One reason, several logics

Humans have used arguments for defending or refuting statements long before the creation of logic as a specialized discipline. This can be interpreted as the fact that an intuitive notion of "logical consequence" or a psychic disposition to articulate reasoning according to this pattern is present in common sense, and logic simply aims at describing and codifying the features of this spontaneous capacity of human reason. It is well known, however, that several arguments easily accepted by common sense are actually "logical fallacies", and this indicates that logic is not just a descriptive, but also a prescriptive or normative enterprise, in which the notion of logical consequence is defined in a precise way and then certain rules are established in order to maintain the discourse in keeping with this notion. Yet in the justification of the correctness and adequacy of these rules commonsense reasoning must necessarily be used, and in such a way its foundational role is recognized. Moreover, it remains also true that several branches and forms of logic have been elaborated precisely in order to reflect the structural features of correct argument used in different fields of human reasoning and yet insufficiently mirrored by the most familiar logical formalisms.

The dawn of paraconsistency: Russia's logical thought in the turn of XX century

The paper deals with the factors which enabled N. A. Vasiliev to put forward in 1910 - 12 the idea of logics free of the laws of contradiction and excluded middle, the idea of metalogic and to construct his imaginary logic as novel non-classical system. It is shown that background of Vasiliev's ideas lies deeply in Russia's culture and particular approach to logical discourse. Several Russian scholars expressed ideas similar to Vasiliev's though not in such explicit form. This period might be called the prehistory of paraconsistency. Real history of paraconsistency starts with N.C.A. da Costa's works.

Remarks on an algebraic semantics for paraconsistent Nelson's logic

In the paper Busaniche and Cignoli (2009) we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson's logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices.

How real are real numbers?

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel (1871-1956).

Da Costa on ontology: a naturalistic interpretation

Da Costa's conception of being modifies that of Quine to incorporate relativization to non-classical logics. A naturalistic view of this conception is discussed. This view tries to extend to logic some ideas of Maddy's naturalism concerning mathematics.

Remarks on abstract Galois theory

This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva (see da Costa, Rodrigues (2007)). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva's notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are made on the universal theory of (set-theoretic) structures.

Truth, laws and the progress of science

In this paper I analyze the difficult question of the truth of mature scientific theories by tackling the problem of the truth of laws. After introducing the main philosophical positions in the field of scientific realism, I discuss and then counter the two main arguments against realism, namely the pessimistic meta-induction and the abstract and idealized character of scientific laws. I conclude by defending the view that well-confirmed physical theories are true only relatively to certain values of the variables that appear in the laws.

Models, truth and realism: assessing Bas van Fraassen's views on scientific representation

This paper is devoted to an analysis of some aspects of Bas van Fraassen's views on representation. While I agree with most of his claims, I disagree on the following three issues. Firstly, I contend that some isomorphism (or at least homomorphism) between the representor and what is represented is a universal necessary condition for the success of any representation, even in the case of misrepresentation. Secondly, I argue that the so-called "semantic" or "model-theoretic" construal of theories does not give proper due to the role played by true propositions in successful representing practices. Thirdly, I attempt to show that the force of van Fraassen's pragmatic - and antirealist - "dissolution" of the "loss of reality objection" loses its bite when we realize that our cognitive contact with real phenomena is achieved not by representing but by expressing true propositions about them.

Expressiveness and definability in circumscription

We investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of Circumscription. First, we give a proof of the failure of the Löwenheim-Skolem Theorem for Circumscription. Then we show that, if the class of P; Z-minimal models of a first-order sentence is Δ-elementary, then it is elementary. That is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. This means that classes of models of circumscribed theories are either elementary or not Δ-elementary. Finally, using the previous result, we prove that, whenever a relation Pi is defined in the class of P; Z-minimal models of a first-order sentence Φ and whenever such class of P; Z-minimal models is Δ-elementary, then there is an explicit definition ψ for Pi such that the class of P; Z-minimal models of Φ is the class of models of Φ ∧ ψ. In order words, the circumscription of P in Φ with Z varied can be replaced by Φ plus this explicit definition ψ for Pi.

Truths ancient and modern

The paper presents a comparison of the theories of truth, and the solutions of the liar paradox, proposed by Thomas Bradwardine (c. 1290-1349), Jean Buridan (c. 1295-1358), and Alfred Tarski (1901-1983).

The role of consistency in empirical science

Any inconsistent theory whose underlying logic is classical encompasses all the sentences of its own language. As it denies everything it asserts, it is useless for explaining or predicting anything. Nevertheless, paraconsistent logic has shown that it is possible to live with contradictions and still avoid the collapse of the theory. The main point of this paper is to show that even if it is formally possible to isolate the contradictions and to live with them, this cohabitation is neither desired by working scientists not desirable for the progress of science. Several cases from the recent history of physics and cosmology are analyzed.

Consistency from the perspective of an experimental systems approach to the sciences and their epistemic objects

It is generally accepted that the development of the modern sciences is rooted in experiment. Yet for a long time, experimentation did not occupy a prominent role, neither in philosophy nor in history of science. With the 'practical turn' in studying the sciences and their history, this has begun to change. This paper is concerned with systems and cultures of experimentation and the consistencies that are generated within such systems and cultures. The first part of the paper exposes the forms of historical and structural coherence that characterize the experimental exploration of epistemic objects. In the second part, a particular experimental culture in the life sciences is briefly described as an example. A survey will be given of what it means and what it takes to analyze biological functions in the test tube.

Formal semantics for propositional attitudes

Contemporary logic is confined to a few paradigmatic attitudes such as belief, knowledge, desire and intention. My purpose is to present a general model-theoretical semantics of propositional attitudes of any cognitive or volitive mode. In my view, one can recursively define the set of all psychological modes of attitudes. As Descartes anticipated, the two primitive modes are those of belief and desire. Complex modes are obtained by adding to primitive modes special cognitive and volitive ways or special propositional content or preparatory conditions. According to standard logic of attitudes (Hintikka), human agents are either perfectly rational or totally irrational. I will proceed to a finer analysis of propositional attitudes that accounts for our imperfect but minimal rationality. For that purpose I will use a non standard predicative logic according to which propositions with the same truth conditions can have different cognitive values and I will explicate subjective in addition to objective possibilities. Next I will enumerate valid laws of my general logic of propositional attitudes. At the end I will state principles according to which minimally rational agents dynamically revise attitudes of any mode.

Matrix- based logic for avoiding paradoxes and its paraconsistent alternative

The present article shows that there are consistent and decidable many- valued systems of propositional logic which satisfy two or all the three criteria for non- trivial inconsistent theories by da Costa (1974). The weaker one of these paraconsistent system is also able to avoid a series of paradoxes which come up when classical logic is applied to empirical sciences. These paraconsistent systems are based on a 6- valued system of propositional logic for avoiding difficulties in several domains of empirical science (Weingartner (2009)).

Varieties of the generality constraint

Since its introduction by Evans (1982), the generality constraint (GC) has been invoked by various philosophers for different purposes. Our purpose here is, first, to clarify what precisely the GC states by way of an interpretive framework, the GC Schema, and second, to demonstrate in terms of this framework some problems that arise if one invokes the GC (or systematicity) without clearly specifying an appropriate interpretation. By utilizing the GC Schema these sorts of problems can be avoided, and we thus propose it as a tool to facilitate argumentation that appeals to the GC.