SORITES, ISSN 1135-1349
Issue #12. May 2001. Pp. 5-32.
From Paraconsistent Logic to Universal Logic
Copyright © by SORITES and Jean-Yves Béziau
From Paraconsistent Logic to Universal Logic*
by Jean-Yves Béziau
by Jean-Yves Béziau
1. Paraconsistent Logic (Paris, 1989-91)
But it was enough to strongly attract me. Why? Some people are attracted by paraconsistent logic, via contradictions, i. e. they think that contradictions are fundamental and therefore are naturally interested in a logic which does not reject them, but deal with them.
This was not my case. I was attracted by paraconsistent logic because I was interested in the question What is logic? Traditionally the principle of contradiction is taken as a fundamental pillar of logic. The idea is that reasoning is not possible without it. Paraconsistency goes against this idea. And if paraconsistent logic is rightly a logic, therefore what are the ground principles of logic, if any?
At this time I was studying logic at the department of mathematics of the University of Paris 7. Daniel Andler was giving there a graduate course on non-classical logics including modal logic, temporal logic, non-monotonic logic, etc. Linear logic was also in the air. But none of these logics attracted me as much as paraconsistent logic did.
It is clear that a logic like e. g. linear logic is far to be as challenging as paraconsistent logic. Informal motivations for linear logic are based on a few attractive and funny examples involving cigarettes and food, but they are not connected with a serious philosophical analysis (much the same as the penguin case for non-monotonic logic). Moreover there is a big gap between these informal motivations and the technical aspects of Girard's logic. Until now there are no convincing intuitive interpretations of linear logic operators.
Linear logic is tightly connected with the so-called structural rules of sequent calculus and it had a key role in the emergence of the new research field of substructural logics.Foot note 1_1 Of course there is a huge amount of nice technical problems related to linear logic. But it is true also for Kripke semantics and matrix theory.
My attention was directed to the common ground between C1 and classical logic. These two logics are very different and my intuition was that the very essence of logic should not lie in any of their specific differences but on their common features.
The semantics for C1 is at first sight very strange, because it is a blend of known and unknown materials. On the one hand it looks like semantics for classical logic because it is bivalent, on the other hand it looks very different because it is not truth-functional and in particular you cannot start with distributions on atomic formulas and then extend them to bivaluations on the whole set of formulas.
The common feature is that both are characteristic functions of maximal sets. In fact when you have a logic, you can always consider the class of characteristic functions of maximal sets, this makes sense even in the case where they are not at the same time homomorphisms, like in the classical case. Moreover this notion of maximal set can be defined in a purely abstract way. Often a maximal set is called a maximal consistent set and its definition depends on negation. But this must be different in paraconsistent logic, because a theory can be inconsistent without being trivial (one can in fact found paraconsistency on this distinction). The common ground notion of paraconsistent logic and classical logic is the abstract notion of non trivial maximal set.
This notion palys a key role in the completeness theorem of many logics. Studying a lot of non-classical logics, I saw plenty of completeness theorems and apparently there was an invariant kernel and this was related to Lindenbaum's extension lemma saying that every consistent set can be extended in a maximal consistent one.
The completeness theorem often appears as a kind of magic link connecting two different ontological fields: proof and truth. A close study of Lindenbaum's lemma helps to understand better this magic. Moreover if the notion of proof is defined with a sequent calculus instead of an Hilbert's style system, the completeness effect is not so spectacular.
Sequent calulus was quite popular at that time in Paris mainly because of linear logic. I remember a course of Girard presenting simultaneously and comparatively classical, intuitionistic and linear systems of sequents and giving very intuitive hints on how sequent calculus works and on the cut-elimination theorem. I got very interested in the subject and learned it thoroughly by myself reading Gentzen's original paper.
Therefore it was natural for me to try to build a sequent system for C1 and prove cut-elimination for it. There had been an aborted tentative in the late sixties for doing this by RaggioFoot note 1_3. I built a set of sequent rules for C1 using an intuitive transposition of semantical conditions. Only four years later I was able to prove a general completeness theorem which explains why this intuitive transposition was working.
I then verified that monstrous rules with three premisses and without the subformula property were not conflicting the sophisticated machinery of the cut-elimination theorem, showing that necessary conditions for this theorem are of a quite general nature, and that in particular the subformula property is not one of them.
My study of C1 was presented in my Master thesis supervised by Daniel Andler, at the department of mathematics of the University of Paris 7.Foot note 1_4 I then started a PhD with him in the same line and projected, with his support, to go to spend sometimes in Brazil with da Costa. I was lucky to meet da Costa just at this time, in January 1991, when he was visiting Paris and I was needing intuitive interest and a formal letter in order to go to Brazil.
I saw da Costa for the first time when he was presenting a memorable lecture in Paris during which, so enthusiastically animated, he performed a spectular jump, nearly breaking his legs. I was introduced to him after the lecture and the contact was quite good. I gave him a kind of abstract of my Master thesis about C1 which he liked very much and we met again several times. He asked me why I was interested in paraconsistent logic and was satisfied with my answer. My trip to Brazil was projected for august.
In fact just before meeting da Costa, my interest had already shifted definitively from paraconsistent logic to general logical stuff (at this time I had no name for this kind of thing, I had heard about general abstract non sense for category theory and I liked the expression).
This had arised mainly due to two influences. The first one was a line of research developed by da Costa himself under the name theory of valuation, which I knew through his paper with A. Loparic: «Paraconsistency, paracompletness and valuation» (see Loparic/da Costa 1984). In this paper there is a first part which is a general form of completeness theorem which is then applied to a particular logic, inspired on C1, which is both paraconsistent and paracomplete (i. e. neither the principle of contradiction nor the principle of excluded middle hold). After easily working out a sequent version for this systemFoot note 1_5 I was eager to understand the essence of this general theorem, which would take me about one year.
The other one was the study of a little book by Curry, Leçons de logique algébrique (Curry 1952). I spent one month with it in the West Indies and came back quite enlightened. In this book Curry presents, among other things, a study of four kinds of negation. To carry out this study he develops a quite general framework based on such general notion as relational algebra and gives results such as a general form of the replacement theorem.Foot note 1_6 Over the years I kept on meditating on Curry's book which was an important source of inspiration for me.Foot note 1_7
2. 1. Saturation and valuation
Arriving there I started to work on a series of problems that will turn into my paper «Recherches sur la logique abstraite: les logiques normales» (B 98h) which is itself a preliminary draft of the first chapter of my math PhD (B 95e), published separately as (B 95f).
At this time I already had decided to work only at the general level, and to use the terminology Abstract Logic to emphasize this and the fact that I was working independently of any specifications of languages and logical operators. I used the expression «logic» both as a generic term and also as a specific term. I defined an abstract logic to be a consequence relation on a given undetermined set. I stated this definition with no axioms for the consequence relation, even if my work was concerned mainly with what I called normal logics in which the three basic properties (reflexivity, monotony, transitivity) hold. My motivation and my terminology were taken from Birkhoff's famous notion of abstract algebra, that I found in Lattice theory (cf. Birkhoff 1940), which is just a set with a family of operations. My idea was already that the basic foundations of logic were not more principles for the consequence relation than principles for connectives, like the principle of contradiction. I reached the idea that we must throw out all principles altogether, that logic is not grounded on any principles or laws. In fact at this time I also launched the notion of Alphabar logics, which are abstract logics for which the law of autodeductibility (a formula is a consequence of itself) does not hold.Foot note 1_8
An intuitive example of such a logic was given to me as an adaptation of da Costa's paraclassical logic.Foot note 1_9
Within this framework of abstract logic I was tackling the general completeness theorem of da Costa's theory of valuation, according to which every logic is two-valued. The central notion in this theorem is the notion of saturated setFoot note 1_10 and not maximal set and I was wondering why. I succeeded to find an answer that satisfied me after a precise and detailed study of the abstract version of Lindenbaum's extension lemma. I distinguised four kinds of Lindenbaum's extensions (two involving the concept of maximal set, two the concept of saturated set), all equivalent in classical logic but that I proved to be all distinct at the abstract level. Moreover I succeeded to prove that the semantics of saturated sets is minimal.Foot note 1_11
I also tried to make a connection between the fact that all saturated sets ar maximal (in such a case a logic is said to be absolute, intuitionistic logic is not absolute) and the presence of certain logical connectives. By doing so I wanted to give an abstract characterization of logical operators. I succeeded to prove that both the existence of a classical negation or of a classical implication imply that a logic is absolute.Foot note 1_12
This work was purely abstract in nature and no concrete examples of logics were given, nor did I dealt with the notion of systems of deductions (rules and proofs). It was complemented by a paper that I wrote with da Costa, which was the fruit of our collaboration during this year.Foot note 1_13
The starting notion of da Costa's theory of valuation is a highly idealized version of an Hilbert's style system of deduction, simply called a calculus. The nature of the objects is not specified and the rules are just pairs with no restrictions of recursivity or cardinality. It is easy to see that in fact such a calculus, due to the definition of Hilbert's style notion of proof is the same thing as a normal abstract logic. Da Costa's definition fits better if one has the intention to apply general stuff to concrete Hilbert's style logical calculi.
By valuation, da Costa means generally any bivaluation, i. e. function which attributes true or false to formulas. His theory of valuation is a kind of generalization of his semantics for C1 (see da Costa/Alves 1977), based on the fact that once truth-functionality is dropped, bivaluations can be used as a semantics for any calculus.
My paper with da Costa has two parts. One dealing with generalities, including some results without proofs about abstract logics, but also some abstract results about rules and proofs, definitions of these extended in order to catch Gentzen's style systems as well as Hilbert's ones. The second part is on applications and shows how concrete cases of completeness can be elegantly and easily obtained from general results. An important point is that with this method it is possible to give a proof of the completeness theorem for classical logic connective by connective (therefore this theorem is the disjoint union of all completeness theorems for classical connectives). There is a sketchy indication of how to apply this method for logics of any order.Foot note 1_14 It also includes da Costa's result about the characterization of truth-functional bivalent logics.
In another paper writing at this time (B 90b) I show how it is possible to generalize da Costa's methods for C1 in order to construct a family of paraconsistent, paracomplete and non-alethic logics. Before arriving in Brazil, I already had the idea of extending naturally C1 in a logic strictly stronger that I called C1+. I didn't wrote at this time a paper devoted exclusively to this logic because I had already lost interest for the study of such or such system for its own sake. Therefore I presented C1+ in a paper among many other logics all generated by the same guiding ideas.Foot note 1_15
In the same paper I also introduced the notion of non truth-functional many-valued semantics. My initial idea was to construct a non truth-functional three-valued semantics for C1 in order to get the subformula property. The equivalence between this semantics and the standard one was given by a theorem showing how to reduce any semantics to a bivalent one.Foot note 1_16 As this example shows, such a reduction theorem does not necessarily mean that non bivalent semantics are useless. They can be useful, for technical reasons or philosophical interpretations.
Da Costa was interested in Bourbaki since his youth. As it is known A. Weil, J. Dieudonné and A. Grothendieck spent each one about two years at the University of São Paulo during post-war time. They contributed strongly to the development of modern mathematics in Brazil. Da Costa's master, E. Farah was a close friend of Weil and the first Brazilian to work on set theory.Foot note 1_17 During the late eighties, da Costa's interest for Bourbaki was renewed by his research program, developed with F. A. Doria, on the axiomatization of Physics, which leads them to various incompleteness results for physical theoriesFoot note 1_18.
Therefore when I arrived in São Paulo, the Bourbakian concept of structure was in the air and da Costa spoke many time about this subject and indicated us bibliographical references such as Corry (1992) which very rightly points out an important heterogeneity between the Bourbakian informal notion of structure as it is presented in «The architecture of mathematics» (Bourbaki 1950) and the formal definition presented in Theory of sets (Bourbaki 1968). In my opinion this duality reflects perefectly that Bourbaki's idea to take the notion of structure as the fundamental notion of mathematics is independent of his formalist option chosen in Theory of sets, which can be considered as an accidental feature motivated by the circumstances of the time and which was later on rejected by his main promoter (see Chevalley 1985). In fact «The architecture of mathematics» ends with a rather anti-formalist tone with the quotation of Lejeune-Dirichlet's motto: «to substitute ideas for calculations».
Learning more about Bourbaki, my impression was that my idea of abstract logic fitted perfectly well with the mathematical spirit of the General expressed by the slogan From the general to the particular.Foot note 1_19 And reflecting on Bourbaki's bright idea which revolutionized mathematics, my idea was to consider, within the architecture of mathematics, logical structures as mother fundamental structures but different from the three Bourbakian ones (algebra, topology, order).
In fact at this time I discovered at the library of the university of São Paulo a book by the French logician Jean Porte published in 1965 (the year I was born) and entitled Recherches sur la théorie générale des systèmes formels et sur les systèmes connectifs, with the same leading idea. Porte wrote:
«Formal systems» considered here will be some mathematical structures (the word «structure» is taken here with a meaning close to the one given by Bourbaki, but slightly different), not much, not less «fundamental» than the class of algebraic structures for example. (Porte 1965, p. 2)
In many other points Porte's objective and methodology were the same as mine with my «abstract logic». His idea was to work in the spirit of abstract modern mathematics avoiding denotational and terminological complications often met in the formalist approach and trying to dissipate confusions by establishing a general framework providing clear stucturalist definitions of the basic notions of logic.
Porte didn't have a name for his general theory but he rejected the name «metamathematics» in particular because, as he wrote (Porte 1965, p. 3) his work was not restricted to formal systems representing mathematical reasoning.
Porte's book includes a lot of results of Polish logic. It is a bright exposition of the main achievements of the Polish school, such as Lindenbaum's results on matrices, at a time when these works were not well known abroad. But the book contains also a lot of Porte's own contributions. It is much in the spirit of the Polish school (as Porte says, p. 4, like Tarski, he will allow himself to use all the methods of reasoning that the standard mathematician uses) improved by a straight Bourbakian structuralist perspective.
However Porte's work appeared to be actually quite distinct from what I was doing, especially due to the fact that he was working with more specific structures (distinguishing three classes of logical structures).Foot note 1_20
The other important discovery for my work was Polish logic. The expression «Polish logic» is ambiguous, in fact it does not denote logic in Poland but a kind of logical studies which are mainly developed in Poland and not well-known abroad. R. Wójcicki has summarized these works in his book Theory of logical calculi (Wójcicki 1988). A first version of this book was written when he was in Brazil in the late seventies and published by Ossolineum under the title Lectures on propositional calculi (Wójcicki 1984).
The connection between da Costa and Polish logicians is old and is due in particular to Jaskowski. Jaskowski is a famous logician of the Lvov-Warsaw school of logic who is known as the creator of natural deduction (independently of Gentzen) and also for his result about intuitionistic logic (soon after Gödel's result showing that intuitionistic logic cannot be characterized by a finite matrix, Jaskowski in his 1936's paper showed that it can be characterized by a infinite class of finite matrices). But in 1948 Jaskowski also wrote a paper which is now considered as an important step in the history of paraconsistent logic. Jaskowski's work on paraconsistent logic was rediscovered by da Costa, and he started, working jointly with some Polish logicians, the study of this forgotten work of Jaskowski. During the seventies da Costa and other Brazilians such as A. I. Arruda went to Poland and Polish logicians, like J. Kotas, L. Dubikajtis and R. Wójcicki, went to Brazil.
Da Costa was therefore acquainted with the main concepts of Polish logic such as matrix theory and the theory of consequence operator. He used to present the consequence operator as an equivalent formulation of his notion of calculus.Foot note 1_21
This is therefore through da Costa that I myself got soon acquainted with the basic notions of Polish logic. It seemed to me that it was very close in spirit to what I was doing and that I should investigate it seriously. Hence, as soon as december 1991, I had already decided that my next destination after Brazil would be Poland.
In August 1992, I went to the IX Latin-American Symposium on Mathematical Logic in Bahía Blanca, Argentina and presented there a little lecture on my work with da Costa on the theory of valuation (B 93c). By coincidence there was there a prominent Polish logician, Stanislas Surma, who presented a very interesting talk (see Surma 93). I had a conversation with him on the train back to Buenos Aires and as I told him I will soon be in Poland, he draw me a map of logic in Poland (names of cities and logicians). Unfortunately the difficult Polish language didn't help my memory and when I arrived in Poland I didn't remember anything.
3. 1. More abstraction
Famous logicians started to work in Wroclaw right after the war, J. Los (whose famous monograph (cf. Los 1949) on matrix theory which is the first extensive exposition of Lindenbaum's results about matrices was published by Wroclaw University Press) and also R. Suszko, J. Slupecki, etc.
I was received there by the director of the department of logic, Jan Zygmunt, a man with a huge knoweldge of the Polish school of logic, keeping the old tradition of this school alive and who turned to be a very good guide for me.
Arriving in Wroclaw I continued to develop the idea of abstract logic by presenting it and discussing it.
I wrote soon two «philosophical» papers in French about my conception of abstract logic: «De la logique formelle à la logique abstraite» (B 94a) and «La logique abstraite au sein de la mathématique moderne» (B 93d), this last one being a lecture presented at Lodz University that I was happy later on to see published in Ruch Filozoficzny, the journal founded by Twardowski in 1911 and where Lukasiewicz in 1920 presented his famous paper on many-valued logic.
Already in Brazil, I had found out that the Polish logician Roman Suszko had also used the expression «abstract logic» in « series of works carried out at the end of the sixties and the beginning of the seventies, together with two American mathematicians D. J. Brown and S. Bloom.Foot note 1_22 By an «abstract logic» he meant a consequence operator defined on an algebra. It was a slight generalization of the notion of structural consequence operator, notion which has been canonized by himslef and J. Los in their famous paper «Remarks on sentential logics» (Los/Suszko 1958).
The basic logical structure which Polish logicians are working with is indeed not really fixed. The fundamental point is to consider a logic as a consequence operator rather than as a set ot tautologies (keeping Tarski's original idea). Properties of this consequence operator may vary as well as the set on which it is defined. The standard approach is to consider rather a structural consequence operator than an abstract logic in the sense of Suszko. Moreover, even if it is not explicitly said, the replacement theorem is also generally required in addition to the theorem of substitution, in Polish terminology: a logic must not be only structural but also self-extensional (see Wójcicki 1988, p. 200).
In fact, as it is known, when Tarski first developed the theory of consequence operator at the end of the twenties (cf. Tarski 1928), he didn't specify the structure of the underlying set, taking such a set to be just a set of «meaningful sentences» in the sense of Lesniewski. Without doubt the notion of consequence operator as developed by Tarski was inspired by topology, which was highly popular at this time in Poland (cf. Kuratowski with whom Tarski worked).
From the Bourbakian perspective, Tarski's original proposal falls into topology and Suszko's abstract logic appears as a «carrefour de structures» (algebraico-topologic), as well as Los-Suszko's notion of structural consequence operator. In all the cases, logical structures are a by-product of the three fundamental Bourbakian structures.
My proposal was clearly distinct, because I was considering logical structures as different from the already known structures and because by so doing I was defining them in a very abstract way, in particular without stating any axioms for the consequence relation.
One can find indeed examples of logic which are not structural such as the logic of P. FévrierFoot note 1_23 or not self-extensional (this is the case of the paraconsistent logic C1). Moreover there is no good philosophical reasons to consider that the domain of a logic should be an algebra. The fact that logical operators are represented by functions is a mathematical representation that can be rejected: in natural language, there are sentences which are distinct negations of one given sentence, therefore negation appears rather as a relation than a function.Foot note 1_24
As for the axioms for the consequence operator, what did Tarski when he developed the theory of consequence was to axiomatize the notion of logical consequence as defined by Hilbert's style notion of proof. For such a notion, Tarski's axioms hold. But when we generalize the notion of proof, this is not necessarily the case.
In fact in Polish logic there seems to have a confusion between proof-theoretical notions and concepts related to the theory of consequence operator. This happens mainly because proof-theory did not develop by its own in Poland but was incorporated within the theory of consequence operator.Foot note 1_25 People working outside of Poland inspired by the theory of consequence operator but substituting a consequence relation denoted by the Fregean symbol |- for the consequence operator, usually denoted by Cn, have went worse into the confusion. These two concepts are in fact equivalent, but the confusion arises when people are mixing the concept of consequence relation together with Gentzen's sequent calculus as a general setting and employing the Fregean symbol as well for Gentzen's sequents, and using the same names for structural rules of sequent calculus and axioms for the consequence relation (reflexivity, monotony, cut).Foot note 1_26 The matter is even worse when one generalizes the consequence relation, keeping the Fregean symbol to denote it, to relation admitting sets of formulas on both sides (the so-called multiple conclusion logic), which resembles even more to Genzten's concept of sequent. This is for example as D. Scott proceeds in (Scott 1974b).
This leads to a confusion between two notions of different natures: the notions of rule and of law. For example when someone calls cut rule the law of transitivity for the consequence relation, he is making a confusion which leads to a misundertsanding of the cut-elimination theorem which shows in fact that a transitive logic can be generated by a sequent system without the cut rule. Scott for example wrote that «In many formalizations a great deal of effort is expended to eliminate cut as a primitive rule; but it has to be proved as a derived rule. In general, cut is not eliminable.» (Scott 1974b, p. 414). But as it is known one cannot prove cut-elimination for LK by proving that cut is a derived rule of LK without the cut rule, simply because the cut rule is not a derived rule of this system. The cut rule in this system is a permissible non derivable rule. The cut-elimination theorem shows therefore that there are some permissible rules which are not derived rules.Foot note 1_27
Discussing this with Zygmunt and trying to clear up all these confusions, I wrote a paper entitled «Rules, derived rules, permissible rules and the various types of systems of deduction» (B 99b) intended to be, among other things, a remake of D. Scott's paper «Rules and derived rules» (Scott 1974a). In this paper I tried to develop a kind of abstract proof theory, defining the central notions of proofs and rules at the abstract level, i. e. independently of the nature of the objects. This has some connections with Hertz's Satzsysteme (Hertz 1929), work which was the basic source of inspiration of Gentzen's sequent calculus and cut rule.
I decided to change the name abstract logic for universal logic at the beginning of 1993. It was a consequence of learning more about universal algebra and its connection with Polish logic.
I soon realized that universal algebra was very popular among people working in Polish logic. Suszko himself considered «abstract logic» to be a part of universal algebra and it seems that it turned out to be a common idea in Poland, as suggests the following comment by S. L. Bloom: «Roman taught us the Polish view of logic -- as a branch of universal algebra (a novel outlook for us)» (Bloom 1984, p. 313).
The connection in fact goes back to the golden years of the Polish school of logic when Tarski and Lindenbaum transformed the notion of matrix, introduced originally by Lukasiewicz for many-valued logic, into a central tool for a general theory of zero-order logics (i. e. sentential logics). By thus doing they were developing universal algebra independently of Birkhoff. We must also recall that logic was first introduced in Poland via algebra of logic, as Wolenski notes (Wolenski 1989, p. 82).
Birkhoff developed his notion of universal algebra to unify two disjoint approaches: Noether's school with groups and rings on one side and algebra of logic and lattice theory on the other side, as well explained in (Birkhoff 1976)Foot note 1_28.
No doubt that there is a strong connection between logic and universal algebra. Algebra of logic is one fundamental root of abstract algebra, because Boole was the first to deal with algebras whose objects are not quantity, and of universal algebra because the laws for logical operators such as involution are totally different from the laws for numbers; one therefore can understand why Birkhoff's unification was not possible by stating some «universal laws» which would hold for all algebras. As explained by Scott (1974b), Tarski developed model theory via the kind of universal algebra which has emerged in Poland as a general metatheory for zero-order logics, which transformed itself in Poland after the war into the mathematics of metamathematics (cf. Rasiowa/Sikorski 1963). Later on universal algebra and model theory were applied back to the general theory of zero-order logics leading to algebraic logic.Foot note 1_29
Despite all these relations between logic and algebra, I think that to consider a general theory of logics as part of universal algebra is wrong. In fact many people who are doing that are confusing universal algebra with the general theory of structures. Polish general approach to zero-order logic is highly mathematized comparatively to a standard Western approach according to which zero-order logic is presented in a rather linguistic informal way. But to make an extensive use of mathematical tools for the study of logic does not necessarily mean algebraization. It is true that algebraic tools are important but they are not the only ones. Moreover, if a wide class of logical structures can be reduced to algebraic structures via factorization, it is not the case of all logical stuctures, in particular those in which there are no non trivial congruence relations (simple logics), like what happens with the logic C1, as shown by Mortensen (1980). In my paper «Logic may be simple» (B 97h) I discuss all this in details and argue that there are no good reasons to reject such simple logics out of the sphere of logic.
As Suszko's notion of abstract algebra was understood as part of universal algebra and as this expression was therefore already used with a different meaning, I thought bettter to shift the terminology and the expression universal logic seems to me perfectly appropriate. Universal logic stands in the same position with regards to the multiplicity of logics as universal algebra with the multiplicity of algebras. Moreover, as my original idea of a naked logical structure was inspired by Birkhoff's definition of algebraic structure, I thought a good idea to use a similar terminology in logic as the one promoted by Birkhoff in algebra, who is «universally» recognized as the father of modern universal algebra.
The terminology «universal logic» shows clearly that universal logic is different from universal algebra (and in particular not part of it), but at the same time shows also the spiritual connection.
I think that the independency of universal logic with regards to universal algebra is much of the spirit of the Polish school of logic itself whose success was borne out the consideration of logic as an autonomous field as recalled by Wolenski and Zygmunt: «the logicians of the Warsaw school always emphasized the autonomy of logic as a discipline and this ideology, regardless of its justification, was one of the pillars of the success of the Warsaw school of logic» (Wolenski/Zygmunt 1989, p. 403).
The paper contains in a first part, a full description of what I mean by universal logic, including reference to the Bourbakian architecture of mathematics and a short story of universal algebra,Foot note 1_30 illustrated, in a second part, by the example of an improved abstract form of the completeness theorem I was able to present, just having found an important result working on my PhD.
This result connects rules of sequent calculus with conditions for bivaluations in such a way that it is possible to instantaneously derive from it various completeness theorems. This result is purely abstract and does not depend neither on self-extensionality nor on truth-functionality. I achieved this result by putting together da Costa's theory of valuation, Lindenbaum-Asser's abstract form of Lindenbaum's extension lemma (Asser 1959) and abstract sequent calculus. The hint of my theorem was given to me by the study of Gentzen's first paper dealing with Hertz's Satzsysteme (Gentzen 1932). This theorem arrived at the right time in order to link works that I was putting together to form my math PhD, which I decided to entitle Recherches sur la logique universelle (Excessivité, négation, séquents) (B 95e).
Excessivity was the word that I had decided to use instead of «saturation» or «maximal relativization», because I thought «saturation» improper due to the fact that this terminology was already used in model theory with a different meaning and «maximal relativization» was much too lengthy. Moreover I found appropriate to introduce a virgin name to denote a concept that my researches had revealed fundamental. In particular the above central result depends on the fact that excessive sets respect rules of a certain class of systems of sequents. To specify this class and also for a general version of the cut-elimination theorem given there, I presented a deconstructional analysis of the sequent calculus.Foot note 1_31 These general results are applied to the paraconsistent logic C1+. In this work I therefore follow the Bourbakian motto «from the general to the particular», in an inverse route that led me from paraconsistent logic to universal logic. My study of negation does not limit to paraconsistent logic but extends to a reformulation of Curry's theory of negation (Curry 1952). I was able to prove an interesting result showing that intuitionistic negation collapses into classical negation if we slightly modify the morphological feature of negation by admitting not only positive negations but also negative ones. I was led to this result by observing that the two forms of the reductio ad absurdum are quite the same (one increases the number of negations, the other one decreases it) and that therefore there were no good reasons that they should induce two different negations (this part of my PhD has been published as B 94d).
At the beginning of 1995, I left São Paulo from Paris and then flew to Los Angeles where I was to stay a couple of months at UCLA. I attended lectures and seminars there both at the philosophy and mathematics departments but I was surprised to see so few logic, I mean logic for its own sake. Y. Moschovakis righlty described me the situation by a joke saying that there was no logic at UCLA, but on one side philosophy of language on the other side a lot of set theory.
Anyway I presented a talk at the math department «Universal logic: some results in abstract completeness.» A Polish logician, emigrated to the US, told me that of course he had heard about the theory of consequence operator when he was in Poland, but that he rejected it due to the fact that it fails to capture non-monotonic logics. I think that this rejection is common nowadays due to the success of these logics. However I don't think that it is a definitive argument against Polish logic. Most of the results of consequence operator theory can be in fact adapted to the non-monotonic case and Wójcicki wrote a paper apparentlty just to prove that (Wójcicki 200?). Non-monotonic logics just show, in my opinion, that Polish logic must be widened into a true universal logic.Foot note 1_32
I wanted also to present a talk at the philosophy department on the comparison between category theory and set theory as foundations of mathematics which would include comments on Bourbaki and universal algebra. But D. A. Martin told me that it would be a mess because on one hand people of the philosophy department would not understand the talk due to their very poor knowledge of this matter and on the other hand only «big names» were able to attract people in a lecture at this department. I realized therefore that analytic philosophy was not so much different than continental philosophy in the sense that in both cases the man is more important that the stuff he is speaking about. The argumentation of the analytic philosopher is not enough rigorous to have a value by its own, independently of who expresses it, as it may happen in mathematics. I realized also that analytic philosophers were using terms from logic without knowing their exact technical meanings, and that therefore they were speaking rather metaphorically, in a way not so much different to Lacan, Deleuze or Derrida.
I left L. A. at beginning of july 1995 at the time when the airport was under threat by the Unabomber and arrived in Paris to defend my math PhD. I left Paris after escaping for short of the bomb who killed many people in the RER subway at Saint-Michel.
For example, people generally think that intuitionistic or modal logic are not many-valued but they are not really able to sustain their assertion, to turn explicit the matrix backround of it. Even less are they able to explain, if by chance they know it, the challenging result according to which it is possible to give a bivalent semantics to most logics including Lukasiewicz's logic L3 (see Suszko 1975).
As regard to paraconsistent logic, generally people just give a negative definition of it, the exclusion of the ex-falso sequitur quodlibet. But such a negative definition is meaningless as long as it is not complemented by a positive one. However there is not only one possible answer because several positive criteria may be incompatible together. I think that we can make a good job in this direction only if we have a general framework which allows us to compare rightly the various logical and metalogical properties. Working on this direction, I was able to show that there are no De Morgan full paraconsistent negations which are self-extensional (B 98b).
In a dialectical interplay, I worked on general problems and particular logics, and developed further paraconsistent logics (B 97g) including a self-extensional one (B 00a). I also used da Costa's theory of valuation to study connectives which are between conjunction and disjunction (B 98i). This is related to some problems in Biology on which I was working with M. V. Kritz at the LNCC. I think that nowadays logic is more and more connected with all the fields of knowledgeFoot note 1_35 and that universal tools will help us to built the right logic for the right situation.
I also started to develop the philosophical side of my universal logic's project which shows in particular that the formalist approach cannot any longer seriously be sustained; see (B 98e), (B 99f).
5. 1. A theory of our time
Several books and papers have been recently written in this spirit presenting various methodologies and technical tools. Gabbay edited a book (Gabbay 1994) which collects a series of essays which are supposed to answer the question What is a logical system?
In his Mathematical Review of it [96k: 03008], Walter Carnielli rightly points out that the book misses a central question, the question of translations between logic. I think he is perfectly right. We must unify the «inconsistent multiplicity» of logics, to use Cantor's expression, in a Category of logics, and study the morphims between logics, of which translations are particular cases. This is certainly a key point for a general theory of logics.Foot note 1_37
Another trouble with the book is the formulation of the question. The shorter question «What is a logic?» would be better. The expression «logical system» tends to focus on logics considered as proof-theoretical formal systems. It is much out of date and too narrow a view for full abstraction, as shown clearly by Barwise and Hammer's paper (Barwise/Hammer 1994) dealing with diagrams, an old visual approach to logic, which was considered in the past heuristical at best, but which has been taken seriously recently.Foot note 1_38
Despite of this, the proof-theoretical tendency is quite strong nowadays, in particular due to the crucial role of sequent calculus in linear logic, and in substructural logics in general. Some people are mixing this framework with the consequence operator's one and this is generating some confusions in the same line has those found in Scott years ago.Foot note 1_39
What is a substructural logic? One can say that it is a sequent system lacking some structural rules or whose sequents have not the same structure as the classical ones (cf. the intuitionistic case). Very good, very clear. But we must distinguish this system from the logic, the consequence relation, it induces. This system can lack weakening rules, and the consequence relation can be monotonic (in fact monotony always holds for a consequene relation induced by a system of sequent, due to the very defintion of «induced», which however can be modified). So what is a non-monotonic logic? Is a non-monotonic logic substructural or not? Is linear logic non-monotonic?
Gabbay in his own approachFoot note 1_40 to the subject does not avoid the confusions. For him, at first, a logical system is a consequence relation together with a proof system generating it, he says that he is compelled to this second part due to «the central role which proof theoretical methodologies play in generating logics» (Gabbay 1996, p. 3). Gabbay takes a second step by dropping monotony for the consequence relation and considering instead of a simple proof system, what he calls a LDS proof system (Gabbay 1996 p. 11).
No doubt that Labelled Deductive Systems is a nice technique which has innumerable applications, as shown by Gabbay in his book. However it is not simple enough to be as general as one may need. Firstly, why dropping just monotony? Secondly this proof-theoretical approach has some drawbacks. For example the complement of the underlying consequence relation of a logical system in the Gabbay's sense, cannot always be considered as the underlying consequence relation of a logical system (i. e. the case of first-order logic). Mathematically speaking, this class of logical systems is not closed under complementation. It is also not closed for a lot of other operations on a class of structures.
The proof-theoretical approach is limited and there are no good reasons to give priority to it. One may prefer a semantical approach. This is the case of Epstein (1990)Foot note 1_41.
Although Epstein and Gabbay's approaches are based on two different methodologies, their works bear the same defects. On the one hand some general machinery is introduced with few important significative abstract results, on the other hand they present a huge quantities of examples to which their general methodologies apply more or less happily. Moreover working only on one side of the logical business, completeness is not a central question, which seems rather odd.
In view of these works we can say that the present state of research in the systematization of logic is much the same as the pre-Birkhoff period of universal algebra, well-illustrated by the «monstrous» book of Whitehead, which collects a lot of things together, without a serious methodology and without important results; as noted by Grätzer, Whitehead «had no results», though he «recognized the need for universal algebra.» (Grätzer 1979, p. vi).
If we want to go further on, I think we must follow the method that has always shown to be right in the history of mathematics: we must jump into abstraction. We must stop for a while to be preoccupied by such or such logic and work the abstraction for its own sake. This is what Birkhoff did with universal algebra and this is what must be done in logic in order to develop a real universal logic.
The general theory of logics which is emerging is of course in some sense contrasting with the traditional line of reserach in logic that can be «labelled» by the expression foundations of mathematics. One can say that the new trend is about foundations of logicFoot note 1_42. Gabbay presents these two tendencies as opposed:
Unfortunately, the traditional logic community are still very conservative in the sense that they have not even accepted non-monotonic reasoning systems as logics yet. They believe that all this excitement is transient, temporarily generated by computer science and that it will fizzle out sooner or later. They believe that we will soon be back to the old research problems, such as how many non-isomorphic models does a theory have in some inaccessible cardinal or what is the ordinal of yet another subsystem of analysis. I think this is fine for mathematical logic but nor for the logic of human reasoning. (Gabbay 1996, pp. 3-4).
But is mathematical logic not the logic of human reasoning? Maybe reasoning about transfinite is beyond computers minds but Cantor's paradise has been investigated by human brains. Anyway, I don't think that the gap between foundations of mathematics and the general theory of logics is so big. There are some connections as shown by the equivalence between the abstract formulation of Lindenbaum's extension lemma and the axiom of choice (see Dzik 1981). Even if in Poland this duality has increased after the war, people like Los and Suszko have made important contributions on both sides, pursuing the way of their masters, and in particular Tarski, who considered logic as a whole.
On the other hand there is philosophy of logic, which has concentrated mainly on questions of reference (related to the famous «denotational» works of Frege and Russell), and which has turned in fact into philosophy of language where technical terms are used only metaphorically, the technical knowledge of these philosophers being in general reduced to truth-tables. And this may generate confusion.
I will give just an example. A lot of «philosophical logicians» are speaking about intensionality and it is standard to say that modal logics are intensional. But how can someone claim that a self-extensional logic is intensional? Explanation: on the one hand there are some people who are doing technical work on modal logic and know that these logics are self-extensional but they don't know exactly what is the problem of intensionality and use this name to conform to the usually way of speaking, on the other hand there are some people who know «Sinn und Bedeutung» and «On denoting» by heart but are not aware that current modal logics are self-extensional and what this means.Foot note 1_43
One (maybe the original) reason why modal logics are called «intensional» is because the modal operators are not truth-functional. But if non truth-functionality may be considered as a necessary condition for intensionality, it is not at all a sufficient condition, as show indeed modal logics which are self-extensional: self-extensionality clearly conflicts with intensionality as the name rightly points out.Foot note 1_44
What is needed for good philosophical discussions about logic is clear definitions of the central features of logic. Therefore we can see why universal logic can be useful if not indispensable. The definitions philosophers need involve mathematical abstract conceptualization rather than symbolic formalisation. This is what they are maybe not aware of after logicism and formalism which gave a deformed vision of mathematics, according to which mathematics is a game which consists mainly of the manipulation of strings of signs following specific rules.
To understand truth-functionality, one must learn matrix theory, to understand such result as Gödel's result showing that intuitionistic logic cannot be characterized by a finite matrix, the reason why intuitonistic logic is said to be non truth-functional. To understand self-extensionality, one must know what is a congruence relation. Someone who doesn't understand these notions cannot seriously speak about the intensionality/extensionality problem.
Universal logic can give a new direction to the philosophy of logic, because it provides via modern mathematics, rigour and abstraction, without which philosophy of logic is only metaphorical discussion, bad poetry in the sense of Carnap.Foot note 1_45
G. Priest thinks that paraconsistent logic is the most important event in logic in the XXth century because it is kicking out a principle which was taken as the basis of reasoning during more than two thousands years. He uses the word transconsistent (Priest 1987) by comparison with the tranfinite's phenomenon (funny enough paraconsistent logic has been used also to defend a finitist point of view, see e. g. (van Bendegem 1993)).
In some sense he is right, the philosophical import of paraconsistent logic cannot be ignored, but I don't think that paraconsistent logic is the new paradigm.Foot note 1_46 What we know nowadays, after paraconsistent logic, is that logic is not founded on the principle of contradiction, that logic is still logic without this principle. In this sense logic is truly transconsistent. Paraconsistent logic has clearly shown that triviality is more fundamental than consistency, as da Costa (1958) already strongly emphasized, and has thus led us to more abstraction.
In the work of VasilievFoot note 1_47, considered with Lukasiewicz as the main forunner of paraconsistent logic, we can find also some bright ideas, although his work is not technical in nature. Vasiliev argued that the principle of contradiction is empirical, that it is not a real fundamental formal principles of logic. He said that his Imaginary Logic which is a logic without the principle of contradiction just showed this, that this principle is accidental, independent (in the same way that Lobatchevski had shown with its Imaginary Geometry that Euclide's parallel postulate is). What Vasiliev said is that logic is grounded as a deeper level, which he called metalogic.Foot note 1_48
Lukasiewicz himself started his investigations which would lead to matrix theory and the general study of zero-order logics by accurate criticisms to Aristotle's defense of the principle of contradiction (Lukasiewicz 1910).
All this shows that paraconsistent logic has played a fundamental role towards universal logic, by dismissing the last and the more sacred principles of logic, showing that logic is grounded at a more abstract level, where no principles hold.
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[ B 96d ] da Costa, N. C. A. and Béziau, J.-Y.: 1996, «Théorie paraconsistante des ensembles», Logique et Analyse, 153-154, 51-67.
[ B 97a ] Béziau, J.-Y.: 1997, «La logique paraconsistante» in da Costa, N. C. A., Logiques classiques et non classiques, Masson, Paris, pp. 237-255.
[ B 97b ] Béziau, J.-Y.: 1997, «Théorie de la valuation» in da Costa, N. C. A., Logiques classiques et non classiques, Masson, Paris, pp. 256-272.
[ B 97c ] Béziau, J.-Y.: 1997, «A lógica universal» in da Costa, N. C. A., O conhecimento científico, Discurso, São Paulo, pp. 146-150.
[ B 97d ] Béziau, J.-Y.: 1997, «Teoria das valorações» in da Costa, N. C. A., O conhecimento científico, Discurso, São Paulo, pp. 150-152.
[ B 97e ] Krause, D. and Béziau, J.-Y.: 1997, «Relativizations of the principle of identity», Logic Journal of the Interest Group in Pure and Applied Logics, 5, 327-338.
[ B 97f ] Béziau, J.-Y.: 1997, «What is many-valued logic?», in Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117-121.
[ B 97g ] da Costa, N. C. A. and Béziau, J.-Y.: 1997, «Overclassical Logic», Logique et Analyse, 157, 31-44.
[ B 97h ] Béziau, J.-Y.: 1997, «Logic May be Simple -- Logic, Congruence and Algebra», Logic and Logical Philosophy, 5, 129-147.
[ B 97i ] Krause, D., J.-Y. Béziau and O. A. S. Bueno: 1997, «Estruturas em Ciência», Boletim da Sociedade Paranaense de Matemática, 17, 91-111.
[ B 98a ] da Costa, N. C. A., J.-Y. Béziau, O. A. S. Bueno: 1998, Elementos de Teoria Paraconsistente de Conjuntos, CLE, Campinas.
[ B 98b ] Béziau, J.-Y.: 1998, «Idempotent Full Paraconsistent Negations are not Algebraizable», Notre Dame Journal of Formal Logic, 39, 135-139.
[ B 98c ] Béziau, J.-Y.: 1998, «Sémantique Universelle», Preprint, Institute for Advanced Studies, University of São Paulo.
[ B 98d ] Béziau, J.-Y.: 1998, «Paranormal Spaces», Preprint, National Laboratory for Scientific Computing, Rio de Janeiro.
[ B 98e ] Béziau, J.-Y.: 1998, «Do Sentences have Identity?», in The Paideia Project -- Proceedings of the XXth World Congress of Philosophy, http://www.bu.edu/wcp/MainLogi.htm.
[ B 98f ] da Costa, N. C. A. and J.-Y. Béziau: 1998, «Définition, Paraconsistance et Théorie des Objets», Theoria, 32, 367-379.
[ B 98g ] Béziau, J.-Y.: 1998, «Applications de la Logique Paraconsistante à la Justice et au Droit», in Anais do V Congresso Brasileiro de Filosofia, vol. 2, Brazilian Institute of Philosophy, São Paulo, pp. 1119-1128.
[ B 98h ] Béziau, J.-Y.: 1998, «Recherches sur la Logique Abstraite: les Logiques Normales», Acta Universitatis Wratislaviensis, Logika, 18, 105-114.
[ B 98i ] Béziau, J.-Y.: 1998, «Neojunction: between Disjunction and Conjunction», Preprint, National Laboratory for Scientific Computing, Rio de Janeiro.
[ B 99a ] Béziau, J.-Y.: 1999, «Classical Negation can be Expressed by One of its Halves», Logic Journal of the Interest Group in Pure and Applied Logics, 7, 145-151.
[ B 99b ] Béziau, J.-Y.: 1999, «Rules, Derived rules, Permissible Rules and the Various Types of Systems of Deduction», in E. H. Haeusler and L. C. Pereira, (eds.), Pratica -- Proof, Types and Categories, PUC, Rio de Janeiro, pp. 159-184.
[ B 99c ] Béziau, J.-Y.: 1999, «A Sequent Calculus for Lukasiewicz's Three-Valued Logic Based on Suszko's Bivalent Semantics», Bulletin of the Section of Logic, 28, 89-97.
[ B 99d ] Béziau, J.-Y.: 1999, «The Future of Paraconsistent Logic», Logical Studies, 2, 1-17, http://www.logic.ru/LogStud/02/No2-01.html.
[ B 99e ] Béziau, J.-Y.: 1999, «What is Paraconsistent Logic?», to appear in D. Batens et al. (eds.), Frontiers in Paraconsistent Logic, J. Wiley, London.
[ B 99f ] Béziau, J.-Y.: 1999, «The Mathematical Structure of Logical Syntax», W. A. Carnielli, and I. M. L. D'Ottaviano (eds.), Advances in Contemporary Logic and Computer Science, AMS, Providence, pp. 3-15.
[ B 99g ] Béziau, J.-Y.: 1999, «A Logical Analysis of Singular Terms», Sorites, 10, 6-14, http://www.ifs.csic.es/sorites/Issue_10/item02.htm.
[ B 99h ] da Costa, N. C. A. and J.-Y. Béziau: 1999, «La Logique Paraconsistante», in J. Sallantin and J. J. Szczeciniarz, La Preuve à la Lumière de l'Intelligence Artificielle, Presses Universitaires de France, pp. 107-115.
[ B 99i ] Béziau, J.-Y.: 1999, «Ruth Barcan Marcus est-elle la Mère du Fils de Wittgenstein? (Considérations Existentialistes sur la Formule de Barcan), Manuscrito, 22, 11-27.
[ B 99j ] Béziau, J.-Y.: 1999, «Was Frege Wrong when Identifying Reference with Truth-Value?», Sorites, 11, 15-23, http://www.ifs.csic.es/sorites/Issue_11/item04. htm.
[ B 00a ] Béziau, J.-Y.: «The Paraconsistent Logic Z -- A Possible Solution to Jaskowski's Problem», paper presented at the Jaskowski's Memorial Symposium -- Torun 1998, submitted.
[ B 00b ] Béziau, J.-Y.: «Sequents and Bivaluations», submitted.
[ B 00c ] Béziau, J.-Y.: «Paraconsistent Logic!», submitted.
[ B 00d ] Béziau, J.-Y.: «S5 is a Paraconsistent Logic and so is First-Order Classical Logic», submitted.
[ B 00e ] Béziau, J.-Y.: «The Philosophical Import of Polish Logic», paper presented at the 11th International Symposium on Logic, Methodology and Philosophy of Science B Cracow 1999, submitted.
[ B 00f ] Béziau, J.-Y. and M. V. Kritz: «Théories et Modèles», submitted.
[ B 00g ] Béziau, J.-Y.: «A Survey of General Abstract Logic», submitted.
Center for the Study of Language and Information
Stanford University, Stanford, CA, 94305-4115, USA
Center for the Study of Language and Information
Stanford University, Stanford, CA, 94305-4115, USA