In the 19th century, there were a number of attempts to bring logic up to date (George Boole, Ernst Schröder, Gottlob Frege, Bertrand Russell and Alfred N. Whitehead). But, apart from the introduction of symbols, and a certain tidying up, there is no real change here. Great claims are made, for example by the linguistic philosophers, but there are not many grounds for them. Semantics (which deals with the validity of an argument) was separated from syntax (which deals with the deductibility of the conclusions from axioms and premises). This is supposed to be something new, when, in reality, it is merely a re-hash of the old division, well known to the ancient Greeks, between logic and rhetoric. Modern logic is based on the logical relations among whole sentences. The centre of attention has moved away from the syllogism towards hypothetical and disjunctive arguments. This is hardly a breathtaking leap. One can begin with sentences (judgments) instead of syllogisms. Hegel did this in his Logic. Rather than a great revolution in thought, it is like re-shuffling cards in a pack.

Using a superficial and inexact analogy with physics, the so-called atomic method developed by Russell and Wittgenstein (and later repudiated by the latter) tried to divide language into its “atoms”. The basic atom of language is supposed to be the simple sentence, out of which compound sentences are constructed. Wittgenstein dreamed of developing a “formal language” for every science – physics, biology, even psychology. Sentences are subjected to a “truth test” based on the old laws of identity, contradiction and the excluded middle. In reality, the basic method remains exactly the same. The “truth value” is a question of “either…or”, “yes or no”, “true or false”. The new logic is referred to as the propositional calculus. But the fact is that this system cannot even deal with arguments formerly handled by the most basic (categorical) syllogism. The mountain has laboured, and brought forth a mouse.

The fact is that not even the simple sentence is really understood, although it is supposed to be the linguistic equivalent of the “building-blocks of matter”. Even the simplest judgment, as Hegel points out, contains a contradiction. “Caesar is a man”, “Fido is a dog”, “the tree is green”, all state that the particular is the universal. Such sentences seem simple, but in fact are not. This is a closed book for formal logic, which remains determined to banish all contradictions, not only from nature and society, but from thought and language as well. Propositional calculus sets out from exactly the same basic postulates as those worked out by Aristotle in the 4th century B.C., namely, the law of identity, the law of (non-)contradiction, the law of the excluded middle, to which is added the law of double negation. Instead of being written with normal letters, they are expressed in symbols. This makes not the slightest difference to the content of the syllogism. Moreover, symbolic logic itself is not a new idea. In the 1680s, the ever-fertile brain of the German philosopher Gottfried Wilhelm Leibniz (1646-1716) created a symbolic logic, although he never published it.

The introduction of symbols into logic does not carry us a single step further, for the very simple reason that they, in turn, must sooner or later be translated into words and concepts. They have the advantage of being a kind of shorthand, more convenient for some technical operations, computers and so on, but the content remains exactly as before. The bewildering array of mathematical symbols is accompanied by a truly Byzantine jargon, which seems deliberately designed to make logic inaccessible to ordinary mortals, just as the priest-castes of Egypt and Babylon used secret words and occult symbols to keep their knowledge to themselves. The only difference is that they actually did know things that were worth knowing, like the movements of the heavenly bodies, something that cannot be said of modern logicians.

Terms such as ” monadic predicates“, ” quantifiers“, ” individual variables“, and so on and so forth, are designed to give the impression that formal logic is a science to be reckoned with, since it is quite unintelligible to most people. Sad to say, the scientific value of a body of beliefs is not directly proportionate to the obscurity of its language. If that were the case, every religious mystic in history would be as great a scientist as Newton, Darwin and Einstein, all rolled into one.

In Moliere’s comedy, Le Bourgeois Gentilhomme, M. Jourdain was surprised to be told that he had been talking prose all his life, without realising it. Modern logic merely repeats all the old categories, but throws in a few symbols and fancy-sounding terms, in order to hide the fact that absolutely nothing new is being said. Aristotle used monadic predicates (expressions that attribute a property to an individual) a long time ago. No doubt, like M. Jourdain, he would have been delighted to discover that he had been using monadic predicates all the time, without knowing it. But it would not have made a scrap of difference to what he was actually doing. The use of new labels does not alter the contents of a jar of jam. Nor does the use of jargon enhance the validity of outworn forms of thought.

The sad truth is that, in the 20th century formal logic has reached its limits. Every new advance of science deals it yet another blow. Despite all the formal changes, the basic laws remain the same. One thing is clear. The developments of formal logic over the past hundred years, first by propositional calculus (p.c.), then by lower predicate calculus (l.p.c.) has carried the subject to such a point of refinement that no further development is possible. We have reached the most comprehensive system of formal logic, so that any other additions will certainly not add anything new. Formal logic has said all that it has to say. If the truth were to be told, it reached this stage quite some time ago.

Recently, the ground has shifted from argument to deducing conclusions. How are the “theorems of logic derived”? This is quite shaky ground. The basis of formal logic has always been taken for granted in the past. A thorough investigation of the theoretical grounds of formal logic would inevitably result in transforming them into their opposite. Arend Heyting, the founder of the Intuitionist School of mathematics, denies the validity of some of the proofs used in classical mathematics. However, most logicians cling desperately to the old laws of formal logic, like a drowning man clutching at a straw. As Morris R. Cohen and Ernest Nagel remarked:

We do not believe that there is any non-Aristotelian logic in the sense in which there is a non-Euclidean geometry, that is, a system of logic in which the contraries of the Aristotelian principles of contradiction and the excluded middle are assumed to be true, and valid inferences drawn from them.[60]

There are two main branches of formal logic today – propositional calculus and predicate calculus. They all proceed from axioms, which are assumed to be true “in all possible worlds”, under all circumstances. The fundamental test remains freedom from contradiction. Anything contradictory is deemed to be “not valid”. This has a certain application, for example, in computers, which are geared to a simple yes or no procedure. In reality, however, all such axioms are tautologies. These empty forms can be filled with almost any content. They are applied in a mechanical and external fashion to any subject. When it comes to simple linear processes, they do their work tolerably well. This is important, because a great many of the processes in nature and society do, in fact, work in this way. But when we come to more complex, contradictory, non-linear phenomena, the laws of formal logic break down. It immediately becomes evident that, far from being universal truths valid “in all possible worlds”, they are, as Engels explained, quite limited in their application, and quickly find themselves out of their depth in a whole range of circumstances. Moreover, these are precisely the kind of circumstances that have occupied the attention of science, especially the most innovative parts of it, for most of the 20th century.

  1. Morris, R. and Nagel, E. An Introduction to Logic and the Scientific Method, p. vii.