Conjuncted: Internal Logic. Thought of the Day 46.1

So, what exactly is an internal logic? The concept of topos is a generalization of the concept of set. In the categorial language of topoi, the universe of sets is just a topos. The consequence of this generalization is that the universe, or better the conglomerate, of topoi is of overwhelming amplitude. In set theory, the logic employed in the derivation of its theorems is classical. For this reason, the propositions about the different properties of sets are two-valued. There can only be true or false propositions. The traditional fundamental principles: identity, contradiction and excluded third, are absolutely valid.

But if the concept of a topos is a generalization of the concept of set, it is obvious that the logic needed to study, by means of deduction, the properties of all non-set-theoretical topoi, cannot be classic. If it were so, all topoi would coincide with the universe of sets. This fact suggests that to deductively study the properties of a topos, a non-classical logic must be used. And this logic cannot be other than the internal logic of the topos. We know, presently, that the internal logic of all topoi is intuitionistic logic as formalized by Heyting (a disciple of Brouwer). It is very interesting to compare the formal system of classical logic with the intuitionistic one. If both systems are axiomatized, the axioms of classical logic encompass the axioms of intuitionistic logic. The latter has all the axioms of the former, except one: the axiom that formally corresponds to the principle of the excluded middle. This difference can be shown in all kinds of equivalent versions of both logics. But, as Mac Lane says, “in the long run, mathematics is essentially axiomatic.” (Mac Lane). And it is remarkable that, just by suppressing an axiom of classical logic, the soundness of the theory (i.e., intuitionistic logic) can be demonstrated only through the existence of a potentially infinite set of truth-values.

We see, then, that the appellation “internal” is due to the fact that the logic by means of which we study the properties of a topos is a logic that functions within the topos, just as classical logic functions within set theory. As a matter of fact, classical logic is the internal logic of the universe of sets.

Another consequence of the fact that the general internal logic of every topos is the intuitionistic one, is that many different axioms can be added to the axioms of intuitionistic logic. This possibility enriches the internal logic of topoi. Through its application it reveals many new and quite unexpected properties of topoi. This enrichment of logic cannot be made in classical logic because, if we add one or more axioms to it, the new system becomes redundant or inconsistent. This does not happen with intuitionistic logic. So, topos theory shows that classical logic, although very powerful concerning the amount of the resulting theorems, is limited in its mathematical applications. It cannot be applied to study the properties of a mathematical system that cannot be reduced to the system of sets. Of course, if we want, we can utilize classical logic to study the properties of a topos. But, then, there are important properties of the topos that cannot be known, they are occult in the interior of the topos. Classical logic remains external to the topos.

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