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:
If P, then Q.
Hereís 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 hereís 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.
Itís 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 doesnít 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 thereís no fire here. This argument has a different form, which is complementary to modus ponens; it too has a Latin name:
If P, then Q.
Therefore, not P.
Weíve already given one example of modus tollens; hereís another example:
If Lizzy was the murderer, then she owns an axe.
Lizzy does not own an axe.
Therefore, Lizzy was not the murderer.
Letís just suppose that the premises are both true. All right, if Lizzy was the murderer, then she really must have owned an axe. And itís a fact that Lizzy does not own an axe. What follows? That she wasnít the murderer.
Suppose you want to say: well, the first premise is false! If Lizzy was the murderer, then she wouldnít necessarily have to have owned an axe; maybe she borrowed someoneís. Now, that might be a legitimate criticism of the argument, but notice that it doesnít mean the argument is invalid. An argument can be valid even though it has a false premise! Remember, we are distinguishing an argumentís 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 theyíre 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 donít 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, letís get back to argument forms. Hereís another argument form:
Either P or 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 statementís disjuncts.
Either I will choose soup or I will choose salad.
I will not choose soup.
Therefore, I will choose salad.
And hereís another example:
Either the Browns win or the Bengals win.
The Browns donít win.
Therefore, the Bengals win.
Basically weíre told that it has to be one or the other thatís true; then weíre told that itís not the one; so we infer that it has to be the other thatís true.