A rule of inference in propositional logic. It is used to conjoin two conditionals when the antecedent of one matches up with the consequent of the other. For example:

If P then Q
If Q then R
Therefore, If P then R.

If Socrates is a man, then he is mortal.
If Socrates is mortal, he is not a god.
Therefore, if Socrates is a man, he is not a god.

Note that this gives you a new if-then statement. We haven't yet shown that Socrates is a man, or that he is mortal, or that he is not a god. We have only shown that if he is a man, then he is mortal.

Sometimes abbreviated to HS.

Back up to Rules of Inference

A question recently appeared in the chatterbox. "Can the hypothetical syllogism be used to join the following statements?"

If the drugs are necessary, we will buy them.
We will get better, if we buy them.

The answer is yes, but you have to switch the second statement around. Somebody messed up the if-then statement. It should actually read "If we buy them, we will get better".

If the drugs are necessary, we will buy them.
If we buy them, we will get better.
Therefore, if the drugs are necessary, then we will get better.

This is exactly the sort of trick that logic professors love to play on their students.

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