Is every argument with false premises and conclusion valid?

Question

I'm trying to determine the validity of the argument tabulated below. Note: Premise 2 is meant to read as (A ∧ ¬A). My bad!

Consider the final two rows: can an argument still be valid even if either (1) all the premises are false but the conclusion is still true or (2) the premises and conclusion are all false?

Furthermore, is an argument automatically invalid if it lacks any row in which the premises and conclusion are all true?

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

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).

Answer 5

Is every argument with false premises and conclusion valid?

can an argument still be valid even if the premises and conclusion are all false?

An argument with a false premise and false conclusion could be valid or invalid.

• Consider the argument 1+1=10; therefore, 1+1=100. It has a false premise and false conclusion, and it is invalid (in the binary system, this argument has a true premise and false conclusion).
• On the other hand, the second example in Bram's answer is a valid argument whose premises and conclusion are also all false.

can an argument still be valid even if all the premises are false but the conclusion is still true?

An argument with a false premise and true conclusion could be valid or invalid.

• Consider the argument some squared number is negative; therefore, every squared number is nonnegative. It has a false premise and true conclusion, and it is invalid (in complex analysis, this argument has a true premise and false conclusion).
• On the other hand, the first example in Bram's answer is a valid argument with false premises and a true conclusion.

is an argument automatically invalid if it lacks any row in which the premises and conclusion are all true?

No; in fact, every argument with inconsistent premises (i.e., one whose premises can be all true in no possible context) is automatically valid.

To be clear: an argument with a false premise needn't be valid (though it is certainly unsound)!

I do understand that an argument is invalid if its truth table contains a row where the premises are all true but the conclusion is false.

Your assertion actually remains correct when strengthened: a propositional-logic argument is invalid if and only if its truth table contains a row with premises all true and conclusion false.

The short of it is that an argument's validity depends on its structure; so, we inspect its entire truth table rather than particular rows (each of which corresponds to a different set of contexts).