Modus Tollens

The companion to modus ponens is modus tollens, the way of denying.

If I say something like, "if P is true, Q is true," and add, "however, Q is not true," I can conclude that P is not true, either.  In short: "If P, then Q. Q is not true, so neither is P."  I know this is so because if P were true, Q would be true.  But Q isn't true, so P can't be.

Here is an example: If it is raining outside, then there are clouds in the sky.  There are no clouds in the sky, so it is not raining.  The first sentence says there must be clouds for it to rain.  The second sentence tells me that there are no clouds, so how could it be raining?  It couldn't be raining, and it isn't raining!

Another example: In order for me to have stabbed the man, I must have been next to him.  However, I wasn't even in the same room.  Therefore, it couldn't have been me!  This example deviates from the grammatical syntax of the other, but follows the same logical form: If P (I stabbed the man), then Q (I was next to him).  Q isn't true (I wasn't in the room) therefore P isn't true (I didn't stab him).

Modus ponens & Affirming the Consequent

One of the most basic rules of validity is known as modus ponens, "the affirming way."  In short, it says that if I have a conditional statement and the first part of the statement is true, the second part must also be true.  This is often written as "If P, then Q.  P is true, so Q must be true."

Here are a few examples:

If it is raining, the ground is wet.  It is raining, so the ground is wet.

If you are human, you are also a mammal.  You are a human, therefore you're a mammal.

If you get 3 strikes, you're out.  You have three strikes, so you're out.

Each example follows the same logical structure.  If (something happens), then (something else will happen).  Since (the first thing happened), I know that (the second thing happens).  I know that the ground is wet because it is raining.  I know that I am a mammal because I am human.  I know that I'm out because I got 3 strikes.

Keep in mind that this does not work in reverse. I do not know that it is is raining just because the ground is wet (someone could be washing their car).  I do not know that I am human because I'm a mammal (I could be a horse).  I do not know that I got 3 strikes because I'm out (I could have hit a pop fly).  If I were to make an argument this way, I would commit the formal fallacy known as affirming the consequent.

The consequent is the Q part of "If P, then Q."  This fallacy effectively says, "If P, then Q.  I know that Q is true, so P must be true."  As shown in the examples above, this isn't true.  There could be other reasons Q is true without P being true.

Valid/Invalid

"Your argument is invalid!" Perhaps you've heard this phrase before.  Perhaps you've used it yourself.

To most people, invalid is synonymous with wrong (i.e. false, not true) and it is wrong because I said so.  However, an invalid argument is not necessarily a false one.  By that I mean that the conclusion may be true even if the argument is invalid.  Likewise, an argument may be valid but have a false conclusion.

Consider an example of the first possibility:

1. All men are mortal.
2. Churchill is mortal.
3. Therefore, Churchill is a man.

Every one of these statements are true.  But the argument is not valid.  There is nothing about the first two statements that leads me to believe the third.

Consider an example of the second possibility:

1. All men are immortal.
2. Churchill is man.
3. Therefore, Churchill is immortal.

Since the first premise is not true, but the argument is valid, my conclusion is false as a result.  If all men were immortal, Churchill, being a man, would be immortal.  But not all men are immortal, and in fact, all men are mortal.

What is a valid argument?  A valid argument is one where if the premises are true and the terms unambiguous, the conclusion must be true.  Your choice is to believe the conclusion or be illogical.  Validity has to do with the form of an argument.  The structure of some arguments force you to believe the conclusion, just as the structure of some buildings force you to enter and leave a particular way (such as through a door rather than a window). We need valid arguments in order to think, debate, and reason clearly.  Otherwise, we are blindly stumbling looking for truth.

In the first example above, we see that all men are mortal and that Churchill is mortal.  But I do not know anything else about Churchill.  I can't conclude anything about him.  Churchill may be the name of my pet British bulldog, and hence not a man, rather than the British leader nicknamed the British Bulldog.  In the second, my knowledge of the first two statements leads me to my knowledge of the third.  I know that Churchill is mortal because he is a man and all men are mortal.

Consider a third possibility where an argument is valid and the conclusion true:

1. All men are mortal.
2. Churchill is man.
3. Therefore, Churchill is mortal.

When the structure, or form, of an argument is valid; the terms are unambiguous, for example, we know what mortal and men mean; and the premises are true; we must believe the conclusion.  We call this a sound argument.  It is impossible to disbelieve a sound argument and remain a rational thinker.