Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 3.2.2 Not Enough Premises

Recall the key role that premises play. The conclusions that we can draw depend entirely on the assumptions (premises or axioms) that we start with. If we change assumptions we’ll get different (possibly fewer, possibly more) theorems. We can now use truth tables to watch this happen.

Let’s return to the Who Drives Me example that we just considered. Let’s change one assumption. Suppose that John can be late for work. Now it turns out that we can no longer prove that Mary must drive me to the store. Let’s see why not. We drop L as a premise. We now try to use our weaker set of premises to prove this new theorem:

[6] \(((J \vee M) \wedge (J \rightarrow L)) \rightarrow M \)

Its truth table is:

The last column is not all T. So [6] is not a tautology. We cannot conclude Mary must drive. It’s not that we know that she must not. We simply don’t know one way or the other.

Recall the Eradicate Ucklufery problem that we suggested a while ago. We gave names to the following statements:

V: Ucklufery (a very nasty tropical disease) is caused by a virus.

E: We might be able to eradicate ucklufery by developing a vaccine against it.

We’ve got one premise:

[1] V  E If ucklufery is caused by a virus, we might be able to eradicate it by developing a vaccine against it.

We want to prove that we might be able to eradicate ucklufery by developing a vaccine. So we want to show that the following is a tautology:

\((V \rightarrow E) \rightarrow E\)

(We only have one premise, so it’s alone on the left of the outer implication.)

Here’s the truth table:

Oops. We don’t have a tautology. We don’t have enough information to be able to conclude that we should work on a vaccine. This example is so simple that it’s easy to see what information we lack. If we knew one more thing, namely that ucklufery is caused by a virus, then we’d know that we should work on a vaccine.