Are quantum logics isomorphic to inconsistent logics?

Question

I have been wondering if, instead of being discarded, inconsistent logics could be explored as a topic proper - chiefly, the various properties different inconsistent logics have; if we simply accepted that it is valid for a logic to be inconsistent, for our purposes.

A logic is consistent if its axioms cannot derive both p and not-p, for any p.

It occurs to me that quantum logics enable propositions to be in a superposition of states, meaning that a proposition takes the “superstate” (True, False) in certain cases.

Now, one might say that being in superposition is not the same as being both true and false at the same time. Rather, it is more like being as of yet undecided.

Still, in terms of mathematical form, if we consider that in our logic, the way we treat propositions of truth-value (True, False), is the same as if the meaning of (True, False) were actually “true and false”.

So I wonder: are quantum logics therefore isomorphic to inconsistent logics, where there exist propositions that are both true and false?

Answer 1

One point of clarification first. To speak of 'inconsistent logics' is somewhat confused. A logic that proves P and not-P for any P would be of no use to anyone. But there are logics that permit a theory to contain both P and not-P for some specific P without explosion. These are called paraconsistent logics. By contrast, in classical logic all contradictions are false and from a contradiction anything follows. Some paraconsistent logics allow that some contradictions are true; all paraconsistent logics do not cause theories to explode when they contain contradictions.

The term 'quantum logic' has been used to describe a number of different systems of logic. Most commonly it refers to propositional logic that lacks the distributivity of conjunction over disjunction. That is:

P and (Q or R) does not entail (P and Q) or (P and R).

This kind of logic does not give rise to contradictions, so it is not paraconsistent.

But there are other systems of quantum logic that are paraconsistent. The IEP article on Quantum Logic has a short section on this. There are also a couple of papers on arXiv that might be of interest: The Paraconsistent Logic of Quantum Superpositions, by Da Costa and de Ronde, and Quantum Logics, by Chiara and Guintini, especially section 13.

Answer 2

Logics that can account for both p and not-p are called paraconsistent logics. But you seem to be looking for a multi-valued logic, which could include paraconsistent logics, but you don't have to accept paraconsistent principles to be able to process values outside of true/false. You should look at quantum logics and multi-valued logics if you want to know more.