One of the more practical parts of Introductory Logic is Lesson 31 on Hypothetical Syllogisms. Hypothetical syllogisms of all kinds are a very common form of reasoning, so we should not only be able to identify them quickly, but we should also learn to use the valid forms confidently.

A hypothetical statement is an “if/then” statement, such as this one:

If a creature is a mammal then it is warm-blooded.

A hypothetical statement consists of two independent statements. The antecedent is the statement after the ‘if’. In the example above, the antecedent is “A creature is a mammal.” The consequent is the statement after the ‘then’. In our example, the consequent is “It is warm blooded.” Notice that the antecedent and the consequent are complete statements. Notice also that they do not include the ‘if’ or the ‘then’ in them.

A mixed hypothetical syllogism starts with a hypothetical premise, then it has a regular premise, and a regular conclusion. The four types of mixed hypothetical syllogisms are named for what occurs in the premise after the hypothetical premise.

If in the second premise you affirm the consequent, the name of the syllogism is simply Affirming the Consequent:

If a creature is a mammal then it is warm blooded.    ← hypothetical
A bird is warm blooded.      ←  affirm the consequent
Therefore, a bird is a mammal.    ←  conclusion

You see from this counterexample that affirming the consequent is an invalid form of reasoning.

If in the second premise you deny the antecedent, the name of the syllogism is simply Denying the Antecedent:

If a creature is a mammal then it is warm blooded.    ← hypothetical
A bird is not a mammal.      ←  deny the antecedent
Therefore, a bird is not warm blooded.    ←  conclusion

You see that denying the antecedent is also invalid.

If in the second premise you affirm the antecedent, this syllogism has the Latin name Modus Ponens, which means something like “Way of putting forward,” because the second premise puts forward the antecedent:

If a creature is a mammal then it is warm blooded.    ← hypothetical
A dog is a mammal.     ← modus ponens (put forward the antecedent)
Therefore, a dog is warm blooded.   ← conclusion

Finally, if in the second premise you deny the consequent, this form has the Latin name Modus Tollens, which means something like “Way of taking back,” because the second premise takes back the consequent

If a creature is a mammal then it is warm blooded.    ← hypothetical
A lizard is not warm blooded     ← modus tollens (take back the consequent)
Therefore, a lizard is not a mammal.   ← conclusion

If you can keep this in mind, you should always be able to remember the names of the different forms of mixed hypothetical syllogisms, whether you are identifying someone else’s or creating your own.