Let A, B and C be propositions. Define ARG(A, B, C) as the following argument:
- Therefore, C.
My goal is to create a formula whose truth value is equivalent to "ARG(A, B, C) is valid". In other words, I am looking for a formula that yields true if and only if the argument is valid.
ARG(A, B, C) is valid if and only if (A ∧ B) → C
Unfortunately, this attempt does not work. Proof: the following argument is invalid (source: third example on wikipedia here), but (A ∧ B) → C yields true (since A is false):
- All men are immortal.
- Socrates is a man.
- Therefore, Socrates is mortal.
ARG(A, B, C) is valid if and only if A ∧ B ∧ ((A ∧ B) → C)
Unfortuntately, this attempt also does not work, because although it works for the example above, this attempt yields false for any argument that has a false premise (and it is known that there are valid arguments with false premises).
I tried to create other attempts but I'm stuck.
Is this impossible? If yes, why (can you give a proof)? If it's possible, what would be the formula?