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# Fallacies: Denying the Antecedent

In this video, Matthew C. Harris explains the fallacy of denying the antecedent, the formal fallacy that arises from inferring the inverse of a conditional statement. He also explains why graduate students might also be humans.

Speaker: Matthew C. Harris, Duke University.
Created by Gaurav Vazirani.

## Want to join the conversation?

• What are modus ponens and modus tollens?
• Expanding slightly on what was said at :
Modus ponens is a valid argument which looks like this:

P1: If P is true, then Q is also true.
P2: P is true.
C1: Therefore, Q is also true.

Hopefully that one's simple enough to understand.

Modus tollens or denying the consequent is a valid argument which looks like this:
P1: If P is true, Q is also true.
P2: Q is false.
C1: Therefore, P is also false.
Since Q is always true if P is true, it follows that P cannot be true if Q is false. Therefore P must be false (as if P was true Q would be true).
• The following argument has the logical form of denying the antecedent, but is it deductively invalid?

P1: If either Mitt Romney is the president of the U.S. or Mitt Romney is the commander in chief of the U.S., then both Mitt Romney is the president of the U.S. and Mitt Romney is the commander in chief of the U.S.
P2: It is not the case that either Mitt Romney is the president of the U.S. or Mitt Romney is the commander in chief of the U.S.
C: It is not the case that both Mitt Romney is the president of the U.S. and Mitt Romney is the commander in chief of the U.S.
• I don't quite agree with elm.parkinson's argument form:
P1: If (p or q), then (p and q).
( IC1: If p, then (p and q). [From P1] )
( IC2: If q, then (p and q). [From P1] )
( IC3: If p, then q. [From IC1] )
( IC4: If q, then p. [From IC2] )
( IC5: If p, then q and if q, then p. [From IC3 and IC4] )
( All of the above intermediate conclusions are unnecessary because P1 itself is unnecessary, but it explains how elm.parkinson interpreted P1. )
P2: Not (p or q)
C: Not (p and q) [From P2]

For all propositions p and q:
If (p and q), then (p or q).
This follows from the definition of the logical conjunction and disjunction operators.

All conditionals have a contrapositive as so:
Conditional: If p, then q.
Contrapositive: If (Not q), then (Not p).
The contrapositive is true if and only if the conditional is true and can always be replaced with the conditional.

Thus, we can take the contrapositive of this conditional to find that for all propositions p and q:
If [Not (p or q)], then [Not (p and q)].
Thus, your argument above is valid as it follows from this conditional.

Although this has the form of denying the antecedent, the fallacy is only present if we reason so. If we do reason that C is true from P1 and P2 by denying the antecedent, then we are making a logical fallacy and our argument would be fallacious, but that doesn't mean that the premises don't imply the conclusion. Reasoning that premises don't imply a conclusion because a certain reasoning that says so is fallacious is a fallacy itself:
http://en.wikipedia.org/wiki/Argument_from_fallacy

I hope this helps someone!
• I'm a bit confused. What's the difference between denying the antecedent and affirming the consequent?
• If John loves Mary, then John will want to marry Mary.*
The above statement is called a conditional.
*John loves Mary
is the antecedent
John will want to marry Mary is the consequent.
Denying the antecedent means denying John loves Mary. In other words John does not love Mary.
Affirming the consequent means asserting John will want to marry Mary.

In symbolic form, let John loves Mary = J and John will want to marry Mary = M
So, the above conditional now becomes J --> M which is read as "If J, then M"
J is the sufficient condition. That is, if J is true, then M is also true.
So we have the valid argument below:
J --> M
J
Therefore, M
Denying the antecedent fallacy occurs if we reason in the following way:
Note that ~J means NOT J and ~M means NOT M
J --> M
~J
Therefore, ~M
John may not love Mary but he may still want to marry her, for money, etc.

Affirming the consequent fallacy occurs if you reason in the following way:
J --> M
M
Therefore, J
John may want to marry Mary but he may not love her.
• At are both Modus Ponens and Modus Tollens valid or invalid? I think they're both valid, just not 100% sure.
(1 vote)
• They are both valid.

For Modus Ponens:
``P1) If A, then BP2) AC) B``

This is a simple conditional. It should be obvious that A causes B because it is in the first premise, so if A is true, then B has to be true also. Here's an example:
``P1) All cats are animals.P2) Tom is a catC) Tom is an animal``

For Modus Tollens:
``P1) If A, then BP2) not BC) not A``

This is the inverse of Modus Ponens. Since A causes B, if B is not true, then A cannot be either. Here's an example:
``P1) All cats are animals.P2) Mark is not an animal.C) Mark is not a cat.``
• P1: If P, Then Q
P2: Not Q
C: Not P

This form of argument is always valid? Right?
• help here:
If life involve quantity,its physical.
life does not involve quantity.
ergo,they are not ultimate.
(1 vote)
• This is still confusing. I don't find Khan helpful at all. You are explaining it the way my professor does and I need it to be changed into something like "if you go shopping and do thus and such." ....
(1 vote)
• Hello,

can a modus Tollens have a false conclusion? If yes is it still a valid argument or not?

Thank you
(1 vote)
• What if the argument is stated like this?
P1: If I have a job next year, it's gonna be a ski instructor
P2: I will not have a job next year.
C1: Therefore I'm not gonna be a ski instructor.