On Non-Classical Formal Systems

Abstract

Classical logic´s adequacy to deal with certain concepts has been put forward in our time as a problem which can be taken as part and parcel of our philosophical tradition. The various questionings of it which have been made as regards its scope of application, and som of its very principles as well, have in our century given rise to, among other things, non-classical systems of logic.
Vasilev, a forerunner in the treatment of many-valued logics, described his work as an attempt to do with Aristotelian logic what Lobatchevsky had done with regard to Euclidean geometry. From our viewpoint this is a correct approach: in the same way as non-Euclidean geometries are more adequate than Euclidean geometry to deal with certain problems, notably in physics, some non-classical logics are better adapted for the analysis of certain ideas and are more fit than classical logic to deal with certain provinces of natural languages or with approximation problems.
A semantic principle which, even though it has been accepted by the main trends of orthodox logic, had been already discussed and questioned by Aristotle in connection with future contingents, is the principle of two-valuedness, according to which:
Every statement is either true or false.
The taking into account of future contingent statements which can be analyzed as indeterminate, giving rise in this way to a third truth value, and the study of the various propositional modalities, is the source from which the questioning of this principle arises, jettisoning the creation of the so-called many-valued logics.
Different many-valued logics can be obtained by dealin with different propositional modalities as, for example, brobabilistic, epistemical, existential, alethic.
From the beginning of their development many-valued logics were extended into systems with an infinite set of truth values; such systems were even considered by Lukasiewicz as those which, together with 3-value systems, hold the most interest from a philosophical standpoint. For many of these systems we have not as yet an adequate attribution of meanings to their truth values. Even if such is the case, they are of interest because of their applications in purely formal realms, e.g., independence proofs.
More recently we have seen the upsurge of fuzzy logics. The inferential process in them is more of a semantical than of a syntactical nature, with an intention to recapture the impreciseness of our usual modes of reasoning rather than to deal with exact reasoning. Their conception, originating in the notion of fuzzy set introduced by Zadeh, intends to take into account the fact that statement within natural languages are neither true nor false in an absolute way but only in a certain degree and in a certain sense.
Different systems of fuzzy logics have been developed in which the set of truth values can be totally ordered or constitute a lattice; there are others in which the truth values are a fuzzy set. In this case, account is taken of the fact that one cannot asign a precise truth value to a vague statement.
In some systems of many-valued and fuzzy logics the principles of excluded third and of non-contradiction are not valid. The former, identified sometimes with the principle of two-valuedness, has been the most questioned of classical principles along history. It can be stated as:
For any statement A, (A ˅ ~A) is valid.
The validity of this principle depends on how we interpret disjuction, negation and what we understand by “true”, as well.
For example, if we take “to be true” as meaning “to be provable in a deductive systems S” and if S is incomplete, then for a statement A not provable in S, neither A non ~A are true in S and hence (A ˅ ~ A) is not true.
By the same token, if we only admit as true something which has been established according to some rules or methods, and these have not yielded a proof neither of A nor of ~ A, then A ˅ ~ A) will again not be true.
This last is the position adopted by intuitionists; for them the truth of a statement has to be established constructively.
The principle of non-contradiction can be stated thus:
For any statement A, ~(A & ~ A) is valid.
From a formal viewpoint, the strongest reason to accept this principle, both in classical and in intuitionistic logic, is that both of them have as a valid formula:
(I) ((A & ~ A) → B), for any B whatsoever.
Hence if the principle were not taken as valid, this would render the system trivial.
Logics in which (I) is not valid can be used as bases for theories, T, in which we can admit, for a certain statement A, both A and ~A without making T trivial. To do this might be of interest from a philosophical standpoint, given a certain interpretation of negation, or from a linguistic viewpoint if we are intereste in natural languages. Implication has also been questioned in its classical interpretation. We can begin by pointing out the so-called paradoxes of material implication which, for some thinkers, follow from the acceptance of the principle
(II) (A → (B → A))
This principle is valid both in classical and in intuitionistic logics. The problem with (II) is that by accepting it and modus ponens, and by taking a statement A which might be contingently true, it follows then that A is necessarily implied by any statement B whatsoever, i.e.:
A ˫ B → A.
In Lewis´ system S4, (II) is not valid.
To remedy this apparent fault of implication, it has been suggested that for A to be implied by B, B has to be relevant for A. What this means is not clear as yet, but a proposal has been made that a minimal condition of relevancy of B for A is that both statements have, at least, a common propositional variable. Certain systems incorporating this notion have already been developed.
[J.A. Robles]