Contradictions, from Consistency to Inconsistency

If something is contradictory, then it is not consistent; but if something is non-contradictory, is it necessarily consistent? If so, there may be nothing between consistency and inconsistency. Thus if we literally apprehend the title of this book, it will be on nothing. However, the title of this book should be understood more broadly. This is because it is not so obvious how we should deal with notions like contradictions, consistency, inconsistency, and triviality. It must not be the case that something is there and is not there at the same time - here is the principle of contradiction in the formulation of Aristotle which, on the one hand, forms the basis of all critical thinking, and on the other hand, it is the object of controversy among the philosophers from Heraclitus through Hegel, to the present day.

We examine an argument advanced by Newton C. A. da Costa according to which there may be true contradictions about the concrete world. This is perhaps one of the few arguments advancing this kind of thesis in full generality in the context of a scientifically-oriented philosophy. Roughly put, the argument holds that contradictions in the concrete world may be present where paradoxes require controversial solutions, solutions which in general are radically revisionary on much of the body of established science. We argue that the argument may be successfully challenged in the face of the actual practice of science; as a consequence, commitment to true contradictions about the world may be correctly dismissed as unnecessary, at least if the route to contradictions is the one advanced in the argument. We finish by highlighting a parallel between da Costa’s argument and another typical dialetheist argument by Graham Priest to the effect that paradoxes of self-reference are true contradictions.

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic \(\mathbf {LP}\). On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics \(\mathbf {TS}\) and \(\mathbf {ST}\), introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively.

Let us assume that a set of sentences explains a phenomenon within a system of beliefs and rules. Such rules and beliefs may vary and this could have as a collateral effect that different sets of sentences may become explanations relative to the new system, while other ones no longer count as such. In this paper we offer a general formal framework to study this phenomenon. We also give examples of such variations as we replace rules of classical deductive logic with rules more in the spirit of da Costa’s paraconsistent calculi, Reiter’s default theories, or even a combination of them. This paper generalizes the notion of epistemic system in [6]. That notion was used to analyze the concept of explanation, using Reiter’s default theories and a specific paraconsistent logic of da Costa. Our proposal is a formal framework, GMD, based on doxastic systems, which allows us to analyze the interaction between theoretical constructs (in this case, explanations), theories and logics. We mention some obstacles, we develop the formal framework, and finally we apply it to the modeling of scientific explanation. Along the way, we try to shed light on different kinds of interaction between paraconsistency and non-monotonicity.

On closer inspection many apparent contradictions turn out to be mere disagreements between distinct sources of information. For example, if a source \(s_1\) says P and a source \(s_2\) says \(\lnot P\), their disagreement would only become an actual contradiction if we naively merged what they say into our own knowledge base.

The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel’s Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. However, we put forward a thesis that ZFC is actually very powerful at some infinite cardinals, but not at all of them. We have to move away from the first few and to look at limits of uncountable cardinals, such as \( \aleph _\omega \). Specifically, we work with singular cardinals (which are necessarily limits) and we illustrate that at such cardinals there is a very serious limit to independence and that many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. In a certain sense, which we explain, the behavior of the set-theoretic universe is asymptotically determined at singular cardinals by the behavior that the universe assumes at the smaller regular cardinals. Foundationally, ZFC provides an asymptotically univocal image of the universe of sets around the singular cardinals. We also give a philosophical view accounting for the relevance of these claims in a platonistic perspective which is different from traditional mathematical platonism.

The present paper is concerned with the presumed concrete or interpreted character of some axiom systems, notably axiom systems for usual set theory. A presentation of a concrete axiom system (set theory, for example) is accompanied with a conceptual component which, presumably, delimitates the subject matter of the system. In this paper, concrete axiom systems are understood in terms of a double-layer schema, containing the conceptual component as well as the deductive component, corresponding to the first layer and to the second layer, respectively. The conceptual component is identified with a criterion given by directive principles. Two lists of directive principles for set theory are given, and the two double-layer pictures of set theory that emerged from these lists are analyzed. Particular attention is paid to set-theoretic truth and the fixation of truth-values in each double-layer picture. The semantic commitments of both proposals are also compared, and distinguished from the usual notion of ontological commitment, which does not apply. The approach presented here to the problem of concrete axiom systems can be applied to other mathematical theories with interesting results. The case of elementary arithmetic is mentioned in passing.

Let \(A^*\) and \(B^*\) be finite sets of continuous events (e.g., physical observables, or random variables) represented by elements of semisimple MV-algebras A and B. Suppose \(\alpha :A^*\rightarrow [0,1]\) and \(\beta :B^*\rightarrow [0,1]\) are coherent books, i.e., maps satisfying de Finetti’s coherence criterion. Suppose all events in \(A^*\) are (logically) independent of all events in \(B^*.\) Let \(C=A\otimes B\) be the semisimple tensor product of A and B. We first prove that if \(a,a'\in A^*\) and \( b,b'\in B^*\) satisfy \(a\otimes b=a'\otimes b'\), then \(\alpha (a)\beta (b)=\alpha (a')\beta (b')\). Thus by setting \(\gamma (a \otimes b)=\alpha (a)\beta (b)\) we obtain a [0, 1]-valued function \(\gamma \) defined on the set \(C^*\) of pure tensors of C of the form \(a\otimes b\) for \(a\in A^*\) and \(b\in B^*\). We then prove that \(\gamma \) is a coherent book on \(C^*\). For the proofs we need the MV-algebraic extension of de Finetti Dutch Book theorem, Fubini theorem, and the Kroupa–Panti theorem (which in turn rests on the preservation properties of the \(\varGamma \) functor, the Stone–Weierstrass theorem and the Riesz representation theorem).

We discuss the implications of a local-global (or global-limit) principle for proving the basic theorems of real analysis. The aim is to improve the set of available tools in real analysis, where the local-global principle is used as a unifying principle from which the other completeness axioms and several classical theorems are proved in a fairly direct way. As a consequence, the study of the local-global concept can help establish better pedagogical approaches for teaching classical analysis.

In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous properties hold throughout that class, for whose members there exists a set of linear algebraic techniques applicable in the study of satisfiability decision problems. In this presentation, we consider as Quantitative Logic Reasoning the tasks performed by propositional Probabilistic Logic; first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic.

We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

We study an array of logics defined on a small set of connectives (including an implication \(\rightarrow \) and a bottom particle \(\bot \)) by modularly considering subsets of a set of inference rules that we fix at the start of the game. We provide complete semantics based on possibly non-deterministic logical matrices and complexity upper bounds for the considered logics. As a consequence of the techniques applied, we also obtain completeness results for the negation-only fragments (obtained by defining the negation connective as \(\lnot p:=p\rightarrow \bot \), as usual) of all the above-mentioned logics, and analyze their possible paraconsistent character.