What is Dialectical Logic?

J . F. A. K. VAN B E N T H E M

W H A T IS D I A L E C T I C A L L O G I C ?

1. I N T R O D U C T I O N

Despite the grand title, this paper is not a survey of the complex field called "dialectical logic". Its purpose is merely to give a critical discussion of a specific kind of dialectical logic which has been put forward quite recently by two relevance logicians called R. K. Meyer and R. Routley (cf. [18] and [19]). Up to the present, dialectical logic has been a rather esoteric subject, due to the generality (or vagueness) of many publications in the area. In sections 2 ("Logic and Dialectic") and 3 ("Dialectical Logic") a general picture is sketched. This situation is greatly improved by authors like the above who provide us with an exposition satisfying normal logical standards, and - especially, with claims which are easily identified - thus form a target for possible refutation. In section 4 ("Relevance Logic to the Rescue") the relevance approach to dialectical logic is outlined and discussed from a logical point of view. The upshot is that this enterprise is viable, technically speaking; although, the burden of proof for certain resounding claims about the foundations of science rests entirely with the authors. But, as for philosophical importance, judgment has to be sus- pended until the proposed formal semantics has received any kind of satisfactory explanation. (At the present stage, the sense in which contradictions may be said to be "true" remains a purely formal one.) Such a suspension of judgment may seem a chilly reception to such an interesting attempt at formalization, especially in view of the fact that we do not make any constructive counter-proposals ourselves. (Cf. the few suggestions at the end of section 5 ("Philosophical Considerations").) Let us, then, repeat the outstanding virtue of these authors: their clarity and logical competence make it easy to locate the crucial issues in their enterprise. It is only in this way that "dialectical logic" can become a scientific subject.

2. L O G I C A N D D I A L E C T I C

Somewhere in the eighteenth century Logic arrived at a cross-roads (cf. Bochefiski [5]). To the left there waited a broad alley of philosophical

Erkenntnls 14 (1979) 333-347. 0165--0106/79/0143--0333 $01.50 Copyrigh t �9 1979 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

334 J . F. A. K. VAN B E N T H E M

speculation, to the right a narrow and difficult path of technical studies into the nature of formal reasoning. A panoramic view of the nineteenth century reveals a steady flow of travellers along the former road - led by Kant and Hegel - while only a few isolated stragglers may be discerned along the latter one - notably Bolzano, Boole and Frege. And yet, as we all know, it is the narrow roads which eventually lead to salvation; in this case the heaven of contemporary symbolic logic.

One of the more enduring products of the broad alley is Hegel's dialectic. Although few people still accept such specific Hegelian results like the famous triad "Being, Nothing, Becoming", his "dialectical method" for developing concepts by means of opposition and subsequent synthesis has been extremely influential. Several aspects of this method seem to call for a confrontation with ordinary logic, especially the ascending spiral of "concept, its negation, the negation of the negation [= the original concept, but at a 'higher level'], etc.". Such a confrontation has hardly taken place in the West, however, because logic and dialectic flourish in separate autarchic subcultures. To be sure, nowadays the logico-analytical subculture does produce work on "logic and dialectic", "formal dialectic" and the like; but more often than not "dialectic" is taken in its classical, pre-Hegelian, sense of "debating or dialogue technique" which is much closer in spirit to modern logic. (Cf. Hamblin [9] and Sz~ibo [21]. Two examples of bridge studies are Krohn [12] and Barth [3].)

In the communist countries of the East no such separation can exist between the logical and dialectical "milieus". In fact, it was a matter of cultural life or death to the "logicians" to establish a modus vivendi. (Cf. Bochefiski & Blakeley [6].) The terms of the resulting compromise may be found in many text books (e.g., Klaus [10]): dialectic describes the development of concepts (the dynamic aspect of thought, so to speak), whereas logic studies historical cross-sections: arguments with fixed concepts (the static aspect of thought). There remain sporadic border clashes, e.g., about the status of contradictions; but on the whole all is quiet at the eastern front.

3. DIALECTICAL LOGIC 9

3.1. The Laws of Logic

In traditional logic, certain laws formed the cornerstones of thought. Foremost among these were the Law of Identity, the Law of the Excluded

W H A T IS D I A L E C T I C A L L O G I C . 9 335

Third and the Law of Contradiction. As philosophical fashions changed, these principles assumed various forms: ontological, epistemological or linguistical. E.g., the Law of Contradiction may be stated in the following forms: "nothing can possess a property as well as its complementary property", "no statement is both true and false", or "it is part of the meaning of conjunction and negation that sentences of the form 'A and not A' cannot be rationally asserted". In a more systematic sense, of course, each universally valid logical principle could be called a "law of logic"; for example the law of double negation ("not not A is equivalent to A") or the law of contraposition ("if A implies B, then not A follows from not B").

The exact status of these laws of logic has been a matter of continuing debate. Positions range from "fundamental laws of thought" to "conven- ient linguistical conventions". All this may leave the laws as such undis- turbed, however; whether "valid" or just "stipulated". But, there is also a long tradition of opposition to specific laws, even starting with Aristotle, who had his doubts about the Law of the Excluded Third. (In our century, Lukasiewicz and also the Dutch intuitionists have kept up the attack on this particular point.) In fact, so-called "alternative logics" - in which certain "classical" laws are abandoned or replaced - have become rather fashionable in logical circles; even receiving a kind of blessing from one of the Church Fathers (cf. Quine [16]). A detailed philosophical discussion of the phenomenon will be found in Haack [8].

The law of Contradiction has been relatively safe from attacks. True, it has been claimed that it fails in Indian logic, but India is far away, and it is not clear in just what sense the law "fails". E.g., if the aim of the relevant Buddhist texts is to block rational thinking, violation of the Law of Contradiction is indeed a very efficient means of doing so (cf. Staal [20]). Within the Western tradition, only dialectic would seem to pose a threat to this principle, as will be seen presently.

3.2. Contradictions

Several classical laws of logic seem to be at stake in the dialectical method. For example, on the "ascending spiral" (not not A) is not just A, but something "higher". Unfortunately, a persistent confusion reigns in this area. Many authors are discussing concepts in one sentence and propositions

in the next. Here we are interested in the propositional reading as well as those conceptual readings which allow for propositional reformulations.

336 J. F. A. K. V A N B E N T H E M

(Accordingly, by "contradiction" we will always mean a propositional contradiction of the form 'A and not A'.) Also, there occurs a so-called "unity of opposites", which presumably means that somehow A and not A can be true ("exist", on the confusion terminology) together. If so, then contradictions are essential to a proper understanding of the development of thought.

The connection with "development" is not accidental. Most popular dialectical examples of "true" contradictions arise in the sphere of movement and change. E.g., Zeno's Paradoxes are a recurrent topic (cf. Lobkowicz [14]). In their barest form, these paradoxes derive a contradiction from the assumption that movement exists. Now, there are three possible rational reactions to such an argument. If one accepts the argument while rejecting the conclusion (like the Eleatics did themselves), then the premiss must be rejected: no movement exists. (As is well-known, the Eleatics took this course, which led to their theory that our senses - reporting movement all the time - deceive us.) If one accepts the argument, but sticks to the premiss, then the conclusion will have to be swallowed: there must be true contradictions. The dialecticians are willing to pay this price. (The third reaction, of course, is to reject the argument itself. And indeed various diagnoses of its invalidity have been put forward throughout history.)

There is a difference in kind between the contradictions mentioned above. Zenonian contradictions seem to be more "objective" : they occur outside, so to speak; whereas the contradictions arising during the dialectical process of thought are more "subjective", occurring inside our minds only. This distinction is current in eastern European literature. Objective contradictions are just there - a favourite example is the duality of light - subjective contradictions are subject to human" intervention: these can be removed. The method used for this purpose is that implicit in Aristotle's formulation of the Law of Contradiction: "no statement can be both true and false in the same respect". In other words, when a contradiction arises, this may be interpreted as a symptom of poverty: a relevant aspect has been neglected. Addition of suitable "parameters" will usually restore consistency, i.e., freedom from contradictions. This procedure may be found at work in many scientific text books. Some explicit examples are to be found in Ashby [2]. (E.g., rubber is elastic, but scientists failed repeatedly to establish elasticity for the rubber molecules.

W H A T IS D I A L E C T I C A L L O G I C ? 337

A contradiction ? No, elasticity at a macro-level is to be distinguished from elasticity at a micro-level.) When viewed in this way, contradictions play a role in the refinement of our theories, not because of their acceptance, but precisely because of their rejection! This insight forms the core of the important treatise Weinberger [23], in which the author claims that this is the rational dialectical sense in which contradictions may be said to be "resolved", "overcome". Some implications for the usual views about logical methods in the foundations of science are spelt out in van Benthem [4]. It should be kept in mind, however, that not all contradictions are "good" contradictions. Some of them are just indications that one has made a mistake, or that the theory used is top-heavy and needs pruning (cf. Quine & Ullian [17]).

Although the above describes the predominant reaction both in the West and in the East, there are some "new logicians" in the West construc- ting various alternative logics in which the Law of Contradiction has been abandoned (cf. da Costa [7] or Routley & Meyer [19]). Their motives are numerous and complex, but one is purely logical curiosity. Suppose one takes dialectical claims about the underlying logic of dialectic (as opposed to classical logic) at face value, can a rigorous logical system be found incorporating just these claims? (Compare Heyting's axiomatization of such a seemingly esoteric subject like intuitionistic logic.) One particular attempt in this vein will be discussed in the next section.

4. RELEVANCE LOGIC TO THE RESCUE

4.1. Introduction

R. Routley and R. K. Meyer, two workers in a rather unorthodox discipline in formal logic called "relevance logic" (cf. [1]), have produced formal dialectical theories for which they claim revolutionary philosophical significance. Their aims, as professed in [19], are quite outspoken:

"To date classical logic - which is Western mainstream logic - has been strongly on the offensive in the ideological warfare between East and West, with many supporters in fact among the Soviets, and dialectical logic has been very much on the defensive. The object of this paper is to try to upset this ideological power structure by furnishing dialectical logic with the

338 J. F. A. K. VAN BENTHEM

framework at least of a viable semantics, and at the same time to shatter the imperviousness of mainstream Western logic, and thereby to assist the cause of that newer, less orthodox and so far minor, logical theory - relevance logic." ([19], p. 1).

The war-like terminology is rather unfortunate. It is downright false that matters of logic have played an important role in the "ideological warfare between East and West", let alone that there was a kind of"classical" fifth column in the East. Responsible logicians with some regard for their colleagues in communist countries should think twice before saying such things. (I suppose I need not elaborate on this.) One might even defend the opposite view that a common (classical) logic has contributed to peaceful scientific communication. (Moreover, one would expect authors who can acquiesce even in contradictions to be of a rather more peaceful disposition.) But, to return to scientific matters, the main claim is that

each position can furnish viable, equally formal, but competing logical theories, and the differences between these positions will come down to philosophical differences about such highly debatable and empirically untestable matters as the consistency of the world. ([19], p. 1).

In [19], an "agnostic" position is defended with respect to this "Consis- tency Hypothesis" - with relevance logic a neutral bystander - but in the later paper [18], Routley shifts to "atheism"; the world is in fact inconsistent! This means that dialectical logic should be preferred to classical logic, and that much of ordinary logic and mathematics (and, hence, science) is to be reconstructed accordingly.

4.2. Formal Dialectical Theories

Let us consider some logical language containing the usual connectives (negation), ^ (conjunction), v (disjunction)and---> (implication).

Various calculi exist axiomatizing the set of classical logical laws involving these connectives. A typical axiom might be ((,4 v B) ---> C) ---> ((A ---> C) ^ (B ~ C)) and a typical rule of inference A, A --~ B I- B (Modus Ponens). Suppose one objects to certain laws thus derived, it is relatively easy to try out different calculi yielding smaller, or incomparable, sets of "laws". Intuitionistic logic provides an example, relevance logic as originally conceived - a description of implication free from traditional "paradoxes"

WHAT IS DIALECTICAL LOGIC? 339

like A --~ (B --~ A) - another. Now, if some of the tenets of dialectical logic are to be accommodated, some of the classical laws or rules will have to be rejected. The most obvious candidate for dismissal is the rule A, ~ A I- B ("Ex falso sequitur quodlibet"). For, it is this rule which brings out the classical undesirability of contradictions: everything follows from them. (Of course, one cannot simply drop this one rule: its supporting rules will have to go as well, like A v B, --TA I- B ("disjunctive syllogism").) This weakening ensures that contradictions may be added to theories without the latter exploding into the set of all statements. Still further weakening may be required to provide for a possible dialectical insight that ~ T A implies A, but not conversely. Routley claims that relevance logic is the natural framework for such investigations. It may be of interest, however, to note that Johansson's minimal logic, which also drops the rule "ex f a l s o . . . " dates already from 1936 (cf. Prawitz [15]). In fact, philosophical objections to this rule are of commmon occurrence.

Having weakened the underlying logic, it now becomes possible to allow specific contradictions as axioms. One example is the Paradox of the Liar, who utters the statement that the very statement he is making is false. This statement can be shown to be both true and false, given the usual explanation of truth, so it could be regarded as a "logical contradiction". Here again, it is instructive to review the possible rational reactions. If one accepts the argument of the Liar Paradox, deriving a contradiction from the assumptions of (1) semantical closure of the language (allowing for self-reference) and (2) the usual truth definition, then there are the following possibilities. One may reject the conclusion, while sticking to (2), and then reject (1) (this is Tarski's course in [22]), or - alternatively - one may stick to (1) and, therefore, reject (2) (this is Kripke's policy in [11]). (Rejection of both (1) and (2) would seem to be overly pessimistic.) On the other hand, one may accept the argument as well as both premises (after all, they always seemed to be obvious) and, then, accept the contradiction: again a dialectical response. (One's first impulse, to attack the argument itself, turns out to be less successful than the above-mentioned procedures.) Adding this single contradiction, say L ^ ~ L , to the logic may generate a whole sequence of contradictions, if one chooses to retain "Aristotle's Law" 7( .4 ^ ~A) . (Note that this is a law of contradiction in the object language; which does not prevent their occurrence.) The important point is that even such a sequence of contradictions need not yield all statements.


It will be clear that there is no end to syntactical creativity here. The real

question rather becomes if a proposed axiomatic calculus has an interesting semantical interpretation. The relevant quotation from [18] is:

Much light is shed on DKQ [a quantified dialectical calculus used by Routley as a basis for his exposition] by its semantical analysis in terms of worlds; for, as will appear, the semantics reveal far more clearly than the corresponding syntactical formulation what is going on philosophically.

Now, the semantics for the dialectical calculi consists of an ordinary

relevance semantics, itself a relative of Kripke's possible worlds semantics, with a dialectical superstructure. The main ideas are as follows. There is a set W of worlds ("situations, theories, scenarios, or whatever"; [19], p. 7),

connected by a ternary ordering relation < (b <a c is to be read as "b precedes c from the perspective of a" ; cf. [18]). (We omit the "regular" worlds as well as the defined "inclusion relation" ~ .) The addition of a valuation V assigning sets V(p) of worlds to proposition letters p turns this

into semantical structures M = ( W, <, V) - all this is our own notation - for which a truth definition may be given as follows.

(~)

M ~ p [a] M ~ A ^ B[a]

M ~ A v B[a] M ~ A ---> B [a]

iff a e V (p) ("p is true at a in M")

iff M ~ A [a] and M ~ B [a] iff M ~ A [a] or M ~ B [a]

iff, for all b, c e W such that b < ~ c , iff M ~ A [b], then M ~ B [c].

Standard relevance logic would add the normal clause for negation:

M ~ ---hA [a] iff not M ~ A [a].

In dialectical logic (as conceived here), this is replaced by:

(!) M k ~ A [a] iff not M ~ A [a*],

where * is an operation on W taking a world to its "reverse" ([18]),

"image" ([19], p. 8). The inserted quotations are not here for pedantic reasons, but because they happen to be the only explanation one gets from the authors. (Probably they regard the notions < and * as self-explanatory.)

It should be added that the full semantics is somewhat more complicated, in that "model structures" divide the set W up into "regular" situations (in which all theorems hold) and non-regular ones. (This idea is familiar from the model theory of non-normal modal logics.) Moreover, there is a distinguished "actual world", which is regular. As far as we can see, these complications do not affect the above points, however.

W H A T IS D I A L E C T I C A L L O G I C . 9 341

Let us now proceed to the vital question which the sober-minded reader

must have been waiting to ask all along: In what way can a contradiction

be true on the semantics ? The answer is simply this:

M ~ A A ~ A [ a ] i f f M ~ A [ a ] a n d n o t M ~ A [ a * ] .

Pending further explanation of the nature of a*, one cannot even begin to

say if this is more than a purely formal trick. Routley himself mentions the possibility of dispensing with * altogether,

in favour of a four-valued approach in terms of " t ruth" (t), "falsity" (f),

"incompleteness" (n) and "inconsistency" (i). The truth table for the semantic negation operator o would presumably become:

t ~ f ~ n ~ i ~

and one would get a valued semantics with a possible key clause

V(M, --7 A, a) = V(M, A, a) ~

Such truth table approaches are even more formal, of course, than the one

criticized above. The above objection does not imply that the semantics is "wrong" in

some technical sense. On the contrary, it is presented in the standard (i.e., the classical) fashion - the meta-language of the paper is based on classical logic - and one can do a lot of mathematics with it. E.g., there are nice correspondences between certain axioms and conditions on < and * needed to ensure their validity. E.g., the validity of the principle (A--~

B) ~ (B ~ ~ A ) depends on, amongst others, a key requirement that

i f b < ~ c, then c* < a b*.

Likewise, a distinction between validity of ~ A ~ A and non-validity of

A ~ ~ A may be introduced by requiring

a** _~ a - i.e., a** < b a for some regular world b,

while dropping the condition a ~_ a**. In this way it becomes possible to state in formal semantical terms what

acceptance or rejection of certain dialectical principles amounts to. (Cf. the similar use of conditions on the alternative relation in the Kripke model theory for modal logics.)

Moreover, the standard Henkin proof technique will yield a completeness theorem. But, all this does not provide for a deeper intuitive understanding. This is a problem with ordinary Kripke semantics already; it


does not become any better here. One could say that, in a sense, Kripke semantics is too versatile. The formal parameters can always be adjusted to new axiomatic systems. (By the way, this would be an interesting claim

to be made precise in some technical sense, and proven.) The resulting completeness proofs tend to make one forget the lack of intuitive explanation of the notions "world", "precedence", etc. In fact, Routley's equation

of "worlds" with "theories" - also implicit in the very formulation of his Consistency Hypothesis - suggests that the Henkin method, in which this equation is a technical matter of course, has been the secret source of inspiration. (After all, Henkin proofs are powerful heuristic instruments.

One can often reverse the usual proof procedure, and extract a suitable semantics for a given axiomatic calculus by inspection of a "tentative" Henkin completeness argument. This is not surprising, because such

arguments consist for a large part in giving a set-theoretic representation

for certain syntactic (Lindenbaum-)algebras of formulas. And there are even more heuristic uses of this approach. A full explanation would fall outside the scope of this paper, however. Suffice it to notice that such heuristic finds should receive an independent motivation later on, if they

are to be of any philosophical significance.) According to its creators, this formal logic has the virtue that

many of the characteristic features of dialectical logic as now commonly conceived by the Soviets can be formally established ([18]),

and

it is now a straightforward matter to inflate DL [another dialectical calculus consid- ered by these authors] so as to validate in the framework some of the most contro- versial dogmas ( . . . ) of Marxism-Leninism ([19], p. 13).

Moreover, it has important implications for classical logic as well.

For, an entirely new approach to the well-known paradoxes becomes available; viz. their plain acceptance. Theories need not be given up any more because of troublesome contradictions; and thus the original set theory of Cantor, or naive pre-Tarskian semantics, may be reinstated. And, what is more, results which seemed to have been put beyond our reach forever by G6del's Incompleteness Theorem, once more become at least a possibility: a decidable (dialectical) predicate logic, a recursively axiomatized complete (dialectical) arithmetic, and much more. Routley's promises in [18] are truly dazzling to us poor refugees from Cantor's Paradise!

W H A T IS D I A L E C T I C A L L O G I C 9. 343

4.3. Logical Comments

It has become clear already that the question is not if syntactical systems of logic can be produced in which contradictions and the like may appear as axioms or theorems without a collapse into triviality. This is a well-known technical possibility. The question of a formal semantics for such a theory is admittedly more interesting, although it comes as no surprise that a flexible tool like the relational possible worlds approach is able to yield one. (Algebraic semantics would be another obvious approach to try.) But it is only at this stage that a decisive question emerges: can any, technical or philosophical, interest be attached to the proposed semantics ? Precisely this question cannot be answered until the authors have told us much more about the intuitions behind their "worlds", "ordering relation" and, especially, their "reversal operation". There remains, then, the matter of the above mentioned logical promises. Could there be an interesting dialectical predicate logic, arithmetic and set theory ? Some conservatives would deny this at once, referring to "the undecidability" of predicate logic, or "the incompleteness" of arithmetic. Against such opponents Routley scores a few points in [18], when he draws attention to the (hidden ?) premises of such insights. E.g., Church's undecidability result states that the set of classically valid predicate-logical formulas is not recursive - and this does not mean that no set of ("valid") formulas can be recursive. But, clearly, the burden of proof lies with the new dialecticians: it is up to them to show (1) that the set of dialectically valid formulas is interesting and (2) that it is recursive. Similar considerations apply to the theorems of G6del and Tarski. E.g., G6del's Incompleteness Theorem states that any recursively axiomatized arithmetical theory which contains enough arithmetic to be able to encode its own proof predicate is in- complete /f consistent. Popular expositions may omit the consistency requirement, but professional logicians, like Routley himself, know that the serious text books do not. Now, Routley claims that G6del's proof does not apply to recursively axiomatized "encoding" arithmetics based upon dialectical logic. These might be "complete" in the sense that, for any arithmetical statement A, A or ~ A (possibly both!) are theorems. This claim is correct: G6del's proof shows that, if the "liar formula" L is a theorem, then so is -TL. By consistency, it is then concluded that L is not provable (though true). In a dialectical theory, this might rather be turned into a proof that both L and 7 L are theorems - and again we have a true

3 4 4 J. F. A. K. V A N B E N T H E M

contradiction, this time inside arithmetic. (Is it, then, impossible to reject certain statements at all in a dialectical theory ? Fortunately, there is a way: derive a statement from it which is known to be outside of the theory, e.g., 0 --- I. Still, the loss of reductio ad absurdum does tie our hands to an annoying degree.) But, again, the burden of proof lies with the revolution- aries: produce some interesting complete recursively axiomatizable arithmetical theory. (To point out that there is any amount of recursively enumerable "complete" - in the new sense - extensions of, say, Peano Arithmetic which do not contain all formulas, is not enough !) Moreover, if the given semantics is to be taken seriously, there is another pressing question, viz. what could be the "reverse" of the standard model of arithmetic? If one is not told at least this, then dialectical arithmetic remains a purely formal exercise.

Now, if all this work is to be carried out successfully, it will probably be a job for technical specialists, highly imbued with the spirit of formal logic. So the old logicians will be needed for quite some time to come: a very comforting thought.

5. P H I L O S O P H I C A L C O N S I D E R A T I O N S

The main conclusion of the preceding sections was that "dialectical logic" is a technical possibility - logically speaking - but that its interest depends entirely upon its philosophical interpretation (which is lacking up to now) and/or its technical fecundity (which is yet to be established). To what extent, then, have the pretensions of [19] and [18] been made good ? Not as far as logic is concerned; but what about dialectic, or "Soviet Logic"? In this respect, it seems extremely dubious that a connection can be established with the main currents in "dialectical" circles. Indeed, the whole idea of formalizing "Soviet Logic" (in the sense of: "Soviet standards of rational- ity") is rather naive, given the fact that this label is no more then the title of a book chapter (cf. [6]) about the science of logic in the Soviet Union. There just is no body of doctrine meriting this description. (One can imagine the consternation of a dialectician when these logical missionaries arrive bringing their bright new formal tools, asking him if this was what he had meant to say all the time, or maybe this, or this .9)

Indeed, how serious are the authors themselves about their new logic ? The tone of their articles is full of missionary zeal; which is understandable,

W H A T IS D I A L E C T I C A L L O G I C ? 345

given the lack of response up to now. But the presentation is - minor errors

apart - well within the classical logical tradition; the Law of Contradiction

included (they are right, the others wrong). How can a dialectical logician convert us when he uses classical logic in the attempt ? Or, for that matter, how could he if he did not use classical logic? These are well-known

philosophical teasers (cf. Haack [8]) which the authors might at least have anticipated. But, even for those who distrust knock-down arguments (like I myself), the above "burden of proof" considerations are conclusive.

A negative appraisal of the present attempt to construct a dialectical logic does not necessarily imply a rejection of communication between

classical and dialectical positions. On the contrary, this border line area of research is both interesting and useful, witness the above-mentioned paper Weinberger [23]. But it is not clear why such studies could not be under-

taken in a quite different spirit. Instead of taking dialectical ideas like the admissibility of contradictions, or the non-equivalence of --1 ~ A and A, at face value (in which case they, naturally, cannot be fitted into classical

logic), one should try a little harder to find an interpretation which does fit into the classical framework. Instantaneous formalization of deviant principles does not point at liberality, but at lack of logical phantasy. (This

point was made by Dana Scott vis-~t-vis many-valued logics.) In this, there is an element of conservatism, which also shows in the idea of a "paradox free" science. The authors think it a virtue that paradoxes need not always worry us any more. A theory may be saved in spite of contradictions, instead of being subjected to the usual "drastic mutilation" ([18]). (Routley

does allow for "inadmissible" contradictions, however; e.g., when one of the statements involved is "not true". We will not probe into this any further, although there appears to be an inconsistency in his position. One wonders: does a new dialectician ca re . . . ?) As a methodolbgical strategy, this procedure is highly conservative. Instead of curing the illness at the source - i.e., the specific inconsistent theory - one gets rid of the symptoms by silencing the messenger - i.e., classical logic - bringing the bad news. The method of 3.2 seems much more promising.

Finally, if much intellectual energy is to be invested in the logical study of the inner mechanism of dialectics, it may well be that a completely different direction is called for. The more interesting aspects of dialectic are not so much tied up with its status as a potential rival system of logic, but as a theory about the formation and development of concepts. (Cf. the

346 ~. F. A . K. V A N B E N T H E M

distinction made at the beginning of 3.2.) One could argue that logic

should be supplemented by a study of the dynamics of reasoning (cf.

Lakatos [13]) in which conceptual change and precisation of arguments are

interwoven. I f dialectics has anything to teach us, it may be here. In all

fairness, however, it should be added that many dialectical thinkers do not

even promise us a trace of method here, but merely a sensitivity to changes

in meaning; at best a point of view.

Groningen State University

R E F E R E N C E S

[1] Anderson, A. R. and Belnap, N. D.: 1975, Entailment. TheLogic of Relevance and Necessity (Princeton University Press, Princeton).

[2] Ashby, W. R.: 1976 ix, An Introduction to Cybernetics (Methuen, London). [3] Barth, E. M. : 1970, 'Enten-Eller. De Logica van Licht en Donker', Algemeen

Nederlands Tijdschrift voor Wijsbegeerte 62, 217-240. [4] Benthem, J. F. A. K. van, 1978, 'Four Paradoxes', Journal of PhilosophicalLogic

7, 49-72. [5] Boche6ski, I. M., 1956, Formale Logik (Karl Alber Verlag, Freiburg and

Mtinchen). [6] Bochefiski, J. M. and Blakeley, T. J. (eds.), 1961, Studies in Soviet Thought,

volume I (Reidel, Dordrecht). [7] Costa, N. C. A. da, 1974, 'On the Theory of Inconsistent Formal Systems',

Notre Dame Journal o f Formal Logic 15, 497-510. [8] Haack, S., 1974, Deviant Logics (Cambridge University Press, London). [9] Hamblin, C. L., 1970, Fallacies (Methuen, LondQn).

[10] Klaus, G., 1959, Einfiihrung in die moderne Logik (Deutscher Verlag der Wissen- schaften, Berlin).

[11] Kripke, S., 1975, 'Outline of a Theory of Truth', Journal of Philosophy 72, 690- 716.

[12] Krohn, W., 1972, Die formale Logik in Hegel's 'Wissenschaft der Logik' (Carl Hanser Verlag, Mtinchen).

[13] Lakatos, I., 1976, Proofs and Refutations (Cambridge University Press, Cam- bridge).

[14] Lobkowicz, N., (ed.), 1959, Das Widerspruchsprinzip in der neueren sowjetischen Philosophic (Reidel, Dordrecht).

115] Prawitz, D., 1965, Natural Deduction (Almqvist and Wiksell, Stockholm). [16] Quine, W. V. O. 1963, third printing, 'Two dogmas of Empiricism', in: From a

logical point of view (Harper and Row, New York). [17] Quinr W. V. O. and Ullian, J. S., 1970, The Web of Belief(Random House,

New York). [18] Routley, R., 1979, 'Dialectical Logic, Semantics and Metamathematics',

Erkenntnis 14, 301-331.

WHAT IS DIALECTICAL LOGIC. 9 347

[19] Routley, R., and Meyer, R. K., 1976, 'Dialectical Logic, Classical Logic, and the Consistency of the World', Studies in Soviet Thought 16, 1-25.

[20] Staal, F., 1975, Exploring Mysticism (Penguin Books, Harmondsworth). [21] Sz~tbo, A. A., 1969, Anfdnge der griechischen Mathematik (Oldenbourg, Mfin-

chen). [22] Tarski, A., 1956, Logic, Semantics, Metamathematics (Oxford University Press,

Oxford). [23] Weinberger, O., 1965, 'Der Relativizierungsgrundsatz und der Reduktions-

grundsatz, zwei Prinzipien des dialektischen Denkens', Nakladatelstvl ~eskoslo- vensk~ Akademie Vdd, Praha.

Manuscript received 1 February 1979

Documents

What is Dialectical Logic?