Intro to Logic: Affirming the Consequent

by Luke Muehlhauser on April 3, 2010 in Intro to Logic

logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

Time for another fallacy! Today we discuss affirming the consequent. It looks like this:

  1. If P then Q.
  2. Q.
  3. Therefore, P.

For example:

  1. If Bill Gates owns Fort Knox, then he is rich.
  2. Bill Gates is rich.
  3. Therefore, he owns Fort Knox.

Nobody would be fooled by the Fort Knox argument, but they might be fooled by an argument like this:

  1. If I have the flu, then I have a sore throat.
  2. I have a sore throat.
  3. Therefore, I have the flu.1

Of course, you may have a sore throat without having the flu, so the above argument is invalid. Now we might say that, together with other facts about you, the best explanation for your sore throat is that you have the flu. But that is a different kind of argument, an abductive argument, of the kind used in science. The above argument is still deductively invalid.

Here is another example of the fallacy of affirming the consequent:

Never has a book been subjected to such pitiless search for error as the Holy Bible. Both reverent and agnostic critics have ploughed and harrowed its passages; but through it all God’s word has stood supreme…. This is proof…that here we have a revelation from God; for…if God reveals himself to man…, he will preserve a record of that revelation in order that men who follow may know his way and will.2

The argument is built like this:

  1. If God reveals himself in the Bible, the Bible will be preserved despite its critics.
  2. The Bible is preserved despite its critics.
  3. Therefore, God reveals himself in the Bible.

But obviously this does not follow. Affirming the consequent of an “If… then” statement cannot not prove the ‘if’ part. Rather, affirming the ‘if’ (antecedent) part of an “If… then” statement can prove the ‘then’ part. Such an argument would look like this:

  1. If Socrates is a man, then Socrates is mortal.
  2. Socrates is a man.
  3. Therefore, Socrates is mortal.

This argument affirms the antecedent, and it is a valid argument form. Not so with affirming the consequent.

The argument about God and the Bible can be reworked into a valid ‘affirming the antecedent’ form:

  1. If the Bible is preserved despite its critics, then God reveals himself in the Bible.
  2. The Bible is preserved despite its critics.
  3. Therefore, God reveals himself in the Bible.

This argument is logically valid because it affirms the antecedent, not the consequent. Now comes the second test: Are the premises true? Nobody will deny premise (2). So, believers and unbelievers will have to argue over whether or not we have good reason to think that premise (1) is true.

Biconditional premises

There is an apparent exception to the rule against affirming the consequent. Consider this argument:

  1. If Disneyland is in California, then Disneyland is in Los Angeles.
  2. Disneyland is in Los Angeles.
  3. Therefore, Disneyland is in California.

The inference here is valid, and yet we appear to have affirmed the consequent! What is different here?

The difference is that premise (1) is “biconditional” kind of premise. For a biconditional premise, either both parts are true, or both are false. They are not independent. Premise (1) states that “If Disneyland is in California, then Disneyland is in Los Angeles.” But because premise (1) is biconditional, it also entails that “If Disneyland is in Los Angeles, then Disneyland is in California.” The above argument works only because we can reverse the conditional. When we do so, the argument becomes:

  1. If Disneyland is in Los Angeles, then Disneyland is in California.
  2. Disneyland is in Los Angeles.
  3. Therefore, Disneyland is in California.3

And now it becomes obvious why the argument is valid: we are affirming the antecedent, which is a valid argument form.

Another example:

  1. If he is not inside, then he’s outside.
  2. He’s outside.
  3. Therefore, he is not inside.4

The reason this works is because the first premise is biconditional, and entails that “If he’s outside then he’s not inside.” Again, what we are really doing is affirming the antecedent:

  1. If he’s outside, then he is not inside.
  2. He’s outside.
  3. Therefore, he is not inside.

Logic can be tricky!

  1. Thanks to Wikipedia for these examples. []
  2. Hillyer Straton, Baptists: Their Message and Mission (1941), p. 49. []
  3. This example from commenter TaiChi. []
  4. This example from Wikipedia. []

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{ 21 comments… read them below or add one }

Exploring the Uknowable April 3, 2010 at 7:20 am

What do you mean by “The Bible is preserved despite its critics”?

If you mean that it’s still around, okay, but if you mean that biblical criticism has unveiled NO reasons to doubt its veracity, then I’m not so sure about that.

Thanks.

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poban April 3, 2010 at 7:24 am

If the Bible is preserved despite its critics, then God reveals himself in the Bible.
————————————
I think this premise is not valid. Just because biblen is preserved despite its criticism doesnt mean that god reveals himself in biblen. And same logik could be applied to Quran, Vedas or Gospels of FSM.

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lukeprog April 3, 2010 at 7:25 am

poban,

A premise cannot be valid or invalid. Premises are true or false. Only arguments can be valid or invalid.

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Erika April 3, 2010 at 7:32 am

Good post! I think that there are a couple reasons people fall into this fallacy. First, as you point out, it may very well be the case that “if P then Q” and “Q” then “P” is more probable.

A second reason people might be prone to this fallacy is that in casual conversation “if P then Q” often is used to mean “P if and only if Q”.

(Come to think of it, a discussion why certain fallacies are so persistent would be an interesting addition to this series.)

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Justfinethanks April 3, 2010 at 8:13 am

Another reason why this kind of fallacy is persistent is that this kind of reasoning is technically needed in order to conduct science.

http://en.wikipedia.org/wiki/Abductive_validation#Abductive_validation

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Ryan April 3, 2010 at 8:14 am

I don’t think affirming the consequent is always fallacious. On page 207 of “29 Evidences for MacroEvolution” Douglas Theobald states:

“all scientific conclusions rely upon the fallacy of affirming the consequent, and in doing so they rely upon inductive extrapolation.”

http://www.talkorigins.org/pdf/comdesc.pdf

But of course it would be absurd to say that scientific conclusions are therefore untrustworthy.

You know, I’ve read that Bayes’ Theorem logically proves “that a hypothesis is confirmed by any body of data that its truth renders probable”
http://plato.stanford.edu/entries/bayes-theorem/

But the problem is that even if when we take all the theories that we are aware of and see which one renders the evidence we have most probable, in most cases it will not be possible to become aware of all the logically possible theories that could explain the evidence, and therefore we can’t know if there is a better theory that we have yet to think of.

A good example is the alleged “fine tuning” of the laws of physics. It is quite concievable that tomorrow someone will come up with a new account of the fine tuning that no one else thought of. Especially in light of the fact that so many accounts of the fine-tuning — Like Smolin’s cosmological natural selection and Paul Davies’ observer selection conjecture — wouldn’t have been the kind of things I would have ever concieved of before I had heard of them.

So, even if we were able to say that Cosmological Natural Selection (for example) was the most probable explanation for the fine-tuning, we’d have to keep in mind that it is only the most probable KNOWN explanation, because there maybe another theory that is even more probable which we are not yet aware of.

So, how can we be justified in believing scientific conclusions? Are we justified in believing them at all? I think so. I define a belief as a proposition which one thinks and acts as if true. A belief is a proposition which you RELY UPON to be true. And what better proposition to rely upon than the best known explanation [of whatever phenomenon]?

If you’re in a position where you need to choose a theory to rely on (as scientists are, often times) then why not rely on the best explanation found to date? That’s my solution to the problem. Does all this make sense? What do you think, Luke?

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Erika April 3, 2010 at 8:23 am

Ryan, I think that Luke actually answers your object.

Now we might say that, together with other facts about you, the best explanation for your sore throat is that you have the flu. But that is a different kind of argument. The above argument is still deductively invalid.

Luke is claiming that affirming the consequent is a fallacious deductive argument. What you are discussing would not, I think, be in an Introduction to Logic series (or maybe not a logic series at all, at that point).

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lukeprog April 3, 2010 at 8:43 am

Erika,

Exactly. That’s why I put that paragraph there.

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Lorkas April 3, 2010 at 10:00 am

It frustrates me when people I’m talking to make the following error in thinking:

1) If X argument is sound, then God exists.
2) God exists.
3) Therefore, X argument is sound.

or, related but framed as affirming the consequent instead:

1) If X argument is sound, then God exists.
2) X argument is not sound.
3) Therefore, God does not exist.

Both are obviously invalid arguments, but believers defend terrible arguments all the time if the arguments are supposed to establish the existence of God. Atheists seem to fall into this trap as well, thinking that disproving all the arguments for God can logically prove that God exists.

To believers: even assuming that you’re right, and God exists, that doesn’t mean that all of the arguments for his existence are sound. Please abandon bad arguments when you see that they are bad arguments.

To atheists: please stop acting as though discrediting an argument for God is the same as demonstrating that God does not exist.

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MauricXe April 3, 2010 at 11:06 am

To atheists: please stop acting as though discrediting an argument for God is the same as demonstrating that God does not exist.

I don’t think that’s the thinking process for atheists, at least a vast majority of them. I believe the typical response from an atheist to a fallacious argument is:

“Thus your argument does not provide evidence for God’s existence.”

Not:

“Thus your fallacious argument disproves the existence of God.”

Most atheist will say that they find a belief in god unwarranted because the arguments for his existence are fallacious and unconvincing. That is not the same as saying, he therefore does not exist because the arguments are fallacious and unconvincing.

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Rob April 3, 2010 at 11:40 am

Typo:

“Affirming the consequent of an “If… then” statement cannot not prove the ‘if’ part.”

Take the “not” out.

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svenjamin April 3, 2010 at 11:57 am

I have thought for a while that affirming the consequent and arguments to the best explanation looked suspiciously similar, so I wrote out the following logically valid (but slightly more complicated) semi-deductive version of an argument to best explanation. I’m sure it has either been done before or disposed of by someone else, but this is what I came up with, for what it’s worth.

When deciding between alternative explanations p and r for X,
where
p–>q
r–>s
q and ~s,
p is the preferred explanation.

It was something like that. I don’t remember exactly.

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TaiChi April 3, 2010 at 2:20 pm

Here’s an example of affirming the consequent that’s valid:

1. If Disneyland is in California, then Disneyland is in Los Angeles.
2. Disneyland is in Los Angeles.
3. So, Disneyland is in California.

It works because the antecedent and consequent are not logically independent of each other. Usually, “p -> q” is true when (i) p is true and q is true, or (ii) p is false but q is true, or (iii) p and q are both false. So affirming q still leaves open whether (i) or (ii) is correct. But if “q ->p”, as it does in the example, then only (i) can be correct, and so the inference is truth-preserving.

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lukeprog April 3, 2010 at 3:19 pm

TaiChi,

Good point, I’ll add that clarification.

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Lorkas April 3, 2010 at 4:42 pm

p–>q
r–>s
q and ~s,
p is the preferred explanation.It was something like that. I don’t remember exactly.  

Well, one reason is that

r->s
!s

can be used to conclude !r. If the two above statements are true, then !r is definitely true as well.

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TaiChi April 3, 2010 at 6:54 pm

Nice update Luke – that’s wonderfully clear.

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Ryan April 3, 2010 at 10:40 pm

Ryan, I think that Luke actually answers your object.Luke is claiming that affirming the consequent is a fallacious deductive argument. What you are discussing would not, I think, be in an Introduction to Logic series (or maybe not a logic series at all, at that point).  (Quote)

I know. Nevertheless, what I mentioned is still often thought of as a problem for abductive reasoning, and I so I wanted to write something about it in the comments.

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lukeprog April 4, 2010 at 2:02 am

Thanks! I do try.

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wissam October 27, 2010 at 1:49 pm

@TaiChi

You’re mistaken. (~p & q) –> ~(p–>q). So if p is false and q true, then the conditional p–>q would be false (denying the antecedent and/or affirming the consequent).

Anyway, biconditionals are written as “if and only if” pq.

luke, how do you keep up with this blog? You must be very busy! Anyway, this shows how enthusiastic you are about philosophy of religion. I am not sure, but I think it also shows that you think that god (or the god question) matters. If this is true, I would disagree. I only do this shit for entertainment; it’s only food for thought.

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wissam October 27, 2010 at 1:55 pm

Oh man. How can I delete my above comment? I reread it and it’s all fucked up.

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Gilbertus December 1, 2011 at 8:07 pm

This just shows that scientific reasoning is fallacioius, as you pointed out with the Flu example. For their can be, logically, an infinity of things with the same exact prediction. We find the prediction is correct, thus we have evidence for an infinity of logically contradictory things with the same consequent.

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