1. Validity and Soundness, Again

2. Validity
Way back before we even introduced the idea of formal logic, we already encountered and became familiar with the basic idea of validity.
Back then we said that an argument (in natural language) is valid just so long as the truth of the premises guarantees the truth of the conclusion. Or alternately, in a valid argument it is impossible for all the premises to be true and the conclusion to be false.

3. Validity, formalized
We are now finally in a position to revisit this concept in the context of formal logic.
What we need is some method of looking at every possible case in which the premises of an argument are true, and checking whether or not the conclusion of the argument is also true in all of those cases.
And we have just such a method – the truth table!

4. Validity and truth tables
Let’s take an example and see how a truth table might help us.
How about the argument P ~ (R ∨ Q) (P ∨ R) ∧ ~Q
Let’s draw out the truth table together on the board.

5. Validity and truth tables
As we can see, the only line of the truth table for which both of the premises are true is line 4. And in line 4, the conclusion of the argument is also true!
So there are no cases for which the premises of the argument is true and the conclusion is false. And since cases just represent possible assignments of truth values, we can say that it is impossible for the premises to both be true and the conclusion false. Which is just to say that the argument is valid.

6. Validity and truth tables
More precisely we can say: To say that an argument (expressed in the language of sentence logic) is valid is to say that any assignment of truth values which makes all the premises of the argument true also makes the conclusion true.

7. Invalidity and truth tables
Just as we can use truth tables to show arguments to be valid, we can use them to show arguments to be invalid.
For example, the argument: P ∨ Q ~P
Let’s do this truth table on the board now too.

8. Counterexamples
Lines 1 and 2 of this truth table give us assignments of truth values for which the premise of this argument is true, and yet the conclusion is false. These lines demonstrate that the argument is invalid.
We call such cases counterexamples. Again this is a refinement of the intuitive idea that we met earlier in the course. A counterexample to a particular argument is just any assignment of truth values for which all the premises of the argument are true and yet the conclusion is false.

9. Validity and counterexamples
With this definition in hand, we can now give an alternative definition of validity.
An argument of sentence logic is valid just in case there are no counterexamples to it.

10. Validity and logical truth
We can go further here. Imagine an argument with three premises (A, B, C) and a conclusion (D). We can rephrase this argument so there is just a single premise that is a conjunction of the three original premises – so we now have A ∧ B ∧ C, therefore D.
So for any argument, we can express it in the form ‘X, therefore Y’. Where X is a compound sentence formed by the conjunction of each of the premises, and Y is the conclusion.

11. Validity and logical truth
A counterexample to an argument of the form ‘X, therefore Y’ is just a case in which X is true and Y is false. I.e. a case in which X ∧ ~Y is true.
So to say that there are no counterexamples to an argument of the form ‘X therefore Y’ is just to say that in all possible cases, X ∧ ~Y is false. Or alternately, that in all possible cases, ~(X ∧ ~Y) is true.

12. Validity and logical truth
But that is just to say that ~(X ∧ ~Y) is a logical truth – remember, that’s the definition of a logical truth, true in all possible cases.
So we can put this all together to say: an argument ‘X, therefore Y’ is valid just in case ~(X ∧ ~Y) is a logical truth.
Since we can express any sentence logic argument in the form ‘X, therefore Y’, this gives us a quite general test for validity.

13. Soundness
We can now also go over the distinction we made earlier in this course between validity and soundness.
As you’ll recall the distinction runs like this: a valid argument is an argument in which the truth of the premises guarantees the truth of the conclusion.
A sound argument is then just a valid argument with all true premises.

14. Validity vs Soundness
Grasping this distinction is important, as it tells us something significant about validity, and why and how much we should care about it.
In particular, it tells us that validity is not everything! Validity is simply a matter of whether or not the conclusion is true if the premises are true. But we don’t just care about this counterfactual truth. We (also) care about actual truth!

15. Validity vs Soundness
When we make an argument we do two things. We say that the premises are true. And then we say that if the premises are true, the conclusion must be true too. And so, we conclude the conclusion.
Validity deals only with the second stage of this process. It does not deal with the first one.

16. Validity vs Soundness
To see this, we can produce some examples of valid arguments with totally false premises and conclusions. Let’s do this now.

17. Validity vs Soundness
None of this means that validity isn’t important. Nor that thinking about it in isolation is not a very useful thing to do.
But it does serve to remind us that validity isn’t everything. When it comes to natural language arguments in particular, there can be perfectly valid arguments which are nevertheless very bad ones because they use implausible premises.

18. A little exercise
Using the definitions we have agreed on in this class, show that the conclusion of a sound argument must be true.