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Reductio ad Absurdum

1. Introduction

The title of this book mentions the concepts of paraconsistency and constructive logic. However, the presented material belongs to the field of paraconsistency, not to constructive logic. At the level of metatheory, the classical methods are used. We will consider two concepts of negation: the negation as reduction to absurdity and the strong negation. Both concepts were developed in the setting of constrictive logic, which explains our choice of the title of the book. The paraconsistent logics are those, which admit inconsistent but non-trivial theories, i.e., the logics which allow one to make inferences in a non-trivial fashion from an inconsistent set of hypotheses. Logics in which all inconsistent theories are trivial are called explosive. The indicated property of paraconsistent logics yields the possibility to apply them in different situations, where we encounter phenomena relevant (to some extent) to the logical notion of inconsistency. Examples of these situations are (see [86]): information in a computer data base; various scientific theories; constitutions and other legal documents; descriptions of fictional (and other non-existent) objects; descriptions of counterfactual situations; etc. The mentioned survey by G. Priest [86] may also be recommended for a first acquaintance with paraconsistent logic. The study of the paraconsistency phenomenon may be based on different philosophical presuppositions (see, e.g., [87]). At this point, we emphasize only one fundamental aspect of investigations in the field of paraconsistency. It was noted by D. Nelson in [65, p. 209]: “In both the intuitionistic and the classical logic all contradictions are equivalent. This makes it impossible to consider such entities at all in mathematics. It is not clear to me that such a radical position regarding contradiction is necessary.” Rejecting the principle “a contradiction implies everything”(ex contradictione quodlibet) the paraconsistent logic allows one to study the phenomenon of contradiction itself. Namely this formal logical aspect of paraconsistency will be at the centre of attention in this book.

2. Minimal Logic. Preliminary Remarks

3. Logic of Classical Refutability

4. The Class of Extensions of Minimal Logic

In this chapter, we assign to every properly paraconsistent extension L of minimal logic an intermediate logic L int and negative logic Lneg called intuitionistic and negative counterparts of L, respectively. It will be proved that the negative counterpart Lneg explicates the structure of contradictions of paraconsistent logic L. We show that both counterparts Lint and Lneg are faithfully embedded into the original logic L. Finally, we investigate a question: to what extent is a logic L ∈ Par determined by its counterparts? As a first step, we study paraconsistent extensions of the logic\({\rm Le' : = Li } \cap {\rm Ln = Lj + }\left\{ { \bot \vee \left( { \bot \to p} \right)} \right\}.\)
The class of extensions of this logic has a nice property that every logic LεLe′ ∩ Par is uniquely determined by its intuitionistic and negative counterparts.

5. Adequate Algebraic Semantics for Extensions of Minimal Logic

The goal of this chapter is to find a representation of j-algebras, convenient for working with logics lying inside the intervals Spec(L1, L2). We have to understand the structure of an arbitrary j-algebra A with given upper algebra A⊥ and lower algebra A⊥. The semantic characterization of Glivenko’s logic considered in Section 5.1 prompts the solution to this problem. The desired representation is described in Section 5.2. In Section 5.3 with the help of the obtained representation we characterize the Segerberg logics and demonstrate its effectiveness in this way. Finally, in Section 5.4 we consider the Kripke semantics and define for j-frames analogs of upper and lower algebras associated with a j-algebra.

6. Negatively Equivalent Logics

In the following, by negative formulas we mean formulas of the form ¬φ. The well-known Glivenko theorem implies, in particular, that in intuitionistic and in classical logic the same negative formulas are provable. This means that intuitionistic and classical logic, as well as all intermediate logic have, in a sense, the same negation. Generalizing this relation between logics we define negatively equivalent logics as logics where the same negative formulas are inferable from the same sets of hypotheses. From the constructive point of view we need negation to refute formulas on the basis of one or another set of hypotheses, therefore, negatively equivalent logics have essentially the same negation. Unlike the class of intermediate logics, the relation of negative equivalence is non-trivial on the class Jhn+ and in this chapter we obtain several interesting results on the structure of negative equivalence classes. Simultaneously, we prove the results on cardinality of intervals of the form Spec(L1, L2).

7. Absurdity as Unary Operator

This chapter finishes the first part of the book devoted to the concept of negation as reduction to absurdity. As was mentioned in Chapter 1, minimal logic lies on the border line of paraconsistency. We have in Lj for arbitrary formulas φand ψ,{ φ, ¬ φ} ├ ¬ψ
This means that although inconsistent Lj-theories may be non-trivial, they are trivial with respect to negation. Any negated formula is provable in any inconsistent Lj-theory.

Strong Negation

8. Semantical Study of Paraconsistent Nelson's Logic

This chapter starts the second part of the book devoted to paraconsistent Nelson’s logic N4 and to the class of its extensions.
The natural first step in the investigation of the class of N4-extensions is to provide an adequate algebraic semantics for the logic N4, i.e., characterizing N4 via a variety of algebraic systems ν such that there exists a natural dual isomorphism between the lattice of N4-extensions and the lattice of subvarieties of ν.

9. N4┴-Lattices

In the previous chapter, we considered two variants of Nelson’s paraconsistent logic, N4 and N4┴, which were defined in different languages. N4┴ is a conservative extension of N4 in the language with the additional constant ┴ allowing us to define in this logic the intuitionistic negation. The addition of this constant results in an easy modification of the semantics. Models of N4 are isomorphic to twist-structures over implicative lattices and due to the fact that implicative lattices do not necessarily have the least element and the lattices modelling N4 do not have the greatest element either. In fact, no constant can be naturally defined in N4. Extending the language with ┴ we obtain the class of models isomorphic to twist-structures over Heyting algebras, i.e., implicative lattices with the least element 0. A twist-structure over bounded lattice is also bounded, it contains necessarily elements (0, 1) and (1, 0), which are the least and the greatest elements. Thus, the semantics for N4┴ is given by the class of bounded lattices. It turns out that the introduction of ┴ has essential consequences for the class of extensions εN4┴. As we will see in Chapter 10, adding the constant ┴ enriches the class of N4┴-extensions as compared to εN4, and, which is more important, provides it with a regular structure close to some extent to the structure of the class of extensions of minimal logic studied in the first part of the book. This is why we will work mainly with the logic N4┴ and with N4┴-lattices. However, all results in this chapter remains true if we replace "N4┴-lattice" with "N4-lattice" and “Heyting algebra” with “implicative lattice”.

10. The Class of N4┴-Extensions

In this chapter, we study the structure of the lattice εN4┴ and discover a definite similarity to the structure of the lattice Jhn+ studied in the first part of the book. It should be noted that differences in the structure of these two classes of logics are also essential. Moreover, we give first applications of the developed theory: the class of extensions of the logic N4┴C obtained by adding Dummett’s linearity axiom (pq) ∨ (qp) to N4┴is completely described; two classical results by L.L. Maksimova, namely, the description of pretabular logics and the description of logics with Craig interpolation property, are transferred from the class of superintuitionistic logics to the class of N4┴-extensions.

11. Conclusion

Discussing the question “Why is paraconsistency worthy?” J.-Y. Béziau [11] emphasized that paraconsistent logic is an important contribution to the theory of negation and to modern logic in general. The distinction between triviality and inconsistency made in paraconsistent logic is similar to the distinction between implication and inference relation and allows one to elucidate new features of traditional logical notions. Suppose that the investigations presented in this book also contribute to the general theory of logical systems. For two explosive logics with the same positive fragment and with essentially different kinds of negation, we investigated how the lattice of extensions of a logic changes when the explosion axiom is deleted, i.e., if we pass from a logic to its paraconsistent analog. It turns out that in both cases the lattices of extensions extend in a rather regular manner. In the class of extensions of a paraconsistent logic, one can distinguish the subclass of explosive logics, i.e., the class of extensions of the original explosive logic; the subclass of logics, which can be used to represent the structures of contradictions in all extensions of the considered paraconsistent logic (see Remark after Proposition 10.2.5). Finally, all other logics can be obtained via a combination of logics from the above two subclasses. The manner of combination can be explicated via a suitable representation theory for algebras modelling the logics under consideration.


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