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Fallacy Friday: Denying the Antecedent

April 30th, 2011 by Matt

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;

Therefore:

3. The grass will be wet.

Put more abstractly, a modus ponens has the form:

1’ If P then Q.

2’ P;

Therefore:

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;

Therefore:

3’’ It is not raining.

This has the form:

1’ If P then Q.

2’’ Not Q;

Therefore:

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;

Therefore:

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.

Example
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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31 responses so far ↓

  • Or one can argue that even if the antecedent is true, the consequent is false.

    In this case

    IF evil exists n the world THEN we do not have a plausible account of moral goodness.

    evil exists in the world

    THEREFORE we do not have a plausible account of moral goodness.

    THEREFORE God does not exist.

    QED

  • I think you missed the class on non sequitur.

    Without a plausible account of moral goodness, we cannot say that evil exists. Therefore your “argument,” such as it is, is self refuting.

  • […] I’ve just been checking the links. The link to the Flannigans now works.  They are continuing their discussion of logical fallacies by discussing denying the antecedent. […]

  • Sinner, I am not sure how your comment addresses the topic of the post.

    But it seems the argument you offer is flawed. The first premise is false. You suggest that if evil exists then there can be no account of moral goodness. But I am inclined to think the word “evil” in fact presupposes existence of evaluative concepts such as good and evil, so if evil existed goodness would also have to exist.

    Second, your conclusion that God does not exist does not follow from the premises you offer. God does not occur in the premises, so the argument is a non sequitur.

  • Sinner,
    Your first premise is wrong, but even if it were true the problem would look like what I outline below:

    Your Argument
    1. If evil exists in the world then we do not have a plausible account of moral goodness.
    2. Evil exists in the world.
    3. Therefore, we do not have a plausible account of moral goodness.

    A Resulting Argument
    1. If we do not have a plausible account of moral goodness, then we are incapable of determining between moral goodness and badness.
    2. We do not have a plausible account of moral goodness.
    3. Therefore we are incapable of determining between moral goodness and moral badness.

    Another
    1. If we are incapable of determining between moral goodness and moral badness, then we are incapable of forming successful arguments based on the reality of moral badness.
    2. We are incapable of determining between moral goodness and moral badness.
    3. Therefore, we are incapable of forming successful arguments based on the reality of moral badness.

    A Third Argument
    1. If the results of an argument leads to the contradiction of one of its premises then the argument fails.
    2. The conclusion of your initial argument, “Therefore, we do not have a plausible account of moral goodness,” leads to the contradiction of the second premise, “evil exists in the world.”
    3. Therefore, your argument fails.

  • A related problem,

    1. An action is evil if and only if it all else being equal its good to avoid it.

    I take 1. to be a conceptual truth. But it entails that evil can only exist if goodness exists. If all statements about X being good are false, then nothing is good to avoid, in which case nothing is evil.

  • It is said that God himself created all. Thus, he in his infinite wisdom, created the Devil, whom is God’s immortal enemy. It would be fair to assume that since God created his own adversary, he can create other Gods.

    Perhaps now there is a God that has not killed millions with natural disasters, both in recent years and in the Bible, a God who in the Bible, which most regard as truth, killed and destroyed hundreds of times more than the Devil himself.

    Perhaps now there is a perfect vision of God. His name is Allah.

  • It is quite simple in my eyes really. God just can’t exist. Answer this, who created god? How was god able to create the universe? I mean if he can do these things, creating such complex things byond any human understanding, surely he can create a perfect world, with no suffering pain or loss. Of course, you may say it’s heaven, but then why place us humans who he loves on earth and not heaven? Why let such evil place like hell exist? It seems like a game doesn’t it? He allows for good and evil to exist, yet he dislikes evil, and doesn’t correct it even though he can do it at will. Just like a little child. I believe religion is just early mans ATTEMPT to explain our existance, as science was not even slightly advanced as today.

  • Chen, I will be gracious and assume you’re not trolling.

    Firstly, God did not begin to exist. God by definition is eternal. Since time began with the universe, then the creator of the universe is likewise the creator of time. Intuitively you should see that someone who creates time is not in turn subject to it.

    Consequently the question, who created God, is a category error, much like, to whom is the bachelor married?

    Could God have created a world different to this one? I suppose he could have created one worse.

    Let’s be picky though. You claim that suffering, pain and loss are morally bad things. From where do you derive this belief in “moral badness.” An atheistic world is one of brute facts. Biological units are conceived, born, go about their business, experience sensory stimulus of various levels of intensity, reproduce, and die. You can measure intensity of sensation, it may be pleasurable or not but you can’t determine if the sensation is morally bad.

    Hell is not an evil place, it is a place for evil. God is good, in superlative quantity. He is described among other things as a consuming fire. For the unregenerate man to enter his presence would be akin to being a balloon full of petrol thrown on a bonfire. It would be messy. God created hell because he won’t force anyone to be in his kingdom if they don’t want to be.

    As for why he doesn’t remove evil immediately? Only someone who has an unrealistic view of their own goodness could ask that. If he decided to remove evil now, would any human being still be standing?

    Science is a tool for examining how the world works. It advances by building on the knowledge of yesterday, often discarding it in the process. Given that, I hold that science cannot be taken as real knowledge of the world, as we cannot know if what we now believe to be true will not be overthrown tomorrow. It is a brittle stick you claim to use against Christianity.

  • Belial, in Jewish tradition the devil is basically God’s prosecuting attorney (which gives you an idea of God’s opinion of lawyers). He did have dominion over the Earth, and seems to have suffered from a swollen head (like most of his devotees).

  • Jason,

    I was obviously referring to Christianity. Don’t change the subject, it shows how ignorant you are to the truth.

    May Allah be merciful.

  • Jason,
    You say, “Hell is not an evil place, it is a place for evil”, but this is essentially the same thing, if there if a place for evil, then it has to be a place of evil.

    You say, “Could God have created a world different to this one? I suppose he could have created one worse” but I talking the opposite of this, I’m talking about creating a BETTER world than earth.

    You say, “You claim that suffering, pain and loss are morally bad things” yes, I do in some way, but my point is, why do they exist? Why cannot a human being or an animal be in peace all the time, be without pain, be without loss (loss as in loss that brings negative emotions).

    You say, “God created hell because he won’t force anyone to be in his kingdom if they don’t want to be” then why create such two big contrasting things? Heaven, or hell. Why not let US create OUR own world where WE get the choice to do what WE want. That way, there is no need for hell, and there is no need for heaven. In fact, why not start out like this? In peace, in our OWN world.

    You say “If he decided to remove evil now, would any human being still be standing?” this is a God who created all, he who can do ANYTHING and EVERYTHING, he can easily rid evil even though we have done it ourselves, he can just make us forget, or come up with another solution that we cannot think off, he is God after all.

    You say “Science is a tool for examining how the world works. It advances by building on the knowledge of yesterday, often discarding it in the process. Given that, I hold that science cannot be taken as real knowledge of the world, as we cannot know if what we now believe to be true will not be overthrown tomorrow. It is a brittle stick you claim to use against Christianity” Science is not a ‘tool’. It is study, it is knowledge. Knowledge that has brought man to where he is now, taken man to space, given man easier life, given man a glimpse of his existance, given man more reason and more inspiration and more ideas to take science itself further. Evolution itself disregards god, and a large percentage of people widely accept it not a theory but fact.

    I am not here to ‘troll’, I am here to debate with people like you and share my views.

  • “It would be fair to assume that since God created his own adversary, he can create other Gods.”

    You are confusing me here, cause this simply doesnt follow. If Satan chooses to oppose God that doesnt make him a god or equal with God or in anyway as powerful as God. That God created an angel who chose to become God’s adversary doesnt mean he was created as an adversary.

    Interesting that you identify yourself with one of Satans underlings a character whose name means “worthless”

  • Jeremy,

    “If Satan chooses to oppose God that doesnt make him a god or equal with God or in anyway as powerful as God.”

    God created all, so he created Satan. Satan files under the term “all”. But I guess God forgot to brainwash Satan, so he started opposing God. Which brings me to a point; why can’t God, who is all powerful, defeat Satan… an entity he himself created. He kills humans every day…

    I choose this name because as you say, we are all worthless in the universe.

  • As I said, you seem to have a superficial understanding of these things Matt.

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

    2’’’ It is not raining;

    Therefore:

    3’’’ The grass will not be wet.

    Sounds good to me but only if you insert the word “probably.” As in:

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

    2’’’ It is not raining;

    Therefore:

    3’’’ The grass will probably not be wet.

    We cannot use logic about the world we live in without the word “probably.” It is a word that is implicit in “matters of fact.” Now it might be the case that someone poured a bucket of water on a section of the grass or that there is dew of the grass, but we cannot use logic to settle this question unless we specific the circumstances, and if we’re talking about a warm summer day then if it’s not raining then the grass is probably not wet.

    Okay, my friend? I really could go through your whole fallacy Friday’s and show you some things. I just don’t have the time right now.

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

    2’’’ It is not raining;

    Therefore:

    3’’’ The grass will probably not be wet.

    Seriously John?

  • We cannot use logic about the world we live in without the word “probably.” It is a word that is implicit in “matters of fact.” Now it might be the case that someone poured a bucket of water on a section of the grass or that there is dew of the grass, but we cannot use logic to settle this question unless we specific the circumstances, and if we’re talking about a warm summer day then if it’s not raining then the grass is probably not wet.

    This comment actually destroys your point, because you give an example where its not raining and yet (given the bucket of water) highly probable the grass is wet. You also note that wether its probable or not that the grass is wet depends on various background facts.

    What I think you getting at is the ideas expressed in confirmation theory, where its stated that if one hypothesis H1 leads you to expect D and another hypothesis H2 leads you to expect not D, and D is the case, then the probability of H1 is raised relative to H2.

  • Haha, yeah John, or it could even be a case that the sprinkler is on. Kind of like it says in the article.

  • Belial
    “I choose this name because as you say, we are all worthless in the universe”

    speak for yourself, mate, but you do have my sympathy if you really feel that about youself.

    Why doesnt God kill Satan, why doesnt He kill you? you seem to be as much in opposition as Satan.

    “You see, at just the right time, when we were still powerless, Christ died for the ungodly. Very rarely will anyone die for a righteous man, though for a good man someone might possibly dare to die. But God demonstrates his own love for us in this: While we were still sinners, Christ died for us. Romans 5

    Since we have now been justified by his blood, how much more shall we be saved from God’s wrath through him! For if, when we were God’s enemies, we were reconciled to him through the death of his Son, how much more, having been reconciled, shall we be saved through his life!”

    So no matter what you think of yourself, God loves you and seeks to be reconciled with you, so much so in fact that He organised the way and means to do so.

  • You seem to have ignored the first condition: the statement, “if p then q” must be a true statement.

    The statement “If God exists then we have a plausible account of (a) the nature of moral goodness and (b) the nature of moral obligation.” is not necessarily true to begin with. You would need to prove that. That is you need to prove that P is true, that Q is true, AND that P implies Q. You haven’t.

  • Matt,

    I realize this is nit-picking, but aren’t you actually talking about the fallacy of “affirming the consequent?” Perhaps it’s just my ignorance, but I’ve never heard of “denying the antecedent.”

    You’re talking about the following, right?

    If P, then Q.
    Q.
    Therefore, P.

  • Never mind. I just reread the relevant part and realized my mistake. You’re saying:

    If P, then Q.
    Not P.
    Therefore, not Q.

  • Glenn asks:

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

    2’’’ It is not raining;

    Therefore:

    3’’’ The grass will probably not be wet.

    Seriously John?

    Yes. Dead. It’s always possible the grass is not exposed to the rain, say it’s a baseball field with a cover on it.

    We cannot use logic to tell us about matters of fact. Probability and circumstance always dictate the probabilities, always.

  • “speak for yourself, mate, but you do have my sympathy if you really feel that about youself”

    Think about your place in such a vast, perfect universe. You are nothing. Humanity, life itself! is nothing but an abnormality of the universe. All we do is destroy everything we come into contact with. We should not exist. And people like you cannot even comprehend this. I understand the meaning of life; that it should not have occured, it is simply a random event in the fabric of the universe.

    Were I given the option to instantly erase life, would I do it? Would you?

    I would.

  • Zaybu,

    Your correct that the a conditional “If God exists then we have a plausible account of (a) the nature of moral goodness and (b) the nature of moral obligation.” Must be true. However, your comment

    You would need to prove that. That is you need to prove that P is true, that Q is true, AND that P implies Q. You haven’t

    Is just mistaken, to show that a conditional is true you do not need to prove the antecedent is true, nor do you need to prove the consequent is true. What you need to show is that if the antecedent were true the consequent would be. That’s compatible with both being false.

    Nor do you have to show the antecedent implies the consequent.

    Take the claim, If I was shot in the head yesterday I would not be here today. That’s a true conditional, it don’t need to prove I was actually shot in the head or that I am not here today to show its true. Nor does the claim I am shot in the head “imply” I would not be here, that’s a causal not a logical relationship. All I have to know is that if people are shot in the head they aren’t around the next day.

    Sorry but denying the antecedent is a fallacy, that’s first year logic. It doesn’t become less a fallacy because its Sam Harris, your an atheist, and the argument criticises Craig.

  • *Facepalm*
    Why do people feel the need to speak with such an arrogant attitude as if they know what they talk about, when they haven’t even grasped simple logic?!
    Matt I don’t know how you don’t get exhausted dealing with people like that!!

  • […] my last Friday Fallacy post, I looked at the fallacy of  denying the antecedent. There I discussed conditional statements, statements of the form “if P then Q”. Examples […]

  • I really liked the example that Divine Command Theory could be compatible with atheism. Would the counterpart for Christians be affirming that moral relativism is compatible with theism? In other words, could we argue that if God does not exist, then there are no objective moral values or duties? In fact, I often do argue this point, as it help atheists see that if they want to affirm the existence of objective moral facts and obligations, then this is only possible on theism. Hopefully I’m not committing a logical fallacy!
    -Neil

  • Neil, one could argue that when one analysises the role moral properties such as rightness and wrongness in our discourse. They are best understood as something like divine commands, seeing nothing like divine commands exist, it follows our discourse is erroneous and refers to something that does not exist.

    I think some philsophers as Mackie, Anscombe, Richard Taylor, actually do argue something a bit like this, arguably so does Jean Paul Satre.

  • “Fallacy Friday: Denying the Antecedent | MandM” was indeed
    a very good article. If perhaps it possessed even more images it would be quite possibly more beneficial.

    Regards -Elana