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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

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  • 3
    "Maybe" is not formalized by propositional logic. Nov 13, 2023 at 10:11
  • Should we ... formalize it? Nov 13, 2023 at 11:30
  • 2
    "Maybe" is formalized via formalizations of either "possibly" or "probably," isn't it? Nov 13, 2023 at 13:00

3 Answers 3

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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.

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

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.

0
1

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.

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