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Subsection 1.4.3 Unstated Premises

A logical argument only makes sense with respect to a particular set of premises. However, in stating real arguments (and proofs), we often omit explicit mention of premises that we can assume everyone accepts. Brevity requires this.

Activity 1.4.5.

Consider:

Stacey can’t be at the beach. I just saw her in her office.

Most of us have no trouble accepting this argument. It does, however, leave out a key premise (without which it isn’t valid): It’s not possible for someone to be in two different places at once. Thus being in the office precludes being at the beach.

When we write proofs in mathematics, we typically omit explicit mention, as premises, of such things as the facts about standard arithmetic and algebra.

Activity 1.4.6.

Prove that, for any real \(x: x2 + 1 > 0.\)

  1. \(x^2 \geq 0\) Since the square of any real number is nonnegative

  2. \(\displaystyle 1 > 0 \)

  3. \(x^2 + 1 > 0 \) Adding [1] and [2]

Notice that we haven’t justified our addition of [1] and [2].

But we must be very careful when we rely on unstated premises:

  • In the real world, it is possible (and, in fact, happens all the time) that things that seem obvious to me may not be accepted as premises by everyone else.

  • In mathematics, it is possible to build very different (but interesting) theories by starting with different sets of premises. So, while few theories are built without the standard rules of algebra, there are, for example, competing theories of geometry.

Consider this dialogue:

Activity 1.4.7.

  1. Your friend Kris will love you for life if you get her Cool Dragon tickets for her birthday.

  2. You’ve got to be kidding. Kris would hate Cool Dragon.

A’s argument makes sense to him if he starts with the premise that, of course, everyone loves Cool Dragon. But B doesn’t share that assumption.

We could have replaced the Cool Dragon example with just about any discussion of modern politics. Of course, sometimes people are irrational. But often, if you examine political disagreements, you’ll see that the various people involved have started with different sets of premises.

Problems 1.4.8.

Consider the following dialogue:

\(A\text{:}\) We need to vote for Smith for mayor.

\(B\text{:}\) No. We absolutely have to vote for Jones.

\(A\text{:}\) But Jones supports the new downtown development plan.

Let’s assume that both \(A\) and \(B\) are making reasonable logical arguments. But they don’t share the same set of premises. Mark each of the following as True if it helps to explain the fact that \(A\) and \(B\) have come to different conclusions. Mark False otherwise.

(a)

\(B\text{,}\) but not \(A\text{,}\) assumes that the development plan will create jobs.

Answer.
True

(b)

\(B\text{,}\) but not \(A\text{,}\) assumes that the development plan will cost too much.

Answer.
False

(c)

\(B\text{,}\) but not \(A\text{,}\) assumes that the development plan will disturb some wildlife habitats.

Answer.
False

(d)

\(A\text{,}\) but not \(B\text{,}\) assumes that the development plan will disturb some wildlife habitats.

Answer.
True

(e)

\(A\text{,}\) but not \(B\text{,}\) assumes that the development plan was crafted without citizen input.

Answer.
True
Solution.

Premises that might lead someone (say A) to be opposed to the development plan (and thus opposed to voting for Jones, who supports it) include:

  • The plan will disturb wildlife habitats and that’s a bad thing.

  • Any plan crafted without citizen input is likely to be bad for the citizens.

Premises that might lead someone (say B) to be in favor of the development plan (and thus in favor of voting for Jones, who supports it) include:

  • The plan will create jobs and that’s a good thing.

  • Destruction of wildlife habitats isn’t an important consideration.