If P, then Q. If Q, then R. Therefore, If P, then R.
The symbolic form looks like this;
If it is raining, then it is cloudy. If it is cloudy,
then I'll be sad. Therefore, If it is raining, then I'll be sad.
How does it work? Notice the "link" between the two
premises. The necessary condition in the first premise is also
the sufficient condition in the second premise. This makes the "chain." So
it follows that if the antecedent is true in the first sentence,
then the consequent in the second sentence must be true.
Why is it called the "chain argument"? The premises
have an independent clause in common. That independent clause must
be the consequent in one premise and the antecedent in the other.
This common clause makes the "link" which forms the chain.
Why is it called a "hypothetical syllogism"? Notice
that all of the sentences are hypothetical claims. Even the conclusion
is a hypothetical claim. A "syllogism" is a form of argument
featuring two premises and a conclusion. The name "hypothetical
syllogism" is a bit of a misnomer because these arguments
can have more than two premises.