Chapter 3 On the Philosophy and Mathematics of the Logics of Formal Inconsistency Walter Carnielli and Abilio Rodrigues Abstract The aim of this text is to present the philosophical motivations for the Logics of Formal Inconsistency (LFIs), along with some relevant technical results. The text is divided into two main parts (besides a short introduction). In Sect.3.2,we present and discuss philosophical issues related to paraconsistency in general, and especially to logics of formal inconsistency. We argue that there are two basic and philosophically legitimate approaches to paraconsistency that depend on whether the contradictions are understood ontologically or epistemologically. LFIs are suitable to both options, but we emphasize the epistemological interpretation of contradictions. The main argument depends on the duality between paraconsistency and paracom- pleteness. In a few words, the idea is as follows: just as excluded middle may be rejected by intuitionistic logic due to epistemological reasons, explosion may also be rejected by paraconsistent logics due to epistemological reasons. In Sect. 3.3, some formal systems and a few basic technical results about them are presented. Keywords Logics of Formal Inconsistency · Contradictions · Philosophy of paraconsistency Mathematics Subject Classication (2000) Primary 03B53 · Secondary 03A05 · 03-01 The first author acknowledges support from FAPESP (São Paulo Research Council) and CNPq, Brazil (The National Council for Scientific and Technological Development). The second author acknowledges support from the Universidade Federal de Minas Gerais (project call 12/2011) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, research project 21308). We would like to thank Henrique Almeida, Marcos Silva and Peter Verdée for some valuable comments on a previous version of this text. W. Carnielli (B) Centre for Logic, Epistemology and the History of Science and Department of Philosophy - State University of Campinas, Campinas, SP, Brazil e-mail: [email protected] A. Rodrigues Department of Philosophy - Federal University of Minas Gerais, Belo Horizonte, MG, Brazil e-mail: [email protected] © Springer India 2015 57 J.-Y. Beziau et al. (eds.), New Directions in Paraconsistent Logic, Springer Proceedings in Mathematics & Statistics 152, DOI 10.1007/978-81-322-2719-9_3 58 W. Carnielli and A. Rodrigues 3.1 Introduction The aim of this text is to present the philosophical motivations for the Logics of Formal Inconsistency (LFIs), along with some relevant technical results. The target audience is mainly the philosopher and the logician interested in the philosophical aspects of paraconsistency.1 In Sect. 3.2, On the philosophy of Logics of Formal Inconsistency, we present and discuss philosophical issues related to paraconsistency in general, and espe- cially to Logics of Formal Inconsistency. We argue that there are two basic and philosophically legitimate approaches to paraconsistency that depend on whether the contradictions are understood ontologically or epistemologically. LFIsarewell suited to both options, but we emphasize the epistemological interpretation of con- tradictions. The main argument depends on the duality between paraconsistency and paracompleteness. In a few words, the idea is as follows: just as excluded middle may be rejected by intuitionistic logic due to epistemological reasons, explosion may also be rejected by paraconsistent logics due to epistemological reasons. In Sect. 3.3, On the mathematics of Logics of Formal Inconsistency, some formal systems and a few of their basic technical results are presented. These systems are designed to fit the philosophical views presented in Sect. 3.2. 3.2 On the Philosophy of the Logics of Formal Inconsistency It is a fact that contradictions appear in a number of real-life contexts of reasoning. Databases very often contain not only incomplete information but also conflicting (i.e. contradictory) information.2 Since ancient Greece, paradoxes have intrigued logicians and philosophers, and, more recently, mathematicians as well. Scientific theories are another example of real situations in which contradictions seem to be unavoidable. There are several scientific theories, however, successful in their areas of knowledge that yield contradictions, either by themselves or when put together with other successful theories Contradictions are problematic when the principle of explosion holds: 1This paper corresponds, with some additions, to the tutorial on Logics of Formal Inconsistency presented in the 5th World Congress on Paraconsistency that took place in Kolkata, India, in February 2014. Parts of this material have already appeared in other texts by the authors, and other parts are already in print elsewhere [12, 13]. A much more detailed mathematical treatment can be found in Carnielli and Coniglio [10] and Carnielli et al. [15]. 2We do not use the term ‘information’ here in a strictly technical sense. We might say, in an attempt not to define but rather to elucidate, that ‘information’ means any ‘amount of data’ that can be expressed by a sentence (or proposition) in natural language. Accordingly, there may be contradictory or conflicting information (in a sense to be clarified below), vague information, or lack of information. 3 On the Philosophy and Mathematics of the Logics of Formal Inconsistency 59 A → (∼A → B).3 In this case, since anything follows from a contradiction, one may conclude anything whatsoever. In order to deal rationally with contradictions, explosion cannot be valid without restrictions, since triviality (that is, a circumstance such that everything holds) is obviously unacceptable. Given that in classical logic explosion is a valid principle of inference, the underlying logic of a contradictory context of reasoning cannot be classical. In a few words, paraconsistency is the study of logical systems in which the presence of a contradiction does not imply triviality, that is, logical systems with a non-explosive negation ¬ such that a pair of propositions A and ¬A does not (always) trivialize the system. However, it is not only the syntactic and semantic properties of these systems that are worth studying. Some questions arise that are perennial philosophical problems. The question about the nature of contradictions accepted by paraconsistent logics is where a good amount of the debate on the philosophical significance of paraconsistency has been concentrated. In philosophical terminology, we say that something is ontological when it has to do with reality, the world in the widest sense, and that something is epistemological when it has to do with knowledge and the process of its acquisition. A central question for paraconsistency is the following: Are the contradictions that paraconsistent logic deals with ontological or epistemological? Do contradictions have to do with reality proper? That is, is reality intrinsically contradictory, in the sense that we really need some pairs of contradictory propositions in order to describe it correctly? Or do contradictions have to do with knowledge and thought? Contradictions of the latter kind would have their origin in our cognitive apparatus, in the failure of measuring instruments, in the interactions of these instruments with phenomena, in operations of thought, or even in simple mistakes that in principle could be corrected later on. Note that in all of these cases the contradiction does not belong to reality properly speaking. The question about the nature of contradictions, in its turn, is related to another central issue in philosophy of logic, namely the nature of logic itself. As a theory of logical consequence, the task of logic is to formulate principles and methods for establishing when a proposition A follows from a set of premises . But a question remains: What are the principles of logic about? Are they about language, thought, or reality? That logic is normative is not contentious, but its normative character may be combined both with an ontological and an epistemological approach. The epistemological side of logic is present in the widespread (but not unanimous) characterization of logic as the study of laws of thought. This concept of logic, which acknowledges an inherent relationship between logic and human rationality, has been put aside since classical logic has acquired the status of the standard account of logical consequence—for example, in the work of Frege, Russell, Tarski, Quine, and many other influential logicians. 3The symbol ∼ will always denote the classical negation, while ¬ usually denotes a paraconsistent negation but sometimes a paracomplete (e.g. intuitionistic) negation. The context will make it clear in each case whether the negation is used in a paracomplete or paraconsistent sense. 60 W. Carnielli and A. Rodrigues Classical logic is a very good account of the notion of truth preservation, but it does not give a sustained account of rationality. This point shall not be developed in detail here, but it is well known that some classically valid inferences are not really applied in real-life contexts of reasoning, for example: from A, to conclude that anything implies A; from A, to conclude the disjunction of A and anything; from a contradiction, to conclude anything. The latter is the principle of explosion, and of course it is not rational to conclude that 2 + 2 = 5 when we face some pair of contradictory propositions. Nevertheless, from the point of view of preservation of truth, given the classical meaning