# Propositional Logic: Truth table for a certain argument form

I'm confused about evaluating a certain argument form with the truth table approach. I think below argument matches the corresponding truth table I made, but if my truth table is right it means the argument is valid right? Since the conclusion is never false with all true premises? But looking at its form its obviously not valid since the conclusion does not follow from the premises. So could anybody point out the error in my understanding or truth table?

• Ignoring what seems to be a random ~a, the conclusion (b) is entailed by a & b. I can see where you're coming from, and I'd like to discuss the issue further if my suspicions are in any way well-founded. This will require, I'm afraid, a better example of what you're driving at: logically valid non sequiturs. Nov 15, 2023 at 11:35

This argument is inconsistent since a&b and ~a cannot both be true. You will never have all the premises true and so it is trivially valid.

Look at columns 3 and 4 for a&b and ~a respectively. There is a saying for this type of situation:

ex falso sequitur quodlibet, 'from falsehood, anything follows'

See here as well on principle of explosion

• Thank you, so to be clear the truth table approach of "if no rows have a false conclusion with all true premises, the argument is valid" does not always work? Nov 14, 2023 at 0:14
• @DamonFernandez it does. It's just trivially true in the case when no rows have all premises true :) Nov 14, 2023 at 0:17
• @DamonFernandez for example - (IF a and ~a THEN b) is always true, but what we call "vacuously true" because the antecedent is false. Nov 14, 2023 at 0:18