophical conception of mechanistic explanation is grounded on a limited number of canonical examples. These examples provide an overly narrow view of contemporary scientific practice, because they do not reflect the extent to which the heuristic strategies and descriptive practices that contribute to mechanistic explanation have evolved beyond the well-known methods of decomposition, localization, and pictorial representation. Recent examples from evolutionary robotics and network approaches to biology and neuroscience demonstrate the increasingly important role played by computer simulations and mathematical (...) representations in the epistemic practices of mechanism discovery and mechanism description. These examples also indicate that the scope of mechanistic explanation must be re-examined: With new and increasingly powerful methods of discovery and description comes the possibility of describing mechanisms far more complex than traditionally assumed. (shrink)

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, (...) notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)

This paper provides a commentary on Truth and Method III.1.

This article argues for a new interpretation of the realist claim that politics is autonomous from morality and involves specific political values. First, this article defends an original normative source: functional normativity. Second, it advocates a substantive functional standard: political institutions ought to be assessed by their capacity to select and implement collective decisions. Drawing from the ‘etiological account’ in philosophy of biology, I will argue that functions yield normative standards, which are independent from morality. For example, a ‘good heart’ (...) is one that pumps blood well, and a good army is one that it is effective at exerting military force. I then interpret realism’s naturalistic conception of politics as an etiological function of social groups: namely making binding collective decisions under persistent disagreement. I conclude that political institutions should be evaluated realistically by how well they perform this task. Finally, I assess trade-offs between this functional political normativity and other moral values. I conclude that justice, fairness, freedom and equality remain obviously important concerns, but only once the basic political function is secured. (shrink)

This chapter subsumes David Marr’s levels of analysis account of explanation in cognitive science under the framework of mechanistic explanation: Answering the questions that define each one of Marr’s three levels is tantamount to describing the component parts and operations of mechanisms, as well as their organization, behavior, and environmental context. By explicating these questions and showing how they are answered in several different cognitive science research programs, this chapter resolves some of the ambiguities that remain in Marr’s account, and (...) shows that many different areas and traditions of cognitive scientific research can be unified under the mechanistic framework. (shrink)

This article argues for a new interpretation of the realist claim that politics is autonomous from morality and involves specific political values. First, this article defends an original normative...