David R. Henderson  

Logic: Denying the Antecedent

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My favorite part of the introductory philosophy course I took at the University of Winnipeg was the segment on logic, especially on logical fallacies. One of the most common logical fallacies is "denying the antecedent."

Here's the example used in my old logic text, Joseph G. Brennan, A Handbook of Logic, Harper and Row, 1957:

If Bill Nietman is a Princeton graduate, he cuts his own hair. (If p, then q.) Bill Nietnam is not a Princeton graduate. (Not p.) Therefore he does not cut his own hair. (Not q.)

As you can see, the fact that p is not true does not mean that q is not true. Many people besides Princeton graduates cut their own hair.

In his response as a commenter this morning, Nathan Smith commits that fallacy. He writes:

"[Carpenter] contends that preserving NATO is unnecessary because the West European nations now have the economic and military resources to protect their own security." Logically, "p because q" implies "if not-q, then not-p." Logically, "NATO is unnecessary because Western European nations can defend themselves" would seem to imply "if Western European nations couldn't defend themselves, NATO would be necessary."

Notice that he writes:

Logically, "p because q" implies "if not-q, then not-p."

In other words, he is denying the antecedent.

I would get more into his substantive argument about foreign policy, but I'm going to stick to my Econlog knitting and focus, as I almost always do, on economics.

I want to reassure readers, though, that we at Econlog do not block people as commenters simply because they disagree with us. I say this because Nathan wrote:

I wouldn't want to provoke Arnold and Dave and Bryan into blocking me from posting comments in future!

There are two misunderstandings packed into one sentence. First, the webmaster, not Arnold or Bryan or I, has the main say about what gets blocked. Second, she has always been clear about the criteria for blocking and among those criteria there's not even a hint that one gets blocked simply for disagreeing.


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CATEGORIES: Economic Methods



COMMENTS (22 to date)
pj writes:

Perhaps Nathan was a bit confused ... Logically, "p implies q" does imply that "if not-q, then not-p." But if his "p because q" is the same as "q implies p," then you are right that he is denying the antecedent. Perhaps he thought got confused thinking "p because q" was the same as "p implies q."

Kevin Dick writes:

p->q is in fact the same as ~q -> ~p. That's the equivalence of contrapositives.

However, I think he messed up his assignments of p and q to the statements he made so he ended up denying antecedents.

Nathan Smith writes:

Consider the following two statements:

1. If it rains tomorrow, then 2+2=4.
2. 2+2=4 because it will rain tomorrow.

(1) is (rather odd and misleading but) true. (2) is false. EconLog readers will, I presume, understand the principle and see the application to the present case.

Yes, I've noticed that Econlog doesn't censor comments that disagree, I just thought there might be a word limit. I got barred from commenting at Brad DeLong's blog a while back for the comfort of his liberal commenters. *Kudos* to EconLog for not following that example.

SB7 writes:

Nathan and Arnold, you both might be better served if you didn't try to reduce this to propositional logic. I think you need some higher order predicates to work this out symbolically.

Sam writes:
preserving NATO is unnecessary because the West European nations now have the economic and military resources to protect their own security

p := "West European nations have the resources to defend themselves"

q:= "Preserving NATO is unnecessary"

(p -> q) <=> (~q -> ~p)

So the only thing you can reasonably derive (at this level) is

"If preserving NATO is necessary, then West European nations are unable to defend themselves."

In order to get what Nathan wants, which is ~p -> ~q, you must start with q -> p, or "Preserving NATO is unnecessary, therefore West European nations can defend themselves"

That sounds like nonsense, so I am forced to agree with SB7 -- better logic tools are needed.

david writes:

But

Logically, "p because q" implies "if not-q, then not-p."
is true. You're confusing denying the antecdent with modus tollens.
kevin writes:

>Logically, "p because q" implies "if not-q, then not-p."

Um, isn't that modus tollens?

kevin writes:

Gaaah never mind.

Jeff writes:

"Pain don't hurt."

--Road House

Lee Kelly writes:

Nathan's argument is also invalid, because material implication and causation are not the same thing: "P causes Q" does not imply "P entails Q," or vice versa.

Carl Edman writes:

So much logical confusion to go around! Let's try to make everything clear.

First, the formalism:

(p => q) => (~p => ~q) is a fallacy related to what is commonly called denying the antecedent.

(p => q) => (~q => ~p) is a valid logical rule sometimes called the contrapositive.

Now into what category does Nathan's statement fall?

Logically, "NATO is unnecessary because Western European nations can defend themselves" would seem to imply "if Western European nations couldn't defend themselves, NATO would be necessary."

Let's assign "Western European nations can defend themselves" as p, and "NATO is unnecessary" as q.

Then the first statement codes as: (p => q). Note that in the formalism, the condition p precedes the conclusion q. In English, you can write it either way. Here in the English, the conclusion "NATO is unnecessary" precedes the condition "Western European nations can defend themselves."

The second statement codes as (~p => ~q). The condition is ~p, "Western European nations couldn't defend themselves." The conclusion is ~q ("NATO would be necessary."). Note that because of the change in conjunction ("because" in the first English sentence, "if" in the second), the order of condition and conclusion switched.

So it looks like Nathan tried to argue the falalcy that "(p => q) => (~p => ~q)"

David R. Henderson writes:

@Carl Edman,
Correct. You said what I said.

drobviousso writes:

I"m really glad that all of my formal logic training came under the hazy glow of compiler.

Consider how much clearer if you formulate as such:
'Logically, "not-p because q" implies...', or even better "if q then not-p".

BTW, I have no clue what the right '""' rules are here.

stanfo's brother writes:

But you CAN get blocked for discussing Sarah Palin in a negative light. I was just kidding before, she is a very, very smart woman and a gifted speaker.


Yes, I'm still bitter... :-)

Patrick R. Sullivan writes:
I got barred from commenting at Brad DeLong's blog a while back for the comfort of his liberal commenters.

Nicely put, that was my guess too; his friends didn't like losing arguments.

James Q Buchanan writes:

I got barred for discussing Barack Obama in a negative light.

It sounds as if you have outsourced responsibility for one of the most important functions of a website to someone who could probably not follow the logic of this thread.

rob writes:

As everyone has pointed out, Nathan would be totally wrong, if he were arguing "if p, then q".

But he's not. He's arguing "p because q", which, in logical terms, is "p if-and-only-if q". And in that case, "if ~q, then ~p", is logically valid. It is not a fallacy.

At least, that's the only thing that makes sense. Either that, or Nathan is indeed wrong, and stubbornly illogical.

A meta-comment to stanfo's brother and James Q Buchanan:

The content of your comments was not the reason for which you may have been blocked here--if you were blocked here. That would have been explained to you in email.

If you were blocked, it was because you engaged in name-calling or ad hominem attacks. Name-calling or ad hominem remarks, whether they are directed toward the left, the right, a politician, an economist, a journalist, or the tires on your car, are against EconLog policies. EconLog comments are expected to address the economic content of the thread and not devolve into personal, unsupported or unnecessary rudeness.

Evidently, neither of you are altogether blocked here, since you are commenting in this thread. Your costs have probably been driven up by whatever blocks you describe. However, you were and remain welcome to email us at webmaster@econlib.org to discuss your comment privileges. If your comment privileges were not restored in full, it means that you either never responded to our email or that you were rude in your response. If you are still grating about it and don't understand it, feel free to email us.

Addenda:

P.S. To "stanfo's brother"--I have located the email I sent to the email address you/stanfo gave and to which you never responded. I do not think you would be proud of your original comment were I to quote it here--which I won't. You are welcome to email us at any time.

P.P.S. To James Q. Buchanan--I'm working on trying to figure out which ban you are talking about and why. Feel free to email us, identifying yourself clearly. We have a lot of folks who use the name Buchanan as their nick. So far, I do not see any responses by you to email we have sent you. Your comments appear to be very rude, with crude language. Are you the same person? Please feel free to email us.

Carl Edman writes:

rob writes:

[Nathan]'s arguing "p because q", which, in logical terms, is "p if-and-only-if q".

"Because" and "If-and-only-if" (iff) are not logically equivalent.

For example, I may believe that "X is a smart man because X won the Nobel Prize in Economics." Now this may or may not be a wise or accurate belief, but it is clearly not the equivalent of "X is a smart man if and only if X won the Nobel Prize in Economics." The latter, which implies that everybody who has not won the Nobel Prize in Economics is not smart, is pretty clearly a different (and rather obviously inaccurate) belief.

Andy writes:

I don't think any of the logical examples are correct. "Because" does not correspond to any of the simple examples here.

"Because" in ordinary English usage implies causation. That is, "P because Q" doesn't mean Q => P. It means Q => P and Caused(Q, P), where Caused() is an obvious predicate.

However, Nathan's inference is still wrong.

Nathan Smith writes:

Thanks Rob, you nailed it.

To Carl Edman: "because" in natural language is a little too vague to be definitely mapped onto a category of logical inference, but I think it always has to mean something like "if and only if," i.e., necessary and sufficient causation. Of course, it's hard ever to identify examples where a cause is strictly necessary and sufficient. If I say, "I didn't buy a Big Mac because I didn't have enough money," it's assumed that (a) I was hungry, (b) there was a McDonalds nearby, (c) I'm willing to eat Big Macs on occasion, and generally (d) if I had had the money, I would have eaten a Big Mac. If I was deep in the Amazon rainforest at the time, or if I'm a vegetarian, lack of money is not a necessary cause, since I couldn't have bought a Big Mac anyway. If someone was offering me a Big Mac for free, lack of money was not a sufficient cause, since I could have eaten the Big Mac even without money. In either of these cases, the statement that "I didn't eat a Big Mac because I didn't have enough money" is at best very misleading and probably just false.

Carl Edman's example, "X is a smart man because X won the Nobel Prize in economics," is a bad one. The first question is, does "because" here refer to causation or inference. The former is absurd: it suggests that winning the Nobel Prize caused X to be a smart man. If inference, then (a) the sentence should more properly read "X must be a smart man because..." and (b) the "because" is used in a sense quite different from that which is under discussion here. It is not as if we didn't know whether NATO was necessary and then along comes this new evidence, that W. Europe can defend itself, which shows that it isn't necessary. Rather, W. Europe's capacity to defend itself-- so the claim goes-- causes NATO to be unnecessary. This claim is false, or at least is a culpably and excessively misleading abuse of the vagueness of natural language to induce a false reading, unless the author would affirm that NATO would be necessary if NATO could not defend itself.

Since Andy doesn't make it clear why he erroneously thinks my inference is wrong, I can't correct him. Alas. But the inference is definitely valid.

Gena writes:

I am glad to see that the consensus seems to be what I've concluded for myself. Nathan really argues "p is identical to q", in which case "not q" leads to "not p". Nathan incorrectly identifies it as "if p then q", and David fails to notice that.

This is correct only if we imply that NATO is the only entity, besides the states themselves, that can protect them, and that the states want protection. In this case "a state cannot protect itself" is equivalent to "a state needs NATO to protect it".

Formally, "a state cannot protect itself" (~p) leads to "a state needs SOMEBODY to protect it" OR "a state does not want protection", and that of course does not lead to (~q).

So, formally, David may be correct if we interpret the original statement this way but it would probably be an incorrect interpretation because Nathan likely implies that states want protection, and NATO is the only entity that can protect the states besides the states themselves.

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