Winning With Mixed Hypotheticals

Why is it important to understand mixed hypothetical reasoning? It is commonly used and misused in everyday thinking. Anyone who wants to learn how to identify truth, faulty reasoning, or bad philosophy would be very wise to know how to use mixed hypothetical syllogisms. Mixed hypotheticals will help your mind learn how to think about things more properly. Our Heavenly Father has intended us to reason more properly, and this can be done when using mixed hypotheticals. Mixed hypotheticals will help you discover more beauty about the heart of logic. Aristotle gave several reasons why mixed hypotheticals are important. They can be used for practical thinking, moral reasoning, speculations, or productivity in technology.

What is a hypothetical? A hypothetical proposition is a declarative compound sentence, “If…then…” The first part beginning with “If” is called the antecedent. The second part of the hypothetical proposition after “then” is called the consequent.

Example of a hypothetical:

If the wind blows, then the paper will fall off the desk.

If (antecedent), then (consequent).

What is a mixed hypothetical syllogism?

A mixed hypothetical syllogism includes three sentences. It includes: 1.) a hypothetical proposition, 2.) affirming or denying the first or second part of the hypothetical, and 3.) concluding with the opposite part of the hypothetical in the affirmative or negative.

There are four possibilities of a mixed hypothetical:

  1. If the wind is blowing, then the paper will fall off the desk. The wind is blowing. Therefore, the paper will fall off the desk. (Valid)
  2. If the wind is blowing, then the paper will fall off the desk. The paper fell off the desk. Therefore, the wind is blowing. (Invalid)
  3. If the wind is blowing, then the paper will fall off the desk. The wind is not blowing. Therefore, the paper will not fall off the desk. (Invalid)
  4. If the wind is blowing, then the paper will fall off the desk. The paper will not fall off the desk. Therefore, the wind is not blowing.  (Valid)

The first possible form above is called Affirming the Antecedent. The second possible form above is called Affirming the Consequent. The third possible form above is called Denying the Antecedent. The fourth possible form above is called Denying the Consequent. As you can see, there are two valid forms and two invalid forms. The invalid forms are invalid because there could be another cause.

Once a person can understand a mixed hypothetical syllogism, it will be much easier to see how it is valid to affirm the antecedent or deny the consequent. In addition, it is invalid to affirm the consequent or deny the antecedent. Let us take a closer look at another example. We will see how useful it can be in a conversation with a skeptic. The following mixed hypothetical is a variation of the Cosmological Argument.

1.) If the universe had a beginning, then the universe had a cause.

2.) The universe had a beginning.

3.) Therefore, the universe had a cause.

As you can see, affirming the antecedent is a valid form of reasoning. In order to accept premise two as a truthful proposition, it will be necessary for an apologist to give strong reasons why we know premise two to be true. A skeptic who denies the antecedent might end up denying the conclusion, but if so, he has used an invalid form of reasoning. Quite often, the skeptic is actually denying the consequent as a means to deny the antecedent, which is a valid form. However, denying the consequent, “the universe has no cause,” is impossible to prove, which leaves the original mixed hypothetical still standing. In a conversation, however, the skeptic will usually try to say premise two is false – in order to invalidate the argument. This is why it is so important for the apologist to give strong reasons why premise two is true; premise two from the mixed hypothetical Cosmological Argument mentioned above.

Skeptics often deny premise two without examining all the reasons why we know premise two is true. If a skeptic wants to be intellectually honest, he must examine all the reasons given by an apologist. We can know the high probability of premise two even if skeptics doubt it.

Peter Kreeft, Socratic Logic (South Bend, Indiana: St. Augustine’s Press, 2004), pp. 290-296.

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