# On Modus Ponens/Tollens Fallacies

Modus Ponens

1. IF p THEN q
2. p
Ergo,
3. q

Valid!

Modus Tollens

1. IF p THEN q
2. ~q
Ergo,
3. ~p

Valid!

Converse Fallacy

1. IF p THEN q
2. q
Ergo,
3. p

Invalid!

Inverse Fallacy

1. IF p THEN q
2. ~p
Ergo,
3. ~q

Invalid.

However ...

As regards the converse fallacy

1. IF p THEN q
2. q
Ergo,
3. Maybe p

Valid!

As regards the inverse fallacy

1. IF p THEN q
2. ~p
Ergo,
3. Maybe ~q

Valid!

Correct/Incorrect/Both/Neither?

Do nfr things

• "Maybe" is not formalized by propositional logic. Commented Nov 13, 2023 at 10:11
• Should we ... formalize it? Commented Nov 13, 2023 at 11:30
• "Maybe" is formalized via formalizations of either "possibly" or "probably," isn't it? Commented Nov 13, 2023 at 13:00

As alluded in the comments, we need to go to modal logic K (for Kripke) to get to something like "maybe":

We have two new concepts in K-modal logic:

◻ =def "it is necessary that" (i.e., True in all possible worlds)

◊ =def "is is possible that" (i.e., True in at least one possible world)

NOTE: ◊A ≡ ¬◻¬A.

And two new principles of inference:

Necessitation Rule: If A is a theorem, so is ◻A
Distribution Axiom: ◻(A →B) → (◻A →◻B)

So, we could translate your first argument:

p→q
q
∴ "maybe" p

as:

(1) p→q
(2) ◻ (p →q) [by Necessitation Rule]
(3) ◻p → ◻q [by Distribution Axiom]
(4) ◻a ⇒ ¬◻¬a ≡ ◊a [by definition of necessity]
(5) ¬◻q → ¬◻p [Modus Tollens]
(6) q
(7) ◻q [Necessitation Rule]

As you can see, there is no requirement that ◊p. The only requirement of the material conditional is that q not be false and p true. But q is true in all possible worlds, so p can be false in all possible worlds and still this holds; therefore, you cannot conclude ◊p.

• I thought that unless there was a "compelling force" 😁 that shows ~◇p, ◇p. 🤔 Commented Nov 15, 2023 at 11:16
• @AgentSmith that’s the easier-to-ask-for-forgiveness-than-permission variant Commented Nov 16, 2023 at 1:32

I'm assuming (1) that you are using classical logic, (2) you are using valid in the usual conventional way to mean an argument that instantiates a form that has no possible counterexample, and (3) that your p, q are metavariables, not atomic propositions.

Your first four are correct, the last two are not. That is, unless you specify the additional condition that your p, q are logically contingent. In which case, "maybe p" means nothing more than p is logically contingent, and "maybe ¬q" also means nothing more than q is logically contingent, so they would be correct but trivially so, since you have assumed it.

If you don't assume that p, q are logically contingent, then there are counterexamples.

1. IF p THEN q
2. q
3. maybe p

has a counterexample when p is a contradiction and q is true.

1. IF p THEN q
2. ¬p
3. maybe ¬q

has a counterexample when p is false and q is a tautology.

If you actually want to formalise 'maybe' you could use the ◇ operator in modal logic. But those two forms remain invalid.

Maybe ~q Valid! Correct/Incorrect/Both/Neither?

Yes, correct, but trivial!

And maybe q, too.

Because, if we assume (p → q) ∧ ¬p, then we don't assume anything about q or about ¬q.

So maybe ¬q, but also maybe q.

Sorry I couldn't find any scholarly reference here.