Constructive Negations and Paraconsistency

239 Pages·2008·2.23 MB·English

by  Sergei P. Odintsov

Constructive Negations and Paraconsistency
TRENDS IN LOGIC
Studia Logica Library
VOLUME 26
Managing Editor
Ryszard Wójcicki, Institute of Philosophy and Sociology,
Polish Academy of Sciences, Warsaw, Poland
Editors
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Roskilde University,
Denmark
Daniele Mundici, Department of Mathematics “Ulisse Dini”,
University of Florence, Italy
Ewa Orłowska, National Institute of Telecommunications,
Warsaw, Poland
Krister Segerberg, Department of Philosophy, Uppsala University,
Sweden
Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,
Germany
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Trends in Logic is a bookseries covering essentially the same area as the journal
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parisons and sources of inspiration is open and evolves over time.
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Sergei P. Odintsov
Constructive Negations
and Paraconsistency
123
Sergei P. Odintsov
Russian Academy of Sciences
Siberian Branch
Sobolev Institute of Mathematics
Koptyug Ave. 4
Novosibirsk
Russia
ISBN 978-1-4020-6866-9 e-ISBN 978-1-4020-6867-6
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Contents
1 Introduction 1
I Reductio ad Absurdum 13
2 Minimal Logic. Preliminary Remarks 15
2.1 Definition of Basic Logics . . . . . . . . . . . . . . . . . . . . 15
2.2 Algebraic Semantics . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Kripke Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Logic of Classical Refutability 31
3.1 Maximality Property of Le . . . . . . . . . . . . . . . . . . . 32
3.2 Isomorphs of Le . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 The Class of Extensions of Minimal Logic 41
(cid:2)
4.1 Extensions of Le . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Intuitionistic and Negative Counterparts
(cid:2)
for Extensions of Le . . . . . . . . . . . . . . . . . . . 45
4.2 Intuitionistic and Negative Counterparts for Extensions
of Minimal Logic . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 Negative Counterparts as Logics of Contradictions . . 52
4.3 Three Dimensions of Par . . . . . . . . . . . . . . . . . . . . . 53
5 Adequate Algebraic Semantics for Extensions
of Minimal Logic 57
5.1 Glivenko’s Logic . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Representation of j-Algebras . . . . . . . . . . . . . . . . . . 59
5.3 Segerberg’s Logics and their Semantics . . . . . . . . . . . . . 62
5.4 Kripke Semantics for Paraconsistent Extensions of Lj . . . . 78
v
vi Contents
6 Negatively Equivalent Logics 81
6.1 Definitions and Simple Properties . . . . . . . . . . . . . . . . 81
6.2 Logics Negatively Equivalent to Intermediate Ones . . . . . . 84
6.3 Abstract Classes of Negative Equivalence . . . . . . . . . . . 88
+
6.4 The Structure of Jhn up to Negative Equivalence . . . . . . 91
7 Absurdity as Unary Operator 101
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Le and L(cid:2) ukasiewicz’s Modal Logic . . . . . . . . . . . . . . . 104
7.3 Paradox of Minimal Logic and Generalized Absurdity . . . . 108
7.4 A- and C-Presentations . . . . . . . . . . . . . . . . . . . . . 113
7.4.1 Definitions and First Results . . . . . . . . . . . . . . 113
7.4.2 Logic CLuN . . . . . . . . . . . . . . . . . . . . . . . 119
1
7.4.3 Sette’s Logic P . . . . . . . . . . . . . . . . . . . . . 123
II Strong Negation 129
8 Semantical Study of Paraconsistent Nelson’s Logic 131
8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 Fidel’s Semantics . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3 Twist-structures . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3.1 Embedding of N3 into N4 . . . . . . . . . . . . . . . 142
8.4 N4-Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.5 The Variety of N4-Lattices . . . . . . . . . . . . . . . . . . . 147
⊥ ⊥
8.6 The Logic N4 and N4 -Lattices . . . . . . . . . . . . . . . 155
⊥
9 N4 -Lattices 159
⊥
9.1 Structure of N4 -Lattices . . . . . . . . . . . . . . . . . . . . 161
⊥
9.2 Homomorphisms and Subdirectly Irreducible N4 -Lattices . 167
⊥
10 The Class of N4 -Extensions 177
10.1 EN4⊥ and Int+ . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2 The Lattice Structure of EN4⊥ . . . . . . . . . . . . . . . . . 185
10.3 Explosive and Normal Counterparts . . . . . . . . . . . . . . 195
10.4 The Structure of EN4C and EN4⊥C. . . . . . . . . . . . . . 201
⊥
10.5 Some Transfer Theorems for the Class of N4 -Extensions . . 211
11 Conclusion 223
Bibliography 227
Index 237
Chapter 1
Introduction
Thetitleofthisbookmentionstheconceptsofparaconsistencyandconstruc-
tive logic. However, the presented material belongs to the field of paracon-
sistency, not to constructive logic. At the level of metatheory, the classical
methods are used. We will consider two concepts of negation: the nega-
tion 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 in-
consistent 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 situ-
ations 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 paraconsis-
tency 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 contradic-
tions are equivalent. This makes it impossibleto 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
1
2 1 Introduction
to study the phenomenon of contradiction itself. Namely this formal logical
aspect of paraconsistency will be at the centre of attention in this book.
We now turn to constructive logic. Constructive logic is the logic of con-
structive mathematics, logic oriented on dealing with the universe of con-
structive mathematical objects. The common feature of different variants
of constructive mathematics is the rejection of the concept of actual infin-
ity and admitting only the existence of objects constructed on the base of
the concept of potential infinity. In any case, passing to constructive logic
from the classical one changes the sense of logical connectives. For example,
Markov [60] defines the constructive disjunction as follows: “The construc-
tive understandingof the existence of a mathematical object corresponds to
the constructive understanding of the disjunction of sentences of the form
“P or Q”. Such a sentence is considered as accepted if at least one of the
sentences P, Q was accepted as true.” Of course, this understanding of dis-
junction does not allow one to accept the law of excluded middle and leads
to the rejection of classical logic. In the setting of constructive logic, there
are two basic approaches to the concept of negation and they are considered
in our investigation.
Since the Brouwer works, the negation of statement P, ¬P, is under-
stood as an abbreviation of the statement “assumption P leads to a con-
tradiction”. Note that this concept agrees well with paraconsistency. The
above understanding of negation does not assume the principle “contra-
diction implies everything” (ex contradictione quodlibet) responsible for the
trivialization ofinconsistent theories.Thefirstformalization ofintuitionistic
logic suggested by A.N. Kolmogorov [44] in 1925 was paraconsistent. In this
work, A.N. Kolmogorov reasonably noted that ex contradictione quodlibet
(in the form ¬p → (p → q)) has appeared only in the formal presentation
of classical logic and does not occur in practical mathematical reasoning.
However, A. Heyting was sure that using ex contradictione quodlibet is ad-
missible in intuitionistic reasoning and he added the axiom ¬p → (p → q)
to his variant Li of intuitionistic logic [35]. Note that adding ex contradic-
tione quodlibet creates some problems with interpretation of Li as calculus
of problems [45]. One cannot consider the implication P → Q as the prob-
lem of reducing the problem Q to the problem P. In Li, the implication
P → Q means that the problem Q can be reduced to the problem P or the
problem P is meaningless. This difficulty was known to A. Heyting, but he
did not considered this as a serious problem. According to A. Heyting [36,
p. 106], “... it (ex contradictione quodlibet — S.O.) adds to the precision of
the definition of implication” and “I shall interpret implication in this wider
sense.”
1 Introduction 3
Only in 1937 I. Johansson [41] questioned the using of ex contradictione
quodlibet in constructive reasoning and suggested the system, which we de-
note by Lj. Axiomatics for Lj can be obtained by deleting ex contradictione
quodlibet from the standard list of axioms for intuitionistic logic, more ex-
actly, Li = Lj+{¬p → (p → q)}. In [41], Johansson proved that many im-
portantpropertiesofnegationprovableintheHeytinglogicLicanbeproved
also in the system Lj. Since that the logic has the name “Johansson’s logic”
or “minimal logic”(see the title of Johansson’s article). Note that, in fact,
Johansson came back to the Kolmogorov’s variant of intuitionistic logic.
More exactly, the implication-negation fragment of Lj coincides with the
propositional fragment of the system from [44]. Kolmogorov considered the
first-orderlogic,butinthelanguagewithonlytwopropositionalconnectives,
implication and negation.
Unfortunately, the logic Lj was for a long time on the borderline of
studies in the field of paraconsistency, which was traditionally motivated by
the following “paraconsistent paradox” of Lj. Although Lj is not explosive,
admits non-trivial inconsistent theories, we can prove in Lj for any formulas
ϕ and ψ that
ϕ,¬ϕ (cid:3)Lj ¬ψ.
This means that the negation makes no sense in inconsistent Lj-theories,
because all negated formulas are provable in them. In this way, inconsis-
tent Lj-theories are positive. It should be noted that studies in the field
of paraconsistency were directed during a long period to searching for “the
most natural system” of paraconsistent logic, which is maximally close to
classical logic (cf. [39, p. 147]). The above paradox obviously shows that Lj
cannotplaytheroleofsuchlogic.However,recently moreattention hasbeen
paidto thestudyof paraconsistent analogs of well-known logical systems.In
this respect, Johansson’s logic Lj is worthy of attention as a paraconsistent
analog of intuitionistic logic Li.
Turning to the second main approach to negation in constructive logic,
the concept of strong negation. Note that the strong negation is namely a
proper constructive negation.
As happens with most fundamental logical concepts, the concept of
strongnegation was developedindependentlybymanyauthorsandwithdif-
ferent motivations. Constructive logic with strong negation was suggested
for the first time by D. Nelson in 1949 [64]. The truth of a negation of
statement in intuitionistic and minimal logic can be stated only indirectly,
via reducing a negated sentence to an absurdity. As a consequence of this,
the negation in these logics has the following feature, unsatisfiable from the