7.3 Validity

Assess this argument:

1.  Pia or Quinn is guilty.
2.  Pia is innocent.
Thus,
3.  Quinn is guilty.

Formalized, the argument looks like this:

1.  PvQ
2.  ~P
Thus,
3.  Q

We used the double bar double arrow, “⇔”, as shorthand for equivalence. We will use double bar single arrow going left to right, “⇒”, as shorthand for validity or logical entailment.

X ⇒ Y means X entails Y.

So we write PvQ, ~P ⇒ Q to mean the premises PvQ and ~P validly entail Q.

In this section we learn how to use a truth table to show this argument is valid.

The first step is to construct a joint truth table for all the premises and conclusion. Keep the premises in order and put the conclusion on the right.

We place a slash mark before the Q to signal that it’s the conclusion. Think of it like the horizontal line that means “thus”. The exclamation marks are part of the Truth Machine software that ask you to identify the counterexample rows.

Even though Q is just an atomic sentence, we repeat it where the complex sentences go because it is part of the argument.

Next you just complete the table:

Lastly, you check to see if it meets the definition of validity, which says: whenever (all) the premises are true, the conclusion is also true.

A quick check reveals there’s only one row where all the premises are T: row 3.

So we just have to verify that the conclusion is T there too. Since it is, we know the argument is valid.

Rows where some premise is F are irrelevant to validity.

Remember: it doesn’t matter what value the conclusion has on any rows other than rows where all the premises are T. Rows where some premise is F are irrelevant to validity, since validity just says what must be the case whenever all the premises are true.

Truth Table Method: Building a joint truth table to assess an argument for validity.

We will call this the truth table method for assessing an argument for validity.

The beauty of the method is that it always works. BOOL is designed just to help us study the logic of and, or and not, and the truth table method always gives the right answer for that purpose.

If an argument is valid because of the Boolean connectives, the method will tell us it’s valid; and if it’s not valid because of those connectives, the method will also tell us it’s invalid.

Just remember that the method is limited in this way: if an argument is valid, but its validity comes from some other part of logic, then the method will say the argument is invalid. Metaphorically: the method only “sees” Boolean logic, but its vision for Boolean logic is perfect.

Your turn. Use the truth table method to assess this argument:

1.  ~(PvQ)
Thus,
2.  ~Q

Your table should have looked like this:

There is only one row where the premise is true (row 4), and the conclusion is true there too, so we know it is valid.

Now try this argument:

1.  P&~Q
2.  ~P
Thus,
3.  R

Your table should have looked like this:

Now, you might have thought: there’s no row on which all the premises are true, so it can’t be valid. Premise 1 is only true on rows 3 and 4, and premise 2 is false on those rows.

But that isn’t the right test. In order to be valid the argument doesn’t need a row where the premises and conclusion are all true. In order to be valid, it just needs no row on which the premises are true and the conclusion false.

We satisfy that here. Hopefully this example made you think about one of the weird cases of validity: it has contradictory premises!

One last practice:

1.  PvQ
2.  ~Qv~R
Thus,
3.  PvR

If you answered that correctly without guessing, then you should know the answer to this:

No looking below until you’ve worked it out yourself!

Your table should have looked like this:

A common reaction students have is: look at rows 2, 3 and 4; the premises are true and the conclusion is true too, so it is valid.

But that is incorrect. We need to know that whenever the premises are all true, the conclusion is also true. So a single example of true premises and false conclusion invalidates the whole argument.

That’s what we see on row 6:

Counterexample: A row on which the premises are true and the conclusion false.

A row like this, with true premises and a false conclusion, is called a counterexample. A single counterexample shows that the argument is invalid. Put another way: a valid argument is one with no counterexample.

In this section you learned the truth table method: how to use truth tables to assess arguments for validity.