# How be so sure that implications are bivalent? (An attempt to resolve paradoxes of material implication)

The material conditional, P→Q defined as ￢P∨Q, is usually thought not to match the usual linguistics, as seen by many paradoxes. The Wikipedia article gives few good examples. I tried to resolve them by giving an alternate logic system.

It is a many-valued logic that incorporates multiple presence of worlds. Every statement may hold in some worlds, and may not hold in the other worlds. ￢P queries for worlds where P doesn't hold, P∧Q queries for worlds where both P and Q hold, and P∨Q queries for worlds where either P or Q holds. A tautology is the statement holding in every world, and a contradiction is the statement holding in no worlds.

The most distinctive feature of this logic is its definition of implication. P→Q queries whether the worlds where P holds is included by the worlds where Q holds. If the inclusion satisfies, the implication is a tautology. Otherwise, the implication is a contradiction. As a consequence, every implication is bivalent, unlike usual statements.

This seems to solve many paradoxes of material implication. To demonstrate:

• P→￢P might be true in classical logic, but unless P is a contradiction, it's always a contradiction in my new logic.

• Let P and Q be completely unrelated statements. P be "Dannyu NDos is in his office," and Q be "Dannyu NDos will go swimming today." Then P→Q is a contradiction. Blatantly false, unlike ￢P∨Q.

• Q→⊥ is a tautology if and only if Q is a contradiction.

Long story short, my logic goes along with Plato's idea, but insists that there are multiple ideas.

However, I also see some problems:

• The main question is, how be so sure that implications are bivalent, when we all are living in one reality? If there really were alternate worlds, how be so sure that everyone in every world will agree whether the implication holds?

• Here, law of excluded middle works, double negations can be eliminated, and proof by contradiction works because (P→⊥)→￢P is a tautology. Does this mean constructivists will refuse to accept my new logic system?

• How did my new logic system succeeded being a relevance logic without being paraconsistent?

• Compare your implication with $\Box(P\rightarrow Q)$ in the modal logic $S5$. Commented Apr 23, 2023 at 5:22
• @NoahSchweber □P is equivalent to ⊤→P. Commented Apr 23, 2023 at 5:26
• This looks like a relatively minor variation on the usual possible-worlds semantics for modal logic. Where you treat a WFF as denoting a set of possible worlds, the usual semantics would say that the WFF is true in that same set of possible worlds. Commented Apr 23, 2023 at 6:37
• Is this logic a connexive logic? Commented Apr 23, 2023 at 16:41

The logic you describe is not many-valued unless you allow for values other than true and false. The fact that you are using many worlds (usually called possible worlds) is pretty standard, but the logic is still bivalent. A proposition holds true at a world or it doesn't, and a proposition holds true at all worlds or it doesn't. Therefore a proposition may be contingently true or false, or it may be tautologically true or tautologically false. That is bivalent: there are still only two truth values. If you really want to use a many-valued logic you need to specify what other truth value is present in your underlying logic.

Your conditional is really just a reinvention of the strict conditional. Your P → Q holds if every P-world is also a Q-world. That is just strict implication, though we would usually explicitly state an accessibility constraint on the possible worlds.

The properties you listed for your → conditional hold for strict implication:

• □(P ⊃ ¬P) is true if and only if P is a contradiction (i.e. □¬P).

• For unrelated, contingent P and Q, □(P ⊃ Q) is false.

• □(Q ⊃ ⊥) holds if and only if Q is a contradiction (i.e. □¬Q).

• "If you really want to use a many-valued logic you need to specify what other truth value is present in your underlying logic." — I mean, if there are two worlds, there are four truth values, and if there are countably infinitely many worlds, there are continuum many truth values, right? Commented Apr 24, 2023 at 4:03
• Those are not distinct truth values, just different circumstances under which propositions are true or false. My car is blue at the actual world, but there is a possible world where it is red. This does not make "my car is blue" have some kind of intermediate truth value, it just means my car might have been red, if I had bought a red car, or if I repainted it. Tautologies hold in all worlds because it is impossible for them to be false. (Unless, that is, we allow for impossible worlds, which some people do.) Commented Apr 24, 2023 at 16:16

Natural language freely uses doubtful statements, which the laws of the excluded middle and bivalence expressly exclude from consideration. It also includes doubtful conditionals, P→Q in which we do not know whether Q follows from P. Classical logic has no direct means to handle these.

In classical logic, the conditional P→Q works to assure valid reasoning by assuring or at least claiming that Q is at least as true as P, and thus that we have introduced no new error by our reasoning process. (Whether there is error in the starting assumptions is a different matter, one of soundness). It is not generally recognized that this is best described as a relation between statements P and Q, not an operation on them. So long as you are constrained to classical two-valued logic, the definition equivalent to ￢P∨Q is the only reasonable choice you have for the "if-then" relation between statements. It works because this relation has the properties of a mathematical ordering. The more successful alternatives to classical logic simply assume the ordering properties of the conditional without trying to define it.

If you are going to use any form of multi-valued logic, you have already rejected bivalence and the law of excluded middle as general laws applying to all statements: It seems pointless to try to preserve them, except perhaps in special cases. There is no compelling reason to stick to ￢P∨Q as the definition of the conditional, either, and in general, this definition doesn't work for multi-valued logic. The question then becomes how to define a conditional that preserves the mathematical ordering characteristic of valid reasoning, and yet allows for multiple truth values. Multivalued logics in general have not yet solved this problem. With few exceptions, the conditionals they define do not have the required ordering properties, and so how to construct chains of valid reasoning is an unsolved problem with much circumlocution involved.

Until you have handled the problem of valid reasoning with multiple truth values in one world, it does not appear to simplify matters by introducing many worlds.

Here are a few uncontroversial starting points if you want to produce a formal logic which is true of the logic of human reasoning:

1. P → ¬P is just obviously false. The case where P is true falsifies P → ¬P.
2. Q → ⊥ is also obviously false. The case where Q is true falsifies Q → ⊥.
3. A contradiction has the form P ∧ ¬P, while an implication has the form P → Q, so an implication is not and could not possibly be a contradiction.
4. If P and Q are completely unrelated statements, then the case where P is true and Q is false falsifies the implication P → Q, so P → Q is just false.
5. Implications hold or do not hold in our world. If x is a mammal, then it is a vertebrate, so Mx → Vx is true in the real world, while the converse Vx → Mx is false. We need logic to understand the real world. Allegedly possible worlds are not going to help us understand logic.