This feels like a basic question or a misunderstanding, but I'm struggling to identify a simple concept having to do with the provability of an argument or proof.
Imagine you have a proof that is well-reasoned and believed true, but then you discover that one of its underlying premises is actually false. Still valid, but unsound.
That false premise is conjuncted with other true premises to create lemmas, which no longer hold. These faulty lemmas are then conjuncted with other premises and lemmas to create further lemmas that no longer hold, and so on, until the conclusion is reached. Clearly, the conclusion no longer holds either. The conclusion itself might be true, it might be false, but it's definitely false that it holds.
My desire is to track those truth values of whether the various steps hold. The point is that these truth values (of whether the lemmas and conclusion hold) travel through the proof identically as they would through AND gates. And yet you would not say that the lemmas or conclusion themselves are false, it's just false that they hold. This is not implication, because the truth table behaves differently - if a proof is based off of the conjunction of two lemmas, and the first lemma does not hold, you would not argue that the proof is true.
In other words, look at the truth table for our favorite Socrates syllogism:
- A: Socrates is a man
- B: All men are mortal
- C: Therefore, Socrates is mortal
|A||B||A ^ B|
I'm not interested in implication (→) here. I am more curious what A ^ B communicates about C. Not the value of C, but the provability of C. Obviously, if B is false, we have not proven Socrates immortal. But A ^ B does mean something about C that is consistent with the above truth table; something that is not implication. It means that the argument for C does not hold if one or more of its premises are false; that it is false that it holds.
What is this logical concept and what symbol should I choose? Are "provable" and "not provable" ( ⊢ and ⊬ ) the best options? i.e.
|A||B||A ^ B||A ^ B ⊢ C||i.e.|
|T||T||T||T||A ^ B ⊢ C|
|T||F||F||F||A ^ B ⊬ C|
|F||T||F||F||A ^ B ⊬ C|
|F||F||F||F||A ^ B ⊬ C|
It certainly doesn't seem I should use does/doesn't imply/entail (→ ↛ ⇒ ⇏ ⊨ ⊭), because their truth tables are incompatible. Like if A^B is false, I can't say that "it is false that A^B implies C", because "False implies C" is literally true according to the implication truth table.
It's like I want a more colloquial sense of "implies" that means "Since A and B are not both true, A^B (False) does not (imply) that C holds." Is there a symbol/term for that?