Fallacy Friday: Denying the Antecedent

This week I will look at the fallacy of denying the antecedent. Before I can elaborate exactly what is involved in this fallacy, it is important to introduce and analyse some valid arguments that are superficially similar.

Modus Ponens
One of the very first valid inferences one learns in logic is modus ponens. To use the well worn example that was repeated ad nauseam when I was learning logic (and one I probably bored my students with too) a paradigmatic example of modus ponens is,

1. If it is raining then the  grass will be wet.

2. It is raining;


3. The grass will be wet.

Put more abstractly, a modus ponens has the form:

1’ If P then Q.

2’ P;


3’ Q.

UnicornModus ponens proceeds with the first premise contending that a conditional statement is true. A conditional statement is a statement about a hypothetical situation; in this case the claim is “if it is raining then the grass will be wet”. Notice that for this conditional to be true, it does not have to actually be raining. On a sunny day it is still true that if it starts raining the grass will be wet. A conditional statement tells us what will be the case if some other thing or event is the case – not what actually is the case.

Conditional statements of the form “if P then Q” have what logicians call an “antecedent” and a “consequent”. P is the antecedent; in the above example the antecedent is the claim, “it is raining”. In a conditional statement one talks about what occurs if the antecedent is true. Q is the consequent; in the example above the consequent is the proposition “the grass will be wet”. The consequent is what is said to be true if the antecedent is correct.

Modus ponens proceeds by first affirming that a conditional statement is true and then affirming the antecedent is true. If both a conditional statement is true and its antecedent is true then it is impossible for the consequent to not also be true. This is obvious upon immediate reflection. If the conditional ‘if P then Q’ is true, and P is true, then Q must also be true. Note, that in a valid modus ponens inference, one affirms the antecedent.

Modus Tollens
A second and related valid inference is modus tollens. Like modus ponens a modus tollens begins by affirming a conditional statement; however, it proceeds by denying the consequent. To use the example above:

1. If it is raining then the grass will be wet.

2’’ The grass is not wet;


3’’ It is not raining.

This has the form:

1’ If P then Q.

2’’ Not Q;


3’’ Not P.

Modus tollens proceeds by noting a conditional statement is true and then denying the consequent of this condition. It follows from this that the antecedent is false. Again this is a valid argument form. If its true that given a certain antecedent obtains that a consequent will follow, and the consequent has not followed, then the antecedent will not obtain.

Both modus ponens and modus tollens formalise valid inferences involving conditional statements. If one has a conditional statement of the form, if P then Q, one can deny the consequent and argue that P is false or one can affirm the antecedent and argue that that Q is true.

Denying the Antecedent
With this background in place we can turn to the fallacy of denying the antecedent. This fallacy occurs when a person denies the antecedent. To return to our example:

1. If it is raining then the grass will be wet.

2’’’ It is not raining;


3’’’ The grass will not be wet.

This argument is invalid because it is possible for the premises to be true and the conclusion false. Imagine it is a hot summer day in Auckland, there is not a cloud in the sky and the sun is beating down; to cool themselves off my children set up a sprinkler on the grass outside and run through it. In this situation the condition ‘if its raining then the grass will be wet’ is true. It is also true that it is not raining yet the grass is wet; it has been drenched by the sprinkler.

This highlights something about conditionals. When one makes a conditional statement, one claims that if the antecedent is true then the consequent is true. One does not, however, necessarily claim that if the consequent is true then antecedent is true. The example above shows this. It is true that rain causes grass to be wet but this does not mean that rain is the only thing that causes wet grass. So one cannot validly claim that a consequent of a conditional is false by arguing that the antecedent is.

This may all sound a bit abstract and the examples of rain and wet grass somewhat trivial. However, it is necessary to use obvious examples to illustrate the logical point. Let us now turn to an example that has been discussed on this blog lately which has generated a reasonable amount of online commentary.

In the recent debate at the University of Notre Dame between Sam Harris and William Lane Craig, Craig offered the following conditional:

1. If God exists then we have a plausible account of (a) the nature of moral goodness and (b) the nature of moral obligation.

As I noted in my review of the debate, one response Harris offered to 1 (b) was to argue that the existence of evil in the world suggests that God does not exist. I also noted that this objection is unsound. Craig’s contention in 1 (b) was a conditional statement that: If God exists then we have a plausible account of the nature of moral obligation. Arguing that God does not exist does not refute this conditional statement since the conditional does not claim that God exists. Just as one can, on a sunny day, make true statements about what would be the case if it were raining, the claim that ‘if God exists then we have a plausible account of moral obligation’ can be true even if God does not exist.

Since the debate, some of Harris’s supporters have suggested Harris’s argument here did provide a compelling reason for rejecting Craig’s claim that there exists a plausible divine command theory account of moral obligation. Craig’s conditional for this was that if God exists then a divine command theory is defensible. However, they contend that God does not exist and so, therefore, a divine command theory is not plausible.

This does not follow and is pretty clearly a case of the fallacy of affirming the antecedent. As both Plantinga and Mark Murphy have noted separately, a divine command theory is, in fact, compatible with atheism. Plantinga notes,

“one might reject theism but accept a divine command ethics, and as a consequence … reject moral realism.”

Similarly, Mark Murphy contends:

“A metaethical theological voluntarist might claim that no normative state of affairs could be made to obtain without certain acts of divine will, but because there is no God, or because there is a God that has not performed the requisite acts of will, no normative states of affairs obtain.”

The point is that one could accept that the most plausible account of moral obligation is that obligations are identical with God’s commands and still deny God exists; and conclude, therefore, that moral obligations do not really exist.  This is no more incoherent than accepting that the best account of the nature of unicorns is that they are magical horses with one horn in the centre of their forehead and then conclude that because no such horses exist that unicorns do not exist.

It should not need belabouring but calling into question the antecedent of Craig’s conditional does not entail a refutation of the consequent. The fact that so many followers of Sam Harris are defending as valid the fallacy of denying the antecedent is mildly amusing but it is not much else.

To summarise, conditional statements are if-then statements; they claim that a consequent is true, if an antecedent is true. One cannot show the consequent is false by denying the antecedent. One can affirm that the antecedent is true and infer, therefore, that the consequent is too, and one can deny the consequent is true and therefore deny the antecedent but denying the antecedent has little effect at all.

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