Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

<-- Previous | Newer --> | Current: 981079073 LarrySanger at Fri, 02 Feb 2001 01:57:53 +0000.


I said that whether an argument is valid or not depends on the form of the argument.  Now, for that reason, in logic, what one studies a lot is different valid argument forms.  So let me give you some examples of valid argument forms.

The first is called by a Latin name: modus ponens.  That makes the argument form sound a lot more daunting than it really is.  Modus ponens is really simple -- perhaps the simplest sort of argument:

Modus Ponens

If P, then Q.


Therefore, Q.

Here’s an example of an argument that fits the form modus ponens:

If love conquers all, then love conquers Attila the Hun.

Love conquers all.

Therefore, love conquers Attila the Hun.

Or here’s another example:

If democracy is the best system of government, then everyone should vote.

Democracy is the best system of government.

Therefore, everyone should vote.

So basically the idea is to make an "if-then" claim -- these are called conditional claims -- and then affirm the "if" part, or the antecedent, and then conclude with the "then" part, or the consequent.

Suppose that we tried instead to affirm the "then" part of the conditional, the consequent, first, and conclude with the "if" part, the antecedent.  Believe it or not, this argument has a name:

Affirming the consequent.

If P, then Q.


Therefore, P.

It’s called "affirming the consequent" because in arguing this way one does indeed affirm the consequent in the second premise ("Q" is the consequent of the conditional claim, "If P, then Q").  This is a fallacy.  If you argue this way you will be making a mistake.  You should be able to see that with an example.  You know that oxygen is required for fire.  So suppose we argue like this:

If there is fire here, then there is oxygen here.  (Since oxygen is required for fire.)

There is oxygen here.

Therefore, there is fire here.

You can see that that way of arguing doesn’t work.  But now consider a slightly different argument:

If there is fire here, then there is oxygen here.

There is no oxygen here.

Therefore, there is no fire here.

Now, that argument does work.  It is perfectly valid: indeed, if there is no oxygen here, then we can conclude validly that there’s no fire here.  This argument has a different form, which is complementary to modus ponens; it too has a Latin name:

Modus Tollens

If P, then Q.

Not Q.

Therefore, not P.

We’ve already given one example of modus tollens; here’s another example: 

If Lizzy was the murderer, then she owns an axe.

Lizzy does not own an axe.

Therefore, Lizzy was not the murderer.

Let’s just suppose that the premises are both true.  All right, if Lizzy was the murderer, then she really must have owned an axe.  And it’s a fact that Lizzy does not own an axe.  What follows?  That she wasn’t the murderer.

Suppose you want to say: well, the first premise is false!  If Lizzy was the murderer, then she wouldn’t necessarily have to have owned an axe; maybe she borrowed someone’s.  Now, that might be a legitimate criticism of the argument, but notice that it doesn’t mean the argument is invalid.  An argument can be valid even though it has a false premise!  Remember, we are distinguishing an argument’s validity from its soundness.  To be sound, an argument has to be both valid and have true premises.  Our Lizzy argument is unsound, because it has a false premise; but it is perfectly valid, because, even though they’re not, if both premises were true, then the conclusion would have to be true.  You see, a lot of beginning logic students get confused about validity, and don’t see how an argument with false premises can be valid.  But they are confusing validity with soundness.  An argument with false premises can be valid, but it cannot be sound.

All right, let’s get back to argument forms.  Here’s another argument form:

Disjunctive syllogism

Either P or Q.

Not P.

Therefore, Q.

The reason this is called "disjunctive syllogism" is that, first, it is a syllogism -- a three-step argument, and second, it contains a disjunction, which means simply an "or" statement.  "Either P or Q" is a disjunction; P and Q are called the statement’s disjuncts.

Either I will choose soup or I will choose salad.

I will not choose soup.

Therefore, I will choose salad.

And here’s another example:

Either the Browns win or the Bengals win.

The Browns don’t win.

Therefore, the Bengals win.

Basically we’re told that it has to be one or the other that’s true; then we’re told that it’s not the one; so we infer that it has to be the other that’s true.