## 1. Truth vs. Proof

Despite its name, the modern theory of knowledge called Intuitionism, developed by L.E.J. Brouwer[1], can be classed as an excessively deductive approach. It was, significantly, originally intended and designed for mathematics, and was thereafter by extrapolation applied to all knowledge[2]. Equating for all intents and purposes the logical modality of proof with that of fact, “Intuitionist logic” rejects the law of the excluded middle (and hence the inference of a positive statement from a double negation).

Arguing that nothing can be claimed to be true if it is not *proved* to be true, Intuitionism claims to accept the law of non-contradiction (since we cannot both prove A and prove not-A), but denies the law of the excluded middle (since we can both fail to prove A and fail to prove not-A). Thus, whereas Aristotle originally formulated these laws with reference to facts (as nothing can be A and not-A, and nothing can be neither A nor not-A), Brouwer focused on proof alone.

Many errors are involved in this change of perspective. For a start, one can refute it on formal grounds: just as we cannot both prove A and prove not-A, we cannot both *dis*prove A (= prove not-A) and *dis*prove not-A (= prove A). The fact that we can be in ignorance of both A and not-A, i.e. uncertain as to which is true and which is false, does not change the fact that A and not-A cannot be both true or, equally, be both false. The two laws are symmetrical and cannot be taken separately.

Note that Aristotle’s approach was to set ontological standards that would serve as epistemological guides, whereas Brouwer tried to place epistemology squarely before ontology. The former implicitly allowed for knowledge not at all dependent on rational processes, viz. knowledge from experience, whereas the latter considered all knowledge as dependent on reasoning, i.e. as purely mental construction.

For classical logic, proof is a conflation of empirical givens and conceptual constructs. To anchor concepts in experience involves deductive methods, but the result is always inductive. If we precisely trace the development of our knowledge, we always find ultimate dependence on empirical givens, generalization and adduction. *There is no purely deductive truth* corresponding to the Intuitionist’s notion of “proved” knowledge. The Intuitionist’s idea of proof is misconceived; it is not proof.

Even an allegedly “purely deductive system” would need to rely on *our experience* of its symbols, axioms and rules. Thus, it cannot logically claim to be purely deductive (or *a priori* or analytic, in Kantian terms), i.e. wholly independent of any experience. Moreover, *our* *understanding* of the system’s significance is crucial. A machine may perform operations we program into it, but these are meaningless without an intelligent human being to consume the results. Brouwer’s assumptions are rife with ignored or hidden issues.

Note too that Brouwer effectively regards “proved” and “not proved” to be exhaustive as well as mutually exclusive. This shows that he implicitly mentally relies on the law of the excluded middle (and on double negation), even while explicitly denying it. Certainly, we have to understand him this way – otherwise, if the terms proved and *un*proved (N.B. not to confuse with *dis*proved) allow for a third possibility, his theory loses all its force. That is, something in between proved and not proved (N.B. again, not to confuse with proved not) would have to somehow be taken into consideration and given meaning!

Brouwer’s denial of the law of the excluded middle is in effect nothing more than a recognition that some knowledge has to be classed as *problematic*. That was known all along, and we did not need to wait for Mr. Brouwer to realize it. The law of the excluded middle does not exclude the possibility of problemacy, i.e. that humans may sometimes not know for sure whether to class something as A or not-A. On the contrary, the law of the excluded middle is formulated on that very assumption, to tell us that when such problemacy occurs (as it often does), we should *keep looking* for a solution to the problem one way or the other.

The law of non-contradiction is similarly based on human shortcoming, viz. the fact that contradictions do occur occasionally in human knowledge; and its function is similarly to remind us to try and find some resolution to the apparent conflict. Note here the empirical fact that we do sometimes both seem to prove A from one angle and seem to prove not-A from another tack. In other words, if we follow Brouwer’s formulation of the law of non-contradiction, that law of thought should also be denied!

The fact of the matter is that what we commonly call proof is something tentative, which may turn out to be wrong. The genius of classical logic is its ability to take even such errors of proof in stride, and lead us to a possible resolution. It is a logic of realism and adaptation, not one of rigid dogmas.

Indeed, if there is anything approaching purely deductive truth in human knowledge, it is the truth of the laws of thought. So much so, that we can say in advance of any theory of knowledge that if it postulates or concludes that any law of thought is untrue – it is the theory that must be doubted and not these laws. Such antinomy is sure proof that the theory is mixed-up in some way (just as when a theory is in disagreement with empirical facts, it is put in doubt by those facts).

In the case of Intuitionism, the confusion involved is a misrepresentation of what constitutes “proof”. Only people ignorant of logic are misled by such trickery. Why on earth would we be tempted to accept Brouwer’s idea of “proof” in preference to the law of the excluded middle (which this idea denies)? Has he somehow “proved” his idea, or even just made it seem less arbitrary, more credible or more logically powerful than the idea of the law of the excluded middle? His view of proof is not even “proved” according his own standards – and it is certainly not proved (indeed it is disproved) by true logic.

Consider the implications of denials of the second and third laws of thought on a formal level. To deny the law of non-contradiction only is to wish to logically treat X and not-X as subcontraries instead of as contradictories. To deny the law of the excluded middle only is to wish to logically treat X and not-X as contraries instead of as contradictories. To deny both these laws is to say that there is no such thing as negation. All the while, the proponent of such ideas unselfconsciously affirms some things and denies others.

Reflect and ask yourself. If X and not-X cannot be contradictories, why should they be contraries or subcontraries? On what conceivable basis could we say that incompatibility (as that between X and not-X) is possible, but exhaustiveness is not; or vice versa? And if nothing can be incompatible and nothing can be exhaustive – what might negation refer to? It is clear that all such proposed antinomial discourse is absurd, devoid of any sort of coherence or intelligence. It is just a manipulation of symbols emptied of meaning.

## 2. Double Negation

The deeper root of Intuitionist logic is of course *a failure to understand the nature of negation*. What does ‘not’ mean, really? How do we get to know negative terms, and what do they tell us? How does negation fit in the laws of thought? I will not go far into this very important field here, having already dealt with it in detail in the past[3]; but the following comments need be added.

Another, related weakness of Intuitionist is *ignorance of inductive logic*. As already stated, Brouwer functioned on an essentially deductive plane; he did not sufficiently take induction into consideration when formulating his ideas. In a way, these were an attempt to get beyond deductive logic; but his analysis did not get broad enough.

This can be illustrated with reference to *double negation*. On a deductive plane, negation of negation is equivalent to affirmation. This is an implication and requirement of the laws of thought. However, on an inductive plane, the matter is not so simple, because negation is always a product of generalization or adduction. That is to say, ‘not’ always means: ‘so far, not’; i.e. it is always relative to the current context of knowledge.

What distinguishes deductive from inductive logic is that in the former the premises are taken for granted when drawing the conclusion, whereas in the latter the uncertainty of the premises and therefore of the conclusion are kept in mind. Thus, deductively: ‘not not X’ means exactly the same as, and is interchangeable with, ‘X’; but inductively: the premise ‘not not X’ tends towards an ‘X’ conclusion, but does not guarantee it.

Since ‘not X’ really means ‘we have looked for X but not found it so far’, it always (with certain notable exceptions) remains somewhat uncertain. On the other hand, a positive, namely ‘X’ here, can be certain insofar as it can be directly perceived or intuited (and in this context, the experience ‘not found’ must be considered as a positive, to ensure theoretical consistency).

If ‘not X’ is always uncertain to some degree, it follows that ‘not not X’ is *even more* uncertain and cannot be equated in status to the certainty inherent in ‘X’ (if the latter is experienced, and not merely a conceptual product). Double negation involves two generalizations or adductions, and is therefore essentially an abstraction and not a pure experience.

Moreover, the expression ‘not (not X)’ inductively means ‘we have looked for the negation of X and not found it’. But since ‘not X’ already means ‘we have looked for X and not found it’, we may reasonably ask the question: is the path of ‘not not X’ the way to find ‘X’ in experience? Obviously not! If we seek for X, we would directly look for it– and not indirectly look for it through the negation of its negation.

Note, too, that having found ‘X’ in experience we would consider ‘not not X’ to follow with deductive force, even though the reverse relation is (as already mentioned) much weaker.

Thus, the problem of double negation posed by Brouwer is a very artificial one, that has little or nothing to do with actual cognitive practice. Not only are the laws of thought nowhere put in doubt by this problem – if we are careful to distinguish induction from deduction – but it is not a problem that would actually arise in the normal course of thought. It is a modern sophistical teaser.

Drawn from *Logical and Spiritual Reflections* (2008), Book 3, chapter 7.

[1] Holland, 1881-1966.

[2] Such extrapolations are unfortunate: since mathematics deals with special classes of concepts (notably numerical and geometrical ones), insights concerning it cannot always be generalized to all other concepts. Inversely, comments concerning logic in general like the ones made here do not exclude the possibility of specific principles for the mathematical field. I am not a mathematician and do not here intend to discuss that subject.

[3] In Chapter 9 of my book *Ruminations*. I strongly recommend the reader to read that crucial essay.