Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 3.1.3 Premises and Theorems

Every proof that we’re going to construct does the same thing. It establishes the truth (validity) of some statement of the following form:

\(( claim_1 \wedge claim_2 \wedge claim_3 \wedge \cdots \wedge claim_n ) \rightarrow conclusion \)

In other words, it “proves” that, if all the given claims are true, the conclusion must also be true.

There are at least four common names for what we’ve just called claims:

• premises

• postulates

• hypotheses

• axioms

When discussing a single argument, taken on its own, without a larger context, it’s common to use the word premises or hypotheses .

Sometimes, however, a single set of claims will be premises to a whole collection of related conclusions (typically called a theory ). Then, the premises are usually called the axioms or postulates of the theory.

For example, one axiom (postulate) for Euclidean geometry is that, “A straight line segment can be drawn joining any two points”. Within that theory, we are not to debate whether this statement is true or false – for the purpose of the theory, we assume it is true. The power of this idea is that a set of carefully chosen axioms may enable us to prove a large body of very useful things. For example, in Euclidean geometry we can prove such things as the Pythagorean Theorem

We will use the four terms premises, postulates, hypotheses and axioms interchangeably.

A theorem is something that we have proved to be true. So, once we’ve completed its proof:

• This expression is a theorem:

  \(( claim_1 \wedge claim_2 \wedge claim_3 \wedge \cdots \wedge claim_n ) \rightarrow conclusion \)

• If we’ve agreed on a set of axioms and we are building a theory on them, then we’ll say (as we just did in the case of the Pythagorean Theorem) simply that this is a theorem:

  conclusion

Big Idea

What’s in common here is that a theorem is something we know to be true.

(Premise) r → w If it’s raining then the sidewalks will be wet.

(Premise) w → s If the sidewalks are wet, they will be slippery.

(Premise) s → c If the sidewalks are slippery it is important to be careful.

(Premise) r It’s raining.

(Theorem) (( r → w ) ∧ ( w → s ) ∧ ( s → c ) ∧ ( r )) → c If premises true, then it is important to be careful.

(Theorem) c It is important to be careful.

Once we have proved a statement of the form:

\(( claim_1 \wedge claim_2 \wedge claim_3 \wedge \cdots \wedge claim_n ) \rightarrow conclusion \)

we can describe what we know in any of these ways:

• The conclusion follows from the set of claims (or premises or postulates or axioms).

• The set of claims logically implies the conclusion.

• The set of claims entails the conclusion.

If you’re wondering why, in this course, there seem to be so many ways to say the same thing, all we can tell you is that logic has been around for a long time. A lot of folks have had their hands in the pie. A lot of terms have cropped up. We have to live with them.

Big Idea

Change your premises, watch your conclusions change.

The reasoning techniques that we’re about to describe don’t say anything about what premises we should start with. The reasoning methods are completely agnostic in that regard. However, and this is a huge “however”, that doesn’t mean that it doesn’t matter what premises we pick. The premises we choose will determine the conclusions that we can draw. Once we choose a set of premises and attempt to produce a proof, we may find ourselves in any one of these situations:

• The premises imply a conclusion that we want to draw. We will be able to produce a proof. This is what happened in the Wet Sidewalks example once we added the premise that it is raining.

• The premises are too weak; they do not imply the conclusion that we’re trying to prove. This is what happened in the Wet Sidewalks example before we added the premise that it is raining. It also happened in the Eradicate Ucklufery example. When this happens, we typically look for additional premises that we are willing to accept and that would enable us to prove our conclusion.

• The premises are wrong. They enable us to prove a conclusion that we believe to be false. For example, nothing in logic would have prevented us from starting with the premise, “If it’s raining, the sidewalks will be dry.” Can you see how this would lead to a conclusion that you’d reject? In this case, we will need to revisit our choice of premises.

• The premises are contradictory. When this happens, as we’ll soon see, it is possible to prove any conclusion. In fact, given contradictory premises, for any statement p , it is possible to prove both p and ¬ p . So, while we’ll have proofs, we won’t have much useful information.