Hypothetical Proposition

A hypothetical proposition is a conditional statement which takes the form: if P then Q. Examples would include:

If he studied, then he received a good grade.
If we had not eaten, then we would be hungry.
If she wore her coat, then she will not be cold.

In all three statements, the first part (If...) is labeled the antecedent and the second part (then...) is labeled the consequent. In such situations, there are two valid inferences which can be drawn and two invalid inferences which can be drawn - but only when we assume that the relationship expressed in the hypothetical proposition is ​true. If the relationship is not true, then no valid inferences can be drawn.

A hypothetical statement can be defined by the following truth table:

Assuming the truth of a hypothetical proposition, it is possible to draw two valid and two invalid inferences:

Affirming the Antecedent

The first valid inference is called affirming the antecedent, which involves making the valid argument that because the antecedent is true, then the consequent is also true. Thus: because it is true that she wore her coat, then it is also true that she will not be cold. The Latin term for this, ​modus ponens, is often used.

Denying the Consequent

The second valid inference is called denying the consequent, which involves making the valid argument that because the consequent is false, then the antecedent is also false. Thus: she is cold, therefore she did not wear her coat. The Latin term for this, ​modus tollens, is often used.

Affirming the Consequent

The first invalid inference is called affirming the consequent, which involves making the invalid argument that because the consequent is true, then the antecedent must also be true. Thus: she is not cold, therefore she must have worn her coat. This is sometimes referred to as a fallacy of the consequent.

Denying the Antecedent

The second invalid inference is called denying the antecedent, which involves making the invalid argument because the antecedent is false, then, therefore, the consequent must also be false. Thus: she did not wear her coat, therefore she must be cold. This is sometimes referred to as a fallacy of the antecedent and has the following form:

If P, therefore Q.
Not P.
Therefore, Not Q.

A practical example of this would be:

If Roger is a Democrat, then he is liberal. Roger is not a Democrat, therefore he must not be liberal.

Because this is a formal fallacy, anything written with this structure will be wrong, no matter what terms you use to replace P and Q with.

Understanding how and why the above two invalid inferences occur can be aided by understanding the difference between necessary and sufficient conditions. You can also read the rules of inference to learn more.