Key notes on Hypothetical-Categorical Syllogisms

This is a kind of mixed syllogism in which the major premise is a hypothetical proposition, the minor premise is a categorical proposition and the conclusion is a categorical proposition. Symbolically it is represented as,

If A is B then C is D.

A is B

Therefore, C is D.

A concrete example is as follows:

If it rains, then the ground will be wet.

It has rained

Therefore, the ground is wet.

In a mixed hypothetical-categorical syllogism, the major premise is a hypothetical proposition having two parts – the antecedent and the consequent. In the example cited above, “it rains” is the antecedent and “the ground will be wet” is the consequent.

The antecedent states a sufficient condition for the truth of the consequent and the consequent states a necessary condition for the truth of the antecedent. In the minor premise, either we affirm the antecedent or deny the consequent.

Accordingly, in the conclusion, either we affirm the consequent or deny the antecedent. There are two valid forms of mixed hypothetical syllogism (1) Constructive Hypothetical-Categorical and (2) Destructive Hypothetical- Categorical.

(1) Constructive Hypothetical-Categorical Syllogism:

Constructive hypothetical syllogism is otherwise called as Modus Ponendo Ponens. It states that by affirming the antecedent, we can affirm the consequent. Consider the following hypothetical-categorical syllogism.

If Asok is in Bhubaneswar, then he is in Orissa.

Asok is in Bhubaneswar.

Therefore, Asok is in Orissa.

In the major premise the antecedent. “Asok is in Bhubaneswar” states a sufficient condition for the truth of the consequent “Asok is in Orissa”. Since the antecedent states the sufficient condition for the truth of the consequent, by affirming the antecedent we may validly derive the conclusion which affirms the consequent of the major premise.

But we cannot argue in the reverse order. In other words, by affirming the consequent in the minor premise we cannot affirm the antecedent in the conclusion. If we affirm the antecedent by affirming the consequent the argument will be fallacious. It will be a fallacious application of Modus Ponens. Consider the following example,

If Asok is in Kolkata, then he is in West Bengal.

Asok is in West Bengal.

Therefore, Asok is in Kolkata.

This argument is invalid because it is possible for both the premises to be true and yet the conclusion to be false. By affirming the consequent of the major premise in the minor

Premise we cannot affirm the antecedent of the major premise in the conclusion. This is fallacious form of Modus Ponens. By affirming the consequent we cannot affirm the antecedent. So the rule of validity of the constructive form of hypothetical-categorical syllogism may be stated as follows:

By affirming the antecedent we affirm the consequent, but not conversely. Violation of this rule leads to the fallacy of affirming the consequent.

(2) Destructive Hypothetical-Categorical Syllogism:

The second form of the hypothetical-categorical syllogism is called Destructive Hypothetical-Categorical syllogism. It is otherwise called Modus Tollendo Tollens. The rule concerning Modus Tollens states that denying the consequent, we deny the antecedent. Consider the following argument.

If Asok is in Kolkata, then he is in West Bengal.

Asok is not in West Bengal

Therefore, Asok is not in Kolkata.

Here the major premise is a hypothetical proposition, the minor premise denies the consequent of the major premise and the conclusion denies the antecedent of the major premise. An argument of this form is valid because the consequent of a hypothetical proposition states a necessary condition for the truth of the antecedent.

Since we deny the consequent in the minor premise, we can deny the truth of antecedent of the major premise in the conclusion.

But the reverse is fallacious. By denying the antecedent, if we deny the consequent, we shall commit the fallacy of denying the antecedent. If we proceed by denying the antecedent in the minor premise, the argument in question will be invalid. To illustrate this, let us consider the following argument.

If Gita is a mother, then she is a woman,

Gita is not a mother.

Therefore, Gita is not a woman.

This argument is obviously invalid because Gita can still be a woman even if she is not a mother. Here we commit the fallacy of denying the antecedent.