Daily Dialectic

Consider a property P : “is green all over, and is not green all over”.

Alice and Bob are friends. Bob reasons about the world, and accordingly conditions his understanding of what is or isn’t logically possible to classical logic. In particular, not only there are no objects with property P, but there cannot be any such objects since P is a logically impossible property. Alice on the other hand, having travelled far and wide, and having seen curiosities such that no classically (as per classical logic) minded philosopher has even dreamed of, accepts paraconsistent logic as the correct way to reason about the world and its many wonders.

Alice and Bob agree that “empty” (or “the empty collection”, denoted (e)) is an absence of any-thing, i.e. it is an absence of any actual object. They also adopt a convention whereby saying that “some-thing is contained by the empty collection” means the same as saying that “that some-thing does not exist”, e.g. “square circles are contained by the empty collection” is just another way of saying that “square circles don’t exist”. This is the convention both Alice and Bob adopt, and agree on. Nota bene, impossibilia (impossible objects) are also contained by the empty collection, for since they can’t exist, in particular they don’t actually exist.

The duo however disagree about what counts as an object. Whereas Bob considers any object with property P as impossible, i.e. an object that cannot possibly exist, Alice doesn’t. As a consequence, within their discussion about the world and its many wonders, the term “empty collection” despite being correctly understood by both friends as having the same extension, is incorrectly assumed to have the same intension. The extension is the same since it is an absence of any object, but its intension is distinct, due to our protagonists’ differences in reasoning about the world, which in turn bear on what counts as possible and what doesn’t.

In particular, for Bob the empty collection has the property of containing all “objects” with property P, since that amounts to saying that such objects don’t exist (since they can’t exist!). Alice would disagree with Bob with regards to such a property — she would negate it having such a property outright, by saying that it’s not the case that the empty collection contains all objects with property P, since (after all) the existence of such objects is consistent with her view of the world. So if she were to accept that the empty collection had such a property, it would amount to saying that the empty collection in not empty after all. Instead, Alice asserts the negation, i.e. that there are some objects with property P that are not contained by the empty collection.

What does this mean? That Alice and Bob aren’t talking about a unique empty collection, but rather two distinct ones. To eliminate confusion they decide to introduce yet another convention whereby they index those distinct empty collections by what makes them distinct — in this case the distinction is conditioned on who is considering the empty collection, i.e. (e)-A, and (e)-B. But since the differences in the intension of those two terms are in virtue of the reasoning system (logic) adopted by either friend, we can conclude that the intension of “empty collection” is logic relative.

Unlike the act of conditioning the meaning of “the empty collection” to contexts which deliver contingent distinctions, e.g. what a caveman, a poet, a chemist or a physicist would consider as “empty”, the difference in the meaning of “the empty collection” in the case of Alice and Bob arises out of how such concepts can be reasoned about in principle. As such it is a distinction in the meaning of “the empty collection” in principle, i.e. in principle that concept has no unique meaning.

Also this is a nontrivial claim, since it’s “possible” (conceivable) that Alice and Bob never met, and so never discovered the discrepancy in the meaning of “the empty collection”. In other words, it’s a nontrivial result since it’s conceivable that there could be rational agents confined to a single reasoning system only, thereby not being capable (lacking the necessary epistemic condition, which is the act of abstracting away from the preferred reasoning system) of seeing the fundamental non-uniqueness of the notion of “the empty collection”.

To sum up: Alice’s notion of “the empty collection” is not the same as Bob’s notion of it is. Concisely speaking (e)-A is not identical to (e)-B, which yields the truth of the claim “(e)-LP is not identical to (e)-classical”.

But let’s assume per impossible that (e)-LP is identical to (e)-classical; hence x is in (e)-LP iff x is in (e)-classical. Also pick the object a (such that Pa) as an LP-possible object (a is an LP possibilia). This is a fair assumption since not all contradictory properties need be impossible in LP, thus allowing objects with such properties to be legitimate possibilia. Hence a is not in (e)-LP, according to the convention adopted by Alice and Bob. But according to the same convention, all x such that Px are elements of (e)-classical, i.e. all x such that Px are classically impossible (are classical impossibilia). In particular a is in (e)-classical, but that means that a is in (e)-LP given the hypothesis (identity assumption), but we assumed that a is not in (e)-LP, which yields a contradiction.

Therefore, adopting classical logic as the one governing this proof (the meta-logic here), and reductio ad absurdum as a valid proof method, it follows that (e)-LP is not identical to (e)-classical, as required.