7.2 Intuitionistic and multivalued logic Main ideas of intuitionism Multivalued logic

Intuitionistic and multivalued logic

“There is no prophet in his own country,” says an old adage. Those whom we now call classics, once stood on a par with their contemporaries, and the latter did not skimp on criticism.

The classical logic did not have time to develop, grow stronger and show its potencies, as it became the object of harsh criticism, coming from different directions. One of the most active in this regard were the Iptuitionists led by the Dutch mathematician L. Brouwer.

Basic ideas of intuitionism

The source of mathematics, considered Brower, is a fundamental mathematical intuition. Not all ordinary logical principles are acceptable to her. This, in particular, is the case with the law of the excluded middle, saying that either the statement itself or one hundred negation is true. This law historically arose in reasoning about finite sets of objects. But then it was also unreasonably extended to infinite sets. When a set is finite, we can decide whether all the objects in it have a certain property by checking all these objects one by one. But for infinite sets such a check is impossible.

Suppose that considering a finite set of numbers, we proved that not all of them are even. Hence, according to the law of the excluded middle, it follows that at least one of them is odd. In this case, the assertion of the existence of such a number can be confirmed by presenting this number. But if the considered set of numbers were infinite, the conclusion about the existence of at least one odd number among them would be unchecked. It would thus remain unclear what the word "existence" itself means in this case.

According to the expression of the German mathematician G. Weil, evidence of existence, based on the law of the excluded middle, notifies the world that the treasure exists, without specifying its location and not allowing it to be used.

Thus, according to intuitionists, the law of the excluded middle is not universal, equally applicable in the discourse on any objects. As Weil says, not without irony, he "may be true for an omnipotent and omniscient being, as if reviewing with a single glance an endless sequence of natural numbers, but not for human logic."

Highlighting mathematical intuition, intuitionists did not attach much importance to the systematization of logical rules. Only in 1930, a student of Brower, A. Geyting, published a paper setting forth a special intuitionistic logic. In this logic, the law of the excluded middle, which is unquestionable for classical logic, does not apply. A number of other laws are also rejected, which allow one to prove the existence of objects that cannot be constructed or calculated. The rejected ones include, in particular, the law of removing double negation (“If it is incorrect that non-A, then L”) and the law of reduction to absurdity, which gives the right to assert that a mathematical object exists, if the assumption of its non-existence leads to a contradiction.

Subsequently, ideas concerning the limited applicability of the law of excluded third and methods of mathematical proof close to him were developed by Russian mathematicians A. N. Kolmogorov, V. A. Glivenko, A. A. Markov, etc. As a result of rethinking the basic premises of intuitionistic logic, constructive logic, which also considers unjustified the transfer of a number of logical principles that are applicable in reasoning about finite sets, to the domain of infinite sets.

Many-valued logic

Classical logic is based on the principle that every statement is either true or false. This is the so-called double-digit principle. Logic itself, admitting only truth and falsehood and not suggesting anything in between, is usually called two-valued. It is opposed by multi-valued systems. In the latter, along with the true and false assertions, various kinds of “indefinite” statements are also allowed, accounting for which at once not only complicates, but also changes the whole picture.

The principle of ambiguity was known to Aristotle, who did not consider it, however, universal and did not extend its effect to statements about the future.

Two hostile fleets are located opposite each other and wait for the morning and with it a suitable wind. Will there be a sea battle tomorrow? It is obvious that it will either take place or not take place. But according to Aristotle, neither of these two predictions is today either true or false. There is still no hard reason either for the battle to happen or for it to happen. Both options are possible equally, and everything will depend on the further course of events. Plans for fleet commanders may change, a storm may occur and fleets sweep across the sea. In the meantime, one cannot say with certainty that the battle will be, or that it will not happen. Both of these statements are possible, but neither of them is now neither true nor false.

The situation is similar with the question whether this cloak will be cut or not. It all depends on the decision of its owner, and it can change at any time.

It seemed to Aristotle that statements about future random events, the onset of which depends on the will of the person, are neither true nor false. They do not obey the principle of ambiguity. The past and the present are unequivocally defined and not subject to change. The future is to some extent free to change and choose.

The approach of Aristotle already in antiquity caused fierce debate. He was highly appreciated by Epicurus, who allowed the existence of random events. The well-known ancient Greek logician Chrysippus, who categorically denied the accidental, did not agree with Aristotle. He considered the principle of ambiguity to be one of the main provisions not only of all logic, but also of philosophy.

At a later time, the position that any statement is either true or false was disputed by many and for many reasons. It was indicated, in particular, that it complicates the analysis of statements about the future, statements about unstable, transitional states, about non-existent objects like “the current king of France”, about objects inaccessible to observation, like “absolutely black body”, etc. d.

In modern logic alone, it was possible to realize doubts about the universality of the principle of two-valuedness in the form of logical systems. This was facilitated by its wide use of methods that do not impede a formal approach to logical problems.

The first multi-valued logics were built independently of each other by the Polish logician J. Lukasevich in 1920 and the American logician E. Post in 1921. Since then, dozens and hundreds of such "logics" have been built and investigated.

J. Lukasiewicz proposed a three-valued logic based on the assumption that statements are true, false and possible, or uncertain. The latter were related statements like: "I will be in Moscow in December next year." The event described by this statement is now not defined either positively or negatively. Hence, the statement is neither true, false, it is only possible.

All the laws of three-valued logic of Lukasiewicz were also laws of classical logic; the reverse, however, did not take place. A number of classical laws were absent in three-valued logic. Among them were the law of contradiction, the law of the excluded third, the laws of indirect evidence, and others. The fact that the law of contradiction was not in three-valued logic did not mean, of course, that it was in some sense contradictory or incorrectly constructed.

E. Post approached the construction of multi-valued logics purely formally. Let 1 mean truth, and 0 means false. It is natural to assume then that the numbers between one and zero denote some degrees of truth decreasing to zero.

This approach is quite legitimate in the first stage. But in order for the construction of a logical system to cease to be a purely technical exercise, and the system itself - a purely formal construction, in the future, of course, it is necessary to give its symbols a certain logical meaning, a meaningfully clear interpretation. The question of such an interpretation is the most complicated and controversial problem of multi-valued logic. As soon as something intermediate is allowed between truth and falsehood, the question arises: what, in fact, do utterances mean that are neither true nor false? In addition, the introduction of intermediate degrees of truth changes the usual meaning of the very concepts of truth and falsehood. It is therefore necessary not only to give meaning to intermediate degrees, but also to reinterpret the very concepts of truth and lies.

There have been many attempts to substantively substantiate multi-valued logical systems. However, it still remains controversial whether such systems are just “intellectual exercise” or do they still say something about the principles of our thinking.

The many-valued logic in no way ns denies and ns discredits the two-digit one. On the contrary, the first allows a clearer understanding of the ideas underlying the second, and is in a certain sense its generalization.