Philosophy and Ideology. Z. A. Jordan 1963
The opinion that there is no sense of the term ‘true’ in which a belief may be true in spite of not corresponding to any fact and that the correspondence relation in which truth consists is that of reflection, meets with some difficulties in accounting for the truths of logic and mathematics. This is one of the reasons why the materialist theory of truth is firmly committed to the view that logic and mathematics have an empirical basis and express empirical generalisations of past experience. According to Lenin, this was ‘proved’ or ‘brilliantly guessed’ by Hegel. Marxist-Leninists in Poland accepted Engels’ and Lenin’s authority on this matter and felt that it provided a sufficient reason for dismissing any objections that have been raised against considering logic and mathematics as empirical science.
The belief that logic and mathematics have an empirical origin and that they are empirical science should not be confused. For we can be in general agreement with the former and reject the latter. Ernest Nagel rightly observed that a genetic account of logical and mathematical operations, highly speculative and dubious as it must be, contributes little or nothing at all to the understanding of the grounds of their present authority. Nothing seems to follow from the observation of J. S. Mill, transformed by innumerable repetitions into a triviality, that experience and the requirements of social life have led to the development of logic and mathematics. It is even extremely doubtful whether what might be historically true, still applies at the present time.
According to the materialist theory of truth, logical and mathematical theorems reflect the external world in an abstract manner in virtue of which their validity is not necessarily established a posteriori. What an abstract reflection is and what it means has never been explained. Even in the very simple case of the sentence ‘ Warsaw is situated on the Vistula’ it had to be conceded that this sentence reflects only metaphorically the fact of Warsaw being situated on the Vistula. When we pass to the theorems of logic and mathematics the reflection relation is said to be not only metaphoric but also abstract, and our power of understanding is strained beyond its imaginative and intellectual resources. Perhaps no important truth is lost, if, for instance, the assertion that Taylor’s theorem reflects objective reality in an abstract fashion remains unexplored. The view adopted in the materialist theory of truth that the abstract character of reflection provides logical and mathematical theorems with a certain measure of independence of experience is more significant, since it allows us, in fact, to disregard the reflection theory up to a certain point. This does not mean that logic and mathematics can be considered as independent structures, a free creation of the logician and mathematician, subject to one single law, namely, that their structures must be free from contradiction. Marxist-Leninists draw a sharp distinction between axioms and rules of transformations on the one hand and theorems on the other. Only the validation of the latter, but not of the former, is independent of experience. The empirical basis of logic, mathematics and, generally, of the deductive sciences, is preserved in their axioms and transformation rules which are invariably validated by an appeal to experience. Their choice is accomplished in accord with certain objective relationships, of which they are an image. The objective reference of axioms and transformation rules is inherited by theorems. It is, therefore, a fallacy to maintain that logic and mathematics are independent of experience, since their axioms are empirical hypotheses and the validity of their inference rules is based on empirical evidence. Not a single instance in support of this opinion has been given, nor has any attempt been made to substantiate the claim that the suggested method of empirical evidence was ever applied to validate logical axioms and rules of inference. One is reduced to a feeling of intellectual embarrassment when the assertion regarding the empirical character of logic and mathematics is considered to be conclusively established by a quotation from Lenin Philosophical Notebooks to the effect that the figures of syllogism have acquired the ‘permanence of preconceived ideas and an axiomatic character’ on the strength of having been repeated millions and millions of times in man’s practical activity.
Perhaps the principal source of the confusion, from which the argument in favour of logic and mathematics being an empirical science is drawn, is the ambiguity of the term ‘practice’. The word ‘practice’ is a synonym of ‘empirical evidence’ in the Marxist-Leninist terminology. But ‘practice ‘applied to logical and mathematical operations does not mean ‘empirical evidence’. It is practice in some other sense, if it is practice at all, that establishes the validity of logical axioms and transformation rules by providing evidence that only materially true theorems can be deduced from a set of axioms by the application of the accepted rules of transformation which always yield true conclusions from true premisses. If it were true that axioms and inference rules are validated in that manner, and this is not the case, it still would not follow that they are thus validated by recourse to empirical evidence. Moreover, if it were true that axioms and inference rules are empirical hypotheses or inductive generalisations, we should be able to indicate not only the confirming evidence, but also that which would falsify them and oblige us to reject them as unwarranted. For various reasons, which have been frequently explained in detail and need not be stated again, this course is not feasible, since no empirical procedure for testing the validity of logical and mathematical principles can be established.
One of the implications of the Marxist-Leninist view under discussion is the rejection of the conception originated by Wittgenstein and vigorously advanced by logical empiricism, according to which all meaningful statements are either analytic or empirical. Marxist-Leninists seem to be aware of the fact that an analytic statement is defined as a statement true or false in virtue of its wording .
Consequently, they should not confuse it either with Kant’s a priori synthetic statement, since it continues to be true irrespective of what experience may reveal, nor with Kant’s analytic statement, since it is not restricted to the class of subject-predicate sentences, discovered by ‘pure thought’, but is derived by arbitrarily fixed rules from arbitrarily fixed axioms and definitions. This expectation is not fulfilled. Although a-priorism should be understood to be the view which accepts synthetic a priori statements, and neither analytic nor empirical sentences are synthetic a priori statements, the distinction between analytic and synthetic truths is presented by the Marxist-Leninists as a revival of a-priorism and sometimes, oddly enough, also of conventionalism.
It is clear that if the existence of analytic statements is accepted and the theorems of logic, of pure mathematics, or of any axiomatic theory are conceived as systems of such statements, the materialist theory of truth in its present form cannot be maintained. For in this case not every true statement is a reflection of objective reality and some statements would have to be recognised as true in virtue of being in ‘agreement with the accepted principles and inference rules’, that is, their truth would depend solely upon language and not upon extralinguistic facts, as required by the materialist theory. Because of this the supporters of the analytic character of the deductive sciences are singled out as particularly vicious opponents who endanger the materialist theory of truth and by the same token materialism as well .
The doctrine that logic and mathematics are of empirical origin constitutes a part of what has been known in Poland under the name of genetic empiricism . Genetic empiricism can roughly be described as the belief that no part of our knowledge is psychologically independent of experience. Empiricism in the genetic sense should be differentiated from methodological empiricism, that is, from the view that no part of our knowledge can be validated a priori. The Marxist-Leninist standpoint concerning the nature of truths of logic and mathematics can be described as methodological empiricism of a narrow kind. Although it concedes that logical and mathematical theorems can be validated without recourse to experience, it maintains that this procedure is unobjectionable owing to the fact that accepted axioms and rules of transformation are validated a posteriori. The truth of axioms and the validity of inference rules, established by empirical evidence, provide the guarantee that theorems validated a priori are materially true propositions also. The class of analytic statements is an empty class, and all statements are factual or synthetic.
The elimination of propositions true by definition is a tour de force that overlooks some elementary facts of the case under examination. If by axioms are meant axioms of a formal axiomatic system, they differ from theorems in one single respect, significant only in relation to an inquiry, namely, they are logically prior to theorems. There is no reason to suppose that this fact alone should impart to axioms a different epistemological character from that possessed by theorems. What is an axiom in one formal system may be a proposition in another. Generally, an axiom is a relatively undemonstrable proposition, that is, undemonstrable in that system; there are no undemonstrable propositions in an absolute sense. The choice of axioms is subject to some restrictive rules, but these rules are concerned with the characteristics of the formal system itself. These facts are too familiar to deserve more than a passing mention.
The difference between a theorem and a rule of transformation is also relative and not absolute. Every theorem may be used as a rule of transformation. Provided that certain rules of inference are introduced axiomatically, all theorems can be reformulated in such a manner that they become rules of transformation. If Marxist-Leninists were right, this would mean that by a mere reformulation an analytic proposition in the restricted sense of this term, which they too recognise, becomes a proposition validated a posteriori.
If by axioms we do not mean axioms of a formal axiomatic system but what is popularly called so, that is, the laws of logic, such as the law of non-contradiction or the law of the excluded middle, they are not undemonstrable assumptions in the strict sense of this expression. In order not to confuse them with the latter, it is perhaps better to call them ‘logical laws’ or ‘logical principles’. Marxist-Leninists considered these logical principles to be empirical laws, valid within certain relatively narrow limits, and invalid otherwise. Later they changed their opinion and have held them to be reliable and universally applicable empirical hypotheses. In this manner they wished to avoid not only the old-fashioned a-priorism and similar speculative vagaries, but also conventionalism, that is, to regard logical principles as arbitrarily made stipulations. Having safely sailed past Scylla, they have run into Charybdis. For if logical principles are empirical generalisations, they must also be subject to empirical refutation. This cannot, however, be justifiably maintained. There are no imaginable empirical circumstances which would invalidate the principles of logic. The empirical impossibility rests on the fact that the manner in which the principles are formulated excludes the logical possibility of any evidence inconsistent with them ever being admissible.
If in colloquial language the proposition ‘nothing can both be so and so and not be so and so’ is an axiom, the proposition ‘every square has four sides’ is an axiom too. For the ground on which they are acknowledged is the same. Marxist-Leninists seem to overlook the fact that every sentence must be formulated in some language, and every language is determined by rules which govern the acceptance of sentences. If there were no such rules, language would be no means of communication, which it can be provided that the use of expressions is not entirely arbitrary. On the other hand, the fact that language has some rules, which set restrictive limits to the ways in which its expressions are used, should not lead to the conclusion that all the rules of language can be exactly specified. This is the case in artificial languages, but not in natural ones. A special class of rules are those which can be regarded as implicit definitions of a particular expression or of a type of expressions. Ajdukiewicz called this class of rules ‘axiomatic rules of meaning’ .
The afore-mentioned instances of axioms are propositions whose acceptance is governed by axiomatic rules of language; no factual observation, experiment, or argument can provide a valid reason for their rejection; they hold, whatever might happen, as long as the rules hold. (The last qualification is necessary to avoid the impression that analytic statements are immune to revision). Analytic statements are accepted in virtue of the rules, in this case of the English language, that determine the use of the words, which they contain. They are accepted because these words are used in just that manner and we cannot apply them otherwise as long as we speak the English language. If the sentences in question are accepted in virtue of the rules of meaning, no experience can either confirm or refute them. They are true by definition and to call them hypotheses in need of validation by experience is not only idle but also misleading.
The rules of language are not absolute. They might be changed slowly and imperceptibly under the pressure of variable linguistic habits or deliberately revised to serve a definite purpose. This is the case in science, which adopts expressions of everyday language and defines them in accordance with its requirements. The use of scientific terms, such as ‘atom’, ‘evolution’, or ‘axiom’ is revised if factual observations or theoretical needs make it necessary. Since the rules of language change, the class of analytic statements in a given language is not fixed once and for all. We cannot speak of analytic propositions, if by a proposition is meant the meaning of a sentence and of its translations into different languages, but only of analytic statements. An analytic statement is not one ‘true in all possible worlds’, as Leibniz wished to define it. But analytic sentences are no invention; they are a linguistic matter of fact, which is just as impossible to ignore as any other non-linguistic matter of fact.
The elucidation of the relation between experience and variable linguistic habits, whether in everyday or scientific language, and the relation between the latter and the class of analytic sentences, presents a problem of enormous complexity. For the idea that we can neatly separate experience and language is utterly fictitious. Some language or other system of signs must be presupposed in order that we may speak of experience at all, and the mediation of language in describing matters of fact and ‘objective reality’ in general cannot be eliminated.
While probably all colloquial languages and all artificial languages so far constructed contain analytic sentences, there is no valid reason to regard them as a necessary part of every language. Suggestions of a limited scope for the elimination of some analytic sentences have been made in the past. Thus, for instance, it has been suggested that the development of a theory of certain sub-atomic phenomena requires a many-and not the two-valued logic. This implies that in such a theory a substitution of the law of non-contradiction or of the law of the excluded middle would not necessarily be an analytic but a factual sentence, to be tested, confirmed or disconfirmed by experience. In principle, a language could be constructed in which no sentence is accepted by virtue of an appropriate axiomatic rule. In such a language there would be no axiomatic rules of meaning and the laws of logic would become empirical hypotheses subject to confirmation by the evidence of experience.
The assertion of the materialist theory of truth that there are no analytic statements and that all statements are empirical, considered in relation to ordinary language, is a false statement. But it is not nonsensical. If a language were constructed in which there were no axiomatic rules of meaning, no sentence of this language would be analytic. Whether a language of this kind would offer any advantages in scientific inquiry is a question which cannot be settled a priori. Until it is actually constructed the existence of analytic sentences cannot be dismissed, for they are a part of the language of which all of us, including Marxist-Leninists, have to make use.