# Hypothetical syllogism

So hypothetical syllogism is:

1. if a then b
2. if b then c
3. so, if a then c .

According to wikipedia, "In short, it states that if one thing happens, another will as well. If that second thing happens, a third will follow it." But why it says if that second thing happens when it is sure that if a happens then b will follow. How you justify this ?

• Because each sentence stands on its own, they do not have to be combined, and until the conclusion is drawn that any of this happens is hypothetical. – Conifold Sep 4 '18 at 20:58

## 3 Answers

Consider the form of Hypothetical Syllogism as argument (or rule of inference) and apply the definition of logical consequence :

when all the premises are true, also the conclusion must be.

1) must be TRUE; thus we cannot have a TRUE and b FALSE.

So, if a is FALSE, also the conclusion 3) is TRUE.

If a is TRUE, by the consideration above, also b must be TRUE.

But 2) must be TRUE also; and with b TRUE the only possibility left is that c is TRUE.

But now we have both a and c TRUE, and the conclusion 3) is again TRUE.

According to wikipedia, "In short, it states that if one thing happens, another will as well. If that second thing happens, a third will follow it."

But why it says if that second thing happens when it is sure that if a happens then b will follow. How you justify this ?

Well, it is not sure that A will happen; so it is not sure that B will. Also there is nothing prohibiting B happening when A does not. But if it does, then C will.

The only guarantees we have been given are the two sentences:

• If thing-A is true, then thing-B is true.
• If thing-B is true, then thing-C is true.

However, should these two sentences be correct, then it must also be so that :

• If thing-A is true, then thing-C is true.

My understanding is that a hypothetical syllogism is in which there are three statements in which the premises and the conclusion are all hypothetical in form. Classically :

((A => B) ∧ (B => C)) ⊃ (A => C).

Your example fits this. Here's another, just to show a variation |:

If P is true, Q is true

If R is true, P is true

therefore, If R is true, Q is true

The traditional 'Scholastic' logic of the Middle Ages drew distinctions between conditional hypotheticals, disjunctive hypotheticals, and conjunctive hypotheticals. All very interesting but the examples above, yours and mine, illustrate the simple hypothetical of your question.

In both examples the conclusion follows logically from the premises. It's possible, of course, to produce hypothetical syllogisms which go wrong and in which the conclusion does not follow from the premises. We try to avoid these ;)-