Is an argument with all false premises and a false conclusion valid?

Question

I have this truth table:

And I'm trying to determine the validity of the argument, which is:

(B v A), (A ^ ¬A) therefore C

I've got one row at the bottom, where (B v A) and (A ^ ¬A) are both false, but C is true. And the one below that has all false premises and a false conclusion.

Maybe this is a dumb question, but can an argument still be valid even if either (1) all the premises are false but still has a true conclusion or (2) all premises and conclusion are false?

Furthermore, does an argument lacking any row in which all premises are true + conclusion is true automatically mean that the argument is invalid?

I would like to make it clear that I do understand that if a truth table contains a row in which all premises are true but the conclusion is false the argument is invalid.

Thanks.

(Note: the table has (A v ¬A), but it is meant to read as (A ^ ¬A). My bad!

Answer 1

I can't quite make sense of the title and your first question -- true or false where, i.e. under which interpretation? What we need to check is that truth is preserved from the premises to the conclusion under all possible interpretations:

An argument is valid iff
there is no row where all premises are true and the conclusion is false.

We can not determine from a single row with false premises whether the argument is valid or not. Rows where at least one of the premises is false count positive towards the validity, the truth value of the conclusion does not matter in these cases. The only thing that must not happen is for there to be a row where all premises are true but the conclusion is false: We have "if and only if a truth table contains a row in which all premises are true but the conclusion is false the argument is invalid".

This means in particular that if there is no row that makes all premises true to begin with, because the premises are contradictory, then there can be no counterexample. In this case, the argument is (vacuously) valid.

So your argument is valid because there is no counter example where all premises are true but the conclusion is false.

Answer 2

$A\land\lnot A\vDash C$ is valid.   There is no interpretation of the literals where the antecedent is true and the consequent is false.

After all, $A\land\lnot A$ is false in every valuation of $A$.   It is a contradiction.

$$\begin{array}{cc|c}A&C&A\land\lnot A\\\hline\top&\top&\bot\\\top&\bot&\bot\\\bot&\top&\bot\\\bot&\bot&\bot\end{array}$$

So there are clearly no valuations where $C$ is false yet $A\land\lnot A$ is true.

This will remain so no matter what other antecedents you add to the sequent.

Answer 3

Maybe this is a dumb question, but can an argument still be valid even if either (1) all the premises are false but still has a true conclusion or (2) all premises and conclusion are false?

1. Yes. Consider:

All animals are blue

All blue things are mortal

Therefore, all animals are mortal

2. Yes. Consider:

All animals are blue

All blue things have 56374 legs

Therefore, all animals have 56374 legs

Answer 4

Here is the truth table of the statement you present in your question:

Source: https://www.erpelstolz.at/gateway/TruthTable.html

Notice that antecedent is always false (column 5). Therefore, by the so-called principle of explosion, the implication (last column) will always be true.

This does not mean that the consequent C (in your example) is always true. As you can see, it is false in the half the cases presented here (column 3).