Constructive mathemathics and its dual

by MrMind 3 min read28th Feb 201330 comments

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I have stumbled upon an interesting and, as far as I know, new concept: thinking about the duality between constructive and paraconsistent logics, I've noticed that while the meta-theory of intuitionistic logic (constructive mathematics) is very well understood and studied, the meta-theory of the dual logic is not. If we understand constructive mathematics from an epistemological point of view, as an accretion of truth from an empty base, we ought to be able to think about a sort of destructive mathematics, that starts from the totality of assertions and proceeds by expunging falsity. This seems to have surprising consequences for things like theism, Tegmark universe(s), the Many World Interpretation and so on, but first I need to cover some background informations. This I will do in the present post, while in the next I'll present the concept and some of its applications.

There is a variety of philosophical programs known as constructive mathematics, but their common denominator is to refuse the classic way of conceiving truth and adhere instead to a concept known as verificational existence. That is, for a mathematical formula A to be accepted as true, there must be a construction (a direct proof) of A. On the same level, for a mathematical formula to be denoted false, a constructivist accepts only a proof of  (this symbol denotes the negation of A). If neither of such proofs exist, then a constructivist refuse to impose a truth value upon A. This has as consequence the refusal of the general formula , ( denotes logical disjunction), valid in classical logic, a principle called in Latin tertium non datur (TND), which means "a third is not given". For a constructivist nonetheless the principle  still holds ( denotes logical conjunction), because it is still viewed as impossible that there exists a valid proof of A and of its opposite. This one is called ex contraditione quodlibet (ECQ), which means "anything from a contradiction".

The simple excision of TND from classical logic gives a logical system called intuitionistic logic (because it was developed under the intuitionistic program of constructive mathematics), which has many, many interesting properties.

A logical calculus developed on these fundamental consideration is aimed at preserving justification rather than truth: intuitionistic proofs, instead of carrying from true formulas to other true formulas, only produce justified formulas from other justified formulas.

Notice though: ECQ and TND are both theorems (or axioms) of classical logic. They are in fact equivalent. ECQ is espressed as , but under the DeMorgan laws, double negation elimination and commutativity of disjunction: , which is the TND.

If then we decide to break the equivalency, it becomes natural to ask: since there's a logic that accepts ECQ but refuses TND (intuitionistic logic), can there be a logic that's a sort of dual, that is it accepts TND but refuses ECQ?

This question, maybe surprisingly, has an affirmative answer, and the resulting plethora of logical systems thus produced are called "paraconsistent logics".
Under this classification, intuitionistic logic can then be said to be a member of the class now known as "paracomplete logics" (although this name is not much used). Paraconsistent logics are a multitude, but one of them is an exact symmetric of intuitionistic logic, known in the literature as dual-intuitionistic logic.

If you reflect on that a little bit, it may seems very strange at first to abandon ECQ. After all, the refusal of contradictions is one of the primary, if not the primary, foundation of rationality. But in formal logic there's also a very cogent and slightly technical reason why contradictions are not allowed: in classical logic, they imply triviality.

A set (possibly infinite) of sentences and formulas in logic is called a theory. It is clear that an empty theory is not very interesting: it literally tells us nothing about the subject at hand. But an equally uninteresting theory is the total theory: the set of all possible sentences and formulas. Since this set makes no distinction on what's true and what's false about the subject of interest, it's as informative as the empty theory. Such a theory is called trivial, and formal systems developed within classical logic strive to avoid contradictions: indeed, from a single formula and its negation you can prove any other formula.
In systems that rely on classical logich then, any contradiction entails triviality, and they are therefore to be avoided.

Paraconsistent logics however depart from this classical setting, and they abandon this principle (sometimes called "principle of explosion").

Be careful though: only the general principle is abandoned. Exactly like in intuitionistic logic, where  is abandoned in general, but if you have constructed a proof of (say) A, then for that particular formula  is valid, in dual-intuitionistic logic if you have constructed a proof of (say) , then  is still valid only for that formula.

What is the meta-theory of dual-intuitionistic logic? How can it be justified and it's at the end somehow useful?

This is where things get interesting, and it's a theme I want to explore in the next post.

Links and references
Over the net, there's more than you could possibly care to learn about constructive mathematics: the usual pointer are Wikipedia's http://en.wikipedia.org/wiki/Constructivism_(mathematics) and http://en.wikipedia.org/wiki/Intuitionistic_logic, while on the SEP side you have http://plato.stanford.edu/entries/mathematics-constructive/ and http://plato.stanford.edu/entries/logic-intuitionistic/.
There is considerable less material on paraconsistent logic, but again you can find http://en.wikipedia.org/wiki/Paraconsistent_logic and http://plato.stanford.edu/entries/logic-paraconsistent/.

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