Question 1

What are *modus ponens* and *modus tollens*?

Accepted Answer

Expanding slightly on what was said at 2:19 :
_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).

Question 2

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.

Accepted Answer

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!

Question 3

I'm a bit confused. What's the difference between denying the antecedent and affirming the consequent?

Accepted Answer

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.

Question 4

At 2:15 are both Modus Ponens and Modus Tollens valid or invalid? I think they're both valid, just not 100% sure.

Accepted Answer

They are both valid.

For Modus Ponens:
```P1) If A, then B
P2) A
C) 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 cat
C) Tom is an animal```

For Modus Tollens:
```P1) If A, then B
P2) not B
C) 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.```

Course: Wireless Philosophy > Unit 1

Fallacies: Denying the Antecedent

Want to join the conversation?

Video transcript