Knots and Blanks: The Pragmatic Foundation of Logical Principles and the Limitations of Scientific Paradigms

Logical principles, in particular the "law of contradiction" and the "law of middle-term excluded," do play different roles at different levels of discourse, in particular, they can be valid sentences in an axiomatic calculus, or methodological requirements (requirements of consistency and completeness) in the metalanguage of formalized systems.

If they are stated as formal laws, they may look like this: ~p V p and ~(p & p). In this case, they are totally interdefinable (through DeMorgan's transformations) and, of course, also equivalent.

If postulated as methodological requirements, the principles are not equivalent (there exist consistent yet incomplete systems). Nevertheless they are still interdefinable, since completeness of a system can be defined in terms of consistency of another system, which keeps a definite logical relationship with the first one.

At a third level of discourse, that of scientific praxis, the principles come apart even farther: they are neither equivalent nor interdefinable. Notwithstanding, they still keep a family resemblance. I propose to call the principles at this level pragmatic imperatives. They deal with scientific paradoxes, of which I identify two different types: knots (conflicting views) and blanks (holes in the scientific pattern). The left-hand side imperative says: Remove all knots! The right-hand side imperative says: Fill all blanks!

My conclusion will be that formal categories are not primordial: pragmatic imperatives are. These knot-removing and blank-filling injunctions are prior to both the metalanguage requirements and the logical laws, and much more important.

Logical principles –in particular the "law of contradiction" and the "law of middle-term excluded"– have attracted my attention ever since I began to study logic. I have wondered at the relationships that hold between them, as well as at the quite diverse roles they may play according to the level of discourse within which they are actually utilized. From the start, one can distinguish very clearly at least two such levels NOTE 1 : the principles may perform the function of valid statements of an axiomatic system or calculus; or else they may act as methodological requirements which such calculus (in fact, any calculi) must meet in order to be acceptable.

At the first level, we havep & ~p, the law of non contradiction. It says that it is not the case that p (any statement whatsoever) and ~p (the negation of such statement); i.e., that in a pair of contradictory statements at least one must be false. And p V ~p, the law of middle-term excluded, which says that either p or not p; i.e, that in a pair of contradictory statements at least one must be true. These formulae are valid sentences of propositional calculus, on the same footing as any other valid sentence, whose true value can be checked by true tables (or any equivalent method), and which can be rigorously derived from the axioms of the calculus by means of a mechanical application of the corresponding rules of inference.

At the second level, one has the requirement of consistency, which applies to axioms and rules of inference in an axiomatic system (propositional or any other calculus); it says: Of any two contradictory sentences (axioms, rules, valid formulae) at least one must be false. One also has the requirement of completeness, which applies to the same elements of axiomatic systems; it says: Of any two contradictory sentences at least one must be true NOTE 2. The combined import of these two requirements is the desideratum of axiomatic method: of any two sentences one of which asserts what the other denies, one must be true and the other false. In other words, any formalizable problem should be decidable either in the affirmative or in the negative, but not both ways.

It is important to grasp the difference between the two levels of discourse. It is not the same to be part of a system, one of its valid formulae, say ~p V p, and to be a requirement for the soundness of the system. The valid formula is a token for a true statement, for instance Either you do not understand this explanation or you do; it is written in the object language. The requirement, on the other hand, is written in the metalanguage: it does not refer to persons or explanations, but to systems of statements (or rather systems of statement-forms). It says: In order for this system to be acceptable it must be possible within it either to prove one formula valid or to prove as valid the contradictory of such formula.

Let us consider closely the relationship between the two principles, "non contradiction" and "middle-term excluded," at the first level, that is to say, when postulated as formal laws –valid schemata of a formalized system. At this level they are rigorously interdefinable, and also equivalent.

They are interdefinable because one is deductively transformable into the other using the rules of definition of the signs employed. In order to transform the first law into the second, it is only necessary to apply DeMorgan's rule: an outer negation of a conjunction may be replaced by two internal negations within a disjunction. Thus, ~(p & ~p) negates outerly p & ~p; one can get rid of the outer negation by creating two internal ones, ~p and ~~p, joined by a disjunction sign: ~p V ~~p. All that remains is getting rid of the double negation, and one obtains the law of middle-term excluded: ~p V p.

By a similar process, it is possible to transform the law of exclusion of middle term into the law of non contradiction, with the application of a complementary DeMorgan's rule, and again de use of "double negation". The disjunction sign in ~p V p is replaced by a conjunction sign, changing at the same time the quality of the literals (negation to affirmation and vice versa) and negating the whole formula, the result being ~(p & ~p).

The formulae are equivalent because all valid laws of the propositional calculus are equivalent: their uniform value is "true for all substitutions of the variables".

It is now clear that the two formal laws are interdefinable and equivalent.

If the principles are postulated as methodological requirements, they are not equivalent, although they could still be said in a rigorous sense to be interdefinable. The non contradiction principle comes to be the methodological injunction: Thou shalt not decide a problem both ways. The exclusion-of-middle-term principle comes to be the methodological injunction: Thou shalt not excuse thyself from deciding a problem.

What is the logical relationship between these two injunctions? If they were directed to the executive of a corporation, would she understand them as equivalent? Presumably, the board would expect that in all cases the executive complies with the first one: she should never decide a matter both ways, at the risk of a grave peril for the business –an inconsistent executive would make an inefficient employee and should definitely be replaced. Nevertheless, the other injunction is not so fatal: the board would expect that in most cases the executive will not excuse herself from deciding the problem, at least in the sense of recommending some further analysis or study. But even the most demanding board would make allowance for some delay or at least for leeway in the timing of the decisions on the part of the executive.

The same holds true for an axiomatic system. An inconsistent system is of no use, since it is incapable of providing the service that one expects from it: being able to tell apart a valid sentence from an invalid one –within an inconsistent system everything is provable, a single inconsistency generates an infinite number of inconsistencies NOTE 3. But there is more: one can even say that every inconsistent system is complete, since it has all truths in the list of valid sentences (valid, of course, according to the rules of the system); this is so, because for every relevant truth of the field either the system has it in the list of axioms to start with, or the truth is derivable within the system (since anything is derivable in an inconsistent system). This being so, there is no way that consistency and completeness could be equivalent.

On the other hand, even if inconsistent systems are of no use, an incomplete system can be of some service, even of great service in certain cases. The more interesting calculi, those sophisticated enough to allow for the formulation of the natural numbers and the rules of addition and multiplication, are presumably consistent, but they are demonstrably incomplete. Arithmetic itself is incomplete (in fact, incompletable); nevertheless, it has been and will continue to be of tremendous service for the survival of mankind. The existence of such consistent-yet-incomplete systems shows unequivocally that the requirement of consistency and the requirement of completeness are not equivalent.

It is clear, then, that at this level –methodological requirements– the principles are not equivalent. Still, they are, in some sense, interdefinable. Completeness of a system can be defined in terms of consistency of another system, which keeps a definite relationship with the first system: a system S (think of it as a list of independent axioms) is complete if and only if the addition of a new independent axiom would produce a system S' which is inconsistent. The import of this definition amounts to the following: if the axioms are independent –none of them logically derivable from any other axiom in the list– being complete requires that the system have every independent truth in the list; one more member could not possibly be a new independent truth (everyone is already there); it could only be a falsity, presumably (if the system has a negation sign) the negation of one of the axioms.

It is time now for us to tackle the third level of discourse, that of scientific praxis –scientific theory and practice in dialectic interaction, as one finds them in reality. Things change drastically at this level. Logical principles come even farther apart here: they neither have the same logical value, nor are they definable in terms of each other. However, they still keep between them a special relationship, which does justify our dealing with them in joint fashion. I am not saying, of course, that formal laws or methodological requirements have no application in scientific praxis. They do, but in subpatterns of that praxis. When taken as a whole, though, scientific praxis is not as good a field for the application of restricted logical principles as formal laws and methodological requirements are. Scientific praxis is good terrain only for the application of the non restricted principles which I am going to call pragmatic imperatives.

As the reader may have guessed, we are going to introduce at this level two pragmatic imperatives, corresponding strictly to the same two kinds of principles that we have been examining so far. The specific novelty is here, among other things, that the principles are now neither equivalent nor interdefinable, but only "dialectically complementary". They relate with the formal laws of non contradiction and middle-term exclusion, and also with the methodological requirements of consistency and completeness, but only through an air of family resemblance. There is no question, however, about their distinctiveness.

Valid laws of logic and methodological requirements presuppose formalized systems of some sort: formal laws apply internally to such systems, methodological requirements apply externally to them. The design of such systems is part of scientific praxis, but scientific praxis is much more than formalization. It includes, at least, ways of coping with problems of unformalized kind and the history of the development of all sorts of scientific systems. Pragmatic imperatives are open versions of the principles of which formal laws and methodological requirements are closed versions. Closed principles deal with formalized systems; open principles, with problems and the development of systems. In particular, pragmatic imperatives are ways of dealing with certain scientific problems of extreme nature, which justifies our calling them paradoxes.

Not all scientific problems are paradoxes; most scientific problems are problems of concept clarification or hypothesis verification, what Thomas S. Kuhn has called paradigm articulation (KUHN 62). Those problems are not of extreme nature, and need not concern us here. Our concern is with paradoxes, which can be of two very different types: they can be either conflicts, negative connections, knots in the scientific net, or rather linguistic gaps, absence of connection, blanks or holes on the scientific pattern.

Let us schematize a little and say that logical principles have left-hand side versions and right-hand side versions. Accordingly, the law of non contradiction and the requirement of consistency will be considered left-hand side versions; the law of middle-term excluded and the requirement of completeness, right-hand side versions. Now, it is possible to single out a pragmatic imperative in a left-hand side version and another pragmatic imperative in a right-hand side version. The left-hand side version of the pragmatic imperative deals with knots. It says:

Be intolerant with knots in the scientific net, try to remove them wherever you find them.

The right-hand side version of the pragmatic imperative deals with blanks. It says:

Try to fill every blank you may find in the scientific net.

It is about time that I give some examples of paradoxes. In fact, we could hardly make any progress from this point without such illustration. I will select paradoxes of the empirical sciences to be sure they are not amenable to treatment by closed logical principles –mathematical paradoxes offering a borderline case between the syntactical and the pragmatical.

First let us take the case of a knot, that is to say a paradox of the left-hand side type, related to formal contradiction and methodological consistency. My example is from the physical sciences and has to do with the emergence of relativity theory. From the results of the Michelson-Morley experiment, together with Newtonian physical theory, the perplexing conclusion follows that the earth both moves and does not. The paradox lies in the clash between experimental evidence (in favor of immobility) and the theoretical context of the experiment, i.e., Newtonian dynamics (which establishes mobility). We face here a tipical knot.

In order to remove the knot, one has to modify the whole intellectual pattern, transforming Newtonian physics into Einsteinian physics, and reinterpreting the results of the experiment accordingly. In particular, it is necessary to make time, space and mass non invariant through an inertial transformation, contrary to Newton's assumptions. The resolution of the paradox comes about, then, by a contextual redefinition of space, time, and mass. A crucial theoretical term is introduced in the process: the constant velocity c. This term is theoretical and not empirical, inasmuch as experience is not allowed to contradict it –nothing can go faster than the velocity of light.

By the introduction of this term and all the meaning variations in key concepts that it implies, a paradoxical paradigm (something akin to a formal contradiction or an inconsistent system) is transformed into a new paradigm, paradox free, (something akin to a consistent system or a formal tautology). This freedom of paradox is, nevertheless, only relative: it is just an index for our present ignorance of all possible articulations between the paradigm and the relevant facts.

It is important to notice, considering the given example, the radical character of the knot-type paradox. The removal of it from the paradigm by simple means, like revision of the calculating process or repetition of relevant observations, is not possible. It is not a matter of error or mistake, not even of hidden or difficult-to-discover ones. The paradox permeates the whole fabric of the intellectual pattern and can only be removed by fundamental change in the theoretical assumptions of the paradigm.

In fact, the existence of the knot amounts to the coexistence of two quite separate subcontexts, incompatible with each other: subcontext A, made out of Newtonian theory plus the interpreted results of the Michelson-Morley experiment, vs. subcontext B, made out of Newtonian theory plus the interpreted results of all other physical experiments. So, the knot tells us that the standing theory has ceased to be unique, and that we are now faced with a dual, inconsistent theory. If we do not like that, and of course we do not, we are bound to strive for a new unified theory, presumably wider or more exact, in which the conflicting aspects of the dual theory can be reconciled.

Let us now turn to the case of a blank, a paradox of the right-hand side type, related to formal middle-term exclusion and methodological incompleteness. This example is from the social sciences and has to do with the residual category in equilibrium sociology, "social disturbance". I take equilibrium sociology as a blanket designation referring to positions like Pareto's, Parsons', general systems theory, functionalist anthropology, or marginalistic economic theory. All these approaches emphasize the self-regulatory character of the social machinery and tend to deal with extreme perturbations of the equilibrium state only indirectly, as irrational residua of the analysis.

Now, the very same phenomena are dealt with directly by other sociological approaches, namely, those of structural-historic inclination. "Social disturbance" is there replaced by "class struggle," which is not a residual category but, on the contrary, a clear-cut concept, central to the theories in question. It is not ruled out, however, that the substitution of the clear concept for the residual category is carried out only at a price: the emergence of a different blank somewhere in the new context.

Neither is this paradox a common error; it is, if you like, even more radical than the knot-type paradox. In the blank-type paradox one is confronted with an exhaustion of the paradigm one is working with. It inevitably becomes incompetent to formulate, let alone solve, a particularly important problem. This omission is especially grave, since the paradigm presents itself as logically closed, without any manifest gap which would move the unsophisticated scientist to revise the paradigm. In fact, the position of the blank is formally filled by a token, the residual category, with no meaning of its own within the blank-infested paradigm, and the blank or hole tends to pass unnoticed. Only the empirically-minded and well-trained scientist will be able to perceive the lack of an empirical closure and will be moved to undertake a radical exploration of the paradigm.

The existence of this type of paradox, which elsewhere I have dubbed blind spots in scientific theories (GUTIÉRREZ 68), calls for an explanation. I dare to offer the following one, of a pragmatical bent. Scientific theories, instruments as they are, must be handled: hence, by hypothesis, they must have a handle –and this handle produces a blind spot, itself unable to perform the function that the instrument is called upon to perform. Think of a hammer or a probe: part of its body is in contact with the hand and is not directly useful in the function of the tool. It is not impossible to manipulate a tool, even its handle, with another instrument (the fixing of a tool would be a good example); in this case, a blind spot or blank is removed at the price of creating a blank or blind spot somewhere else, in another instrument. One could say that the handle identifies with the handler or user, or at least points to the subject of the action, who is always to be taken into account when describing the function of an instrument NOTE 4.

The same goes for scientific theories. In a general sense, the blind spot of a theory is identical with the radical perspective (point of view) of the theorist, with the purpose that determines its content. Thus, the exploitative relationship is invisible to the ideologist of the capitalistic society: exploitation is a blank in functionalism, general systems theory or marginalistic economics. Handle, perspective, point of view, blind spot, blank: they all coincide with the praxis of the scientist, the purpose which pragmatically determines its intellectual pattern. Man belongs in language; its radical purpose (meaning) is defined by the language he uses. That is what the existence of blanks or blind spots in paradigms amounts to.

Here again, in the case of blanks, we are confronted with different and inconsistent contexts. They are not, as in the case of the knot paradox, coexistent as a dual paradigm in internal conflict. They do no fight each other: they ignore each other! If one recognizes the blank, not letting himself be deceived by logical closure (the token presence of the residual category) one then begins to long for a complementary paradigm in which what is residual could function as a clear-cut concept. This acknowledgment of the exhaustion of a paradigm can arise only from pragmatic reasons; the repeatedly-felt inadequacy of the standing paradigm for dealing with some important practical problem which seriously annoys the scientist.

If the complementary paradigm is discovered, its inconsistency with the original one is manifest: their axioms are not-jointly tenable (think of Marxist and marginalistic economic theory); but this inconsistency is aggravated by the reciprocal inability of the complementary paradigms to deal with, even to formulate, some of the problems characteristic of the other paradigm. So, it is not mainly inconsistency what is at stake, but rather incompleteness. And incompleteness of a radical kind, since, as already shown, it has to do with the pragmatic necessity theories have of being handled by a user. No theory (seems to be the moral of the story) not even the most abstract one, stands in a vacuum. Its pragmatic context is always there to support and limit it.

If one stops at this point and considers the way already traveled, one is struck by the fact that the movement from formal logic to scientific praxis was marked by a progressive slackening of the literal application of our injunctions. Let us see.

The formal law, due perhaps to the very abstraction of its formulation, is totally obligatory: a contradiction should be excluded from a line of proof, as for instance in the case of reductio ad absurdum; and the contradictory of an assumption should be accepted in the same reduction since there is absolutely no middle term between p and ~p.

Next, the methodological requirements are not so binding: incomplete systems are found in the market, and, in the last analysis, one can even buy an inconsistent system if one is willing to pay the price –the risk that a contradiction will arise some time or other.

Lastly, at the level of scientific praxis, the situation is even more lax: the pragmatic imperatives are not even literally compliable– they are to be obeyed in a rather indirect fashion. Thus, to remove a knot (Michelson-Morley's, for instance) one has to abandon the original frame of reference, the very pattern in which one discovered to knot in the first place. On the other hand, in order to fill a blank (for instance the marginalistic blindness toward social conflict) one has also to move to a completely different paradigm, this time a dialectically complementary one. It is not the blank that one has literally to fill; one has to move back and forth from one paradigm to its dialectical complement, and one "fills" one's eye by this rapid change of reference frames. Only alternately changing your view between the two complementary paradigms will you be able to see the whole picture of reality.

The apparent laxity of pragmatic imperatives is misleading. The fact that they are not to be obeyed literally but contextually does not make them any less important o strong. On the contrary, it only emphasizes that they are not abstract entities but traits deeply rooted in concrete reality. Each particular situation, each context, generates its own knots and blanks. But this context dependency is reversible: knots and blanks (in fact, the attempt to remove them) determine the emergence of ever new contexts. This being so, the consequences of solving a knot or filling a blank can be of immense proportions: they may originate whole novel paradigms, forms of thought radically different, even totally new ways of comprehending reality.

Let us summarize: paradoxes (blanks and knots) are context dependent, as in the final analysis all meaning is. This statement must, itself, be understood in context. My ultimate context should be the pragmatist's view of thinking: to think is trying to make sense of appearances, assembling them in fact or imagination in different ways. They may make sense for specific purposes and not for other purposes. Sense is context dependent and context is purpose dependent.

Contexts may be either natural –the normal surroundings of the appearances one is studying– or theoretical –the rational framework in which one accommodates such appearances. Both kinds of contexts are pragmatically determined, since a purpose of some sort, conscious or unconscious, always contributes to the design of what one sees or theorizes.

Extreme scientific problems (paradoxes) are disturbances (knots or blanks) in a paradigm. Paradoxes are bound to occur in any intellectual pattern of a certain degree of complexity. There are blanks and knots in this writing. Follow the pragmatic imperatives: fill the blanks and dissolve the knots, by rewriting (by the way, this is my own rewriting, in 1997, of a 1975 paper!). In this sense, thought is always in a state of flux: rewriting is essential to writing, rethinking is essential to thinking. In fact, there is no such thing as first-hand thinking, ever!

But it is fair to add that rethinking is creative as such, in the sense that the removal of knots or the filling of blanks gives rise to new intellectual patterns, even to new paradigms. By rethinking I communicate with my former self, which is necessarily different from the self of my second thought. To rethink or rewrite something is to attempt the introduction of a certain content into a fresh configuration: that of my current I, which is different from the pattern my I were before this attempt, precisely because then I had not yet made this attempt (purposes define contexts and contexts define meanings).

What is said of self communication is valid also to communication with others: to speak with, to dialogue, is to try to introduce my meaning into the intellectual pattern of my friend. This attempt will change my friend, will change the meaning, and will change me, too. The process by which these transformations are mainly carried out, I claim, is the removal of knots and the filling of blanks, that is, the solution of extreme problems. To solve paradoxes, then, is to transform language and consequently to transform persons, others and oneself.

Knots and blanks are context dependent. This means that what is a paradox in one context may be a clear concept in another context; and vice versa, what is a clear concept in one context may be a paradox in another context (clear concepts are also context dependent).

The main message of paradoxes is this: there is a plurality of languages. Extreme problems tell precisely that: they speak about several languages, conflicting of failing. Paradoxes reveal the depth of language; they presuppose plurality of paradigms: knots, because they break up a context into incompatible, yet consistent, patterns; blanks, since the wordless in one language are nameable in another language.

Theoretical terms and residual categories are tokens which stand for knots and blanks; they have either prior or posterior meaning. If two subcontexts conflict, a knot arises; the subcontexts give the knot prior meaning. A knot is removed by designing a new consistent context in which the knot is replaced by a theoretical term. This theoretical term is a token of the former conflict and inherits the prior meaning of the knot. If a symbolic net does not cover a situation, a blank comes into being; a new pattern may eventually give the blank posterior meaning. A blank is filled by introducing a residual category, which is a token of a complementary context that has to be discovered and will eventually give the category its posterior meaning.

Theoretical terms correlate with a contextual redefinition which removes a paradox by incorporating an experimental result as theoretical principle (example: velocity c in relativity theory). The meaning of velocity c is prior; it can be fully understood only by him who is aware of the conflict story implied in the Michelson-Morley experiment. Residual categories correlate with the exhaustion point of a paradigm and coincide with the scientist's praxis (example: social disturbance coincides with the oppressor's role in the social context). The meaning of "social disturbance" is posterior: in order to be gained one has to have at one's disposal the structural-historic conception of society, epistemologically complementary to the marginalistic conception. The ideology of the capitalistic oppressor is unable to give prior meaning to its own oppressive praxis –a blind spot.

There are, I think, at least two general implications which can be drawn from the above analysis. The first one is the following: Logical principles and methodological requirements are not prime categories, they are only subordinate criteria. Linguistic or pragmatic imperatives are primordial. Neither logic nor methodology is a main pattern, they are only subpatterns of language (abstract contexts). The knot-removing and the blank-filling imperatives are prior to the laws of non contradiction and middle-term excluded, as well as to the requirements of consistency and completeness.

I venture the following extrapolation: even scientific method (hypothetical deductive method) is subordinate: the prime method is contextualization, the putting in context of fragments of meaning –which is based on the criterion that thought does not tolerate isolated symbols: it invents a context if none is available NOTE 5. The development of this idea would, I think, would merit an independent paper.

The second general implication is this: the user of the language belongs in the language, in a double sense. The history of the user, the languages he has used before, determines the meaning of the language, via prior meaning of theoretical terms. On the other hand, the praxis of the user is an integral part of the language, via posterior meaning of residual categories. We belong in language, we cannot think or even be (human) without language. Most of us entertain the illusion that we sometimes see through language, that we somehow manage to reach a non significant reality. But, I suspect, beyond language is always (another) language (significant reality) NOTE 6.

There exists plurality of languages, we can play a variety of language games, move from one significant reality to another, transform or even create languages. But games have rules, and the game of games has this rule of rules: not all transformations are possible, as frivolous thinking would have it. You may pass, to realize particular purposes, from one context to a different one, but you pay a price in terms of blanks and knots, in terms of paradoxes. An all-purpose, paradox-free, perfect language simply cannot exist.

NOTE 1: It is possible, for instance, to discern an intermediate role at the level of inference rules. In a "natural deduction" system of my design, I use a non-contradiction rule to "destroy" formulae and an exclusion-of-middle-term rule to "create" formulae (GUTIÉRREZ 67).

NOTE 2: See TARSKI 54, Chapt. VI, for a nice presentation of this topic.

NOTE 3: This is made clear by the following argument:
A standard procedure for proving a statement is to assume the opposite of what one intends to prove. From the original accepted statements plus the new assumption, it follows some conclusion. If the augmented set of statements is inconsistent, we have to imply that the assumption is no good and false. But this will always be the case if the original set of statements was already inconsistent to start with: after the derivation, it will continue being so. That is why, according to rigorous concepts of derivation, consistence and other logical notions, we are entitle to say that within an inconsistent system one can prove anything.

NOTE 4: See POLANYI 64, chapt. 4, for a good substantiation of this thesis.

NOTE 5: If the available context is somehow blocked, the same applies. Psychoanalysis tells us that much: in rationalization the original context is blocked by censorship hence we produce a substitutive context. Sociology passes the same message: ideology is social rationalization - and we have social censorship and also social substitutive context: current mythology.

NOTE 6: In the last analysis beyond all articulate languages there is praxis, significant action.