An idea to prevent the principle of explosion

Question

The principle of explosion is the law according to which any statement can be proven from a contradiction.

This principle is proved as follows. Assume that P and ¬P are true. Let Q be an arbitrary proposition. By the disjunction introduction, P∨Q is true. By the disjunctive syllogism, Q is true.

Because of this principle, if there is a contradiction in mathematics, then all propositions become true and mathematics becomes trivial.

I have come up with an idea to prevent the trivialization.

How about prohibiting the use of the disjunctive syllogism for P∨Q when we know that both P and ¬P are true? I think this is a legitimate limitation.

1. Does this prevent the principle of explosion?
2. Does this prevent a correct proof from working?

Answer 1

The principle of explosion is a feature of classical logic. It is baked in. You cannot remove explosion from classical logic without turning it into a completely different logic.

There are many ways to prove explosion and you would somehow have to block all of them. It can be proved from disjunction introduction and disjunctive syllogism as you show. It can also be proved using reductio ad contradictione, using conjunction introduction and elimination, from the rule of implication, and in other ways. It can also be proved using model theory and using modal logic.

There are non-classical logics that lack explosion. These are collectively known as paraconsistent logics. These include the entire family of relevance logics, minimal logic, dual intuitionistic logic and the logic of paradox. Some of these are motivated by the desire to tolerate the possibility of true contradictions. Others are motivated by a requirement for the premises of a valid argument to be relevant to the conclusion in a specific way. So if you wish to avoid explosion, you can, by using a different logic.

But why would you want to avoid it? In classical logic no contradictions are true. In fact, in modern usage, the term 'contradiction' has come to have the conventional meaning of a logical falsehood. So within classical logic, explosion serves only to show hypothetically what would follow if we were to suppose something that is inconsistent. A contradiction would have the bad consequence of making a theory trivial by proving every proposition and its negation. Far from being a defect in logic, explosion acts as an enforcer. It insists that you respect the law of non-contradiction on pain of triviality.

As someone who likes to use non-classical logics, I can tell you that logics without explosion are more difficult to work with than classical logic. Also explosion is useful in mathematics. If a theory is inconsistent, then by explosion all sentences within its language are theorems. So by contraposition, if there is even one sentence that is not a theorem then the theory is consistent. Gentzen used this to prove the consistency of Peano arithmetic by showing that with some modest assumptions, it is impossible to prove 0=1.

Answer 2

An idea to prevent the principle of explosion

How about prohibiting the use of the disjunctive syllogism for P∨Q when we know that both P and ¬P are true? I think this is a legitimate limitation.

I have a great deal of difficulty with prohibiting the use of something in logic. One of the glories of logic is its fearlessness in taking the reasoner places that they do not necessarily want to go.

The risk of causing an explosion has its uses: it places a solid boundary around the limits of valid reasoning. You don’t want the result of having it both ways? Then don’t go there.

Please edit your post to explain why your proposal offers a "legitimate limitation."

Answer 3

It seems like you are trying to retain Dialetheism (that there are P's such that P and ¬P) by attempting to geomander your way out of the principle of explosion. To me this seems a little misguided, as you are now writing axioms of logic that are contingent on the dialetheic property of propositions, and evaluating claims of the form 'P is a dialetheia' can be problematic in itself, thus building a weird form of undecidability in the core of the disjunctive syllogism.

To answer your questions:

1 - Does this prevent the principle of explosion?

Yes, probably? But at what cost? You are excluding one of the steps making the principle of explosion, so like other types of paraconsistent logic you will likely face subtle trade-offs coming from the now more convoluted disjunctive syllogism.

2 - Does this prevent a correct proof from working?

This is a more subtle question because what is a 'correct proof'? I don't believe that you can make a fully self-consistent construction of logic, and tweaking and adding sub-clauses to rules of logic will induce a conception of correctness that may or may not be intuitive. This is only my opinion, but I unfortunately think seeking such alternatives is a bit of a wild goose chase, every tweak you make will create intractable ripples down the line, and the more frankensteinian the modifications the more the theory will break in weird and convoluted ways (simply accepting dialetheism creates headache-inducing Russel-like paradoxes from wondering whether 'P is dialetheia' is itself dialetheia..).

P.S.: Even if I am a little skeptical of the whole endeavour, looking more deeply into how such a fix would fits in types of paraconsistent logic could be pretty interesting in itself. And this whole thing is like 'this seems to work' until someone finds a daming counterexample, so here's to hoping there is no edge-case with this construction.

Answer 4

How about prohibiting the use of the disjunctive syllogism for P∨Q when we know that both P and ¬P are true? I think this is a legitimate limitation.

I think the new rule is a little hard to define formally. When we try to write it out, it's something like:

P∨Q, ¬P, ¬P, therefore Q

But the second ¬P has different in meaning than the first. We usually define rules of inference positively. "If we have the following premises, we additionally have the conclusion." This one, however, is "If we have the following premises, and we CANNOT have this other premise, then..." I'm not sure that works, and seems like it has the potential to cause problems.

One potential fix is to introduce a new symbol for "P obeys the LEM". Say we define LEM(P) as the premise that "P, and it cannot be shown that ¬P". But I think that if you introduce something like that, you get into a recursion issue where having the premise LEM(P) doesn't prohibit you from having the premise ¬LEM(P), and so to establish that you'd need LEM(LEM(P)), and so on?

Answer 5

The principle of explosion is the law according to which any statement can be proven from a contradiction.

The Principle of Explosion (PoE) is definitely not a law of logic.

It is only one of a series of rather absurd consequences of taking "p implies q" to be the same thing as "not p or q".

So, it is easy to make the absurdity of the PoE disappear. We just have to dismiss the PoE as bad logic. This implies that we also dismiss as false the theory which says that "p implies q" is the same thing as "not p or q".