# Formal Fallacies

## Affirming the Consequent and Denying the Antecedent

on
Share:

We will close out the logical fallacy series with two of the most common fallacies that occur in arguments about origins: affirming the consequent and denying the antecedent. These are formal fallacies because the mistake in reasoning stems from the structure (the form) of the argument. It is well worth the effort to study formal fallacies and their corresponding terminology because these two fallacies are extremely common—perhaps the two most common fallacies committed by evolutionists.

Formal deductive arguments can be put into a symbolic notation with letters representing the propositions. Consider the proposition, “If it is snowing, then it must be cold outside.” This proposition has the basic form: “If p, then q.” Any proposition that has that form (“if p, then q”) is called a “hypothetical proposition.” This is because it’s not asserting either p or q; it is merely stating that if p hypothetically were true, then q would have to be true as well. In a hypothetical proposition the first part (p) is called the antecedent, and the second part (q) is called the consequent. In our example, “it is snowing” is the antecedent, and “it must be cold outside” is the consequent.

If an argument has two premises, only one of which is hypothetical, then it is called a “mixed hypothetical syllogism.” Here is an example:

1. If it is snowing, then it must be cold outside.
2. It is snowing.
3. Therefore, it is cold outside.

In this argument, the first premise (if p, then q) is hypothetical. The second premise (p) is not hypothetical; it asserts that it is indeed snowing. And the conclusion is q. Since the second premise affirms that p (the antecedent) is true, this type of argument is called “affirming the antecedent” and is perfectly valid. (Recall, “valid” means that if the premises are true, so is the conclusion). The Latin name for this type of argument is modus ponens, which means the “method of affirming.”

## Affirming the Consequent

There is a fallacy that is very similar to modus ponens and has this form:

1. If p, then q.
2. q.
3. Therefore, p.

We can see that this is a fallacy by substituting phrases for p and q.

1. If it is snowing, then it must be cold outside.
2. It is cold outside.
3. Therefore, it must be snowing.

But clearly just because it is cold outside does not necessarily mean that it must be snowing. So, this argument is invalid. Since the second premise affirms that the consequent (q) is true, this fallacy is called “affirming the consequent.” Here are some common examples:

1. If evolution were true, we would expect to see similarities in DNA of all organisms on earth.
2. We do see similarities in DNA of all organisms on earth.
3. Therefore, evolution is true.

The evolutionist making such an argument has failed to recognize that creationists would also expect to see similarities in DNA of all organisms, since the original kinds were made by the same Creator.

1. If the big bang is true, then we would expect to see a cosmic microwave background.
2. We do see a cosmic microwave background.
3. Therefore, the big bang must be true.

This big bang supporter has failed to consider other possible causes for the cosmic microwave background. His argument is an example of the fallacy of affirming the consequent.

Another mixed hypothetical syllogism has the following form:

1. If p, then q.
2. Not q.
3. Therefore, not p.

This is a valid argument as can be seen by substituting the phrases for the symbols.

1. If it is snowing, then it must be cold outside.
2. It is not cold outside.
3. Therefore, it is not snowing.

Since the second premise denies that the consequent (q) is true, this valid argument is called “denying the consequent” or, in Latin, modus tollens, which means the “method of denying.”

## Denying the Antecedent

As before, there is an argument that is superficially similar to modus tollens, but is actually a fallacy. It has this form:

1. If p, then q.
2. Not p.
3. Therefore, not q.

We can see that this is fallacious by substituting the phrases for the symbols:

1. If it is snowing, then it must be cold outside.
2. It is not snowing.
3. Therefore, it is not cold outside.

But clearly, it could be cold outside and still not snow. So, the argument is invalid. Since the second premise denies that the antecedent (p) is true, this fallacy is called “denying the antecedent.” Here are some examples:

1. If we found dinosaurs and humans next to each other in the same rock formation, then they must have lived at the same time.
2. We do not find them next to each other in the same rock formation.
3. Therefore, they did not live at the same time.

This denies the antecedent and is fallacious. There could be several reasons why dinosaur fossils are not normally found next to human fossils; perhaps dinosaurs and people typically did not live in the same area (as one hypothetical explanation).

1. If God were to perform a miracle in front of me right now, then that would prove He exists.
2. God is not performing a miracle in front of me right now.
3. Therefore, He doesn’t exist.

Again, this denies the antecedent. God is under no obligation to perform a miracle at the whim of one of His creations. Nor is it likely that the atheist would accept a given miracle as legitimate anyway—preferring to trust that future studies will reveal that the event is explainable by natural law.

## Summary

 (1) If p, then q. (2) p. (3) Therefore, q. valid: modus ponens (1) If p, then q. (2) q. (3) Therefore, p. fallacy of affirming the consequent (1) If p, then q. (2) Not q. (3) Therefore, not p. valid: modus tollens (1) If p, then q. (2) Not p. (3) Therefore, not q. fallacy of denying the antecedent

## Conclusions

It is the obligation of the Christian to be rational—to pattern our thinking after God’s (Isaiah 55:7–8). We are to be imitators of Him (Ephesians 5:1) and to think in a way that is consistent with God’s logical nature (Romans 12:2).

Not only do we belong to God as his creations, but He has redeemed us by His Son. Our commitment to Christ, therefore, must extend to all aspects of our life. We are to love the Lord with all our heart, soul, strength, and mind (Luke 10:27).

We hope that you have enjoyed the Logical Fallacies series and that the information presented here will help in your defense of the faith. For more information on logical fallacies—including many not covered in this series—consider reading The Ultimate Proof of Creation, which has two chapters on how to spot fallacies. A good textbook on logic or logical fallacies may also be helpful, even if it is not written from a Christian perspective.1 Christian apologist Dr. Greg Bahnsen also has a lecture series on logic and critical thinking that may be very helpful and is available from the Covenant Media Foundation.

## Footnotes

1. I recommend Introduction to Logic—an excellent textbook on logic by Copi and Cohen. I also recommend With Good Reason by S. Morris Engel, which is a book on informal fallacies.