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Geometry: The Law of Detachment

The Law of Detachment

The first type of valid argument I will discus is commonly known as the law of detachment. To illustrate the law of detachment, consider the following example.

Example 1: Assume that the two premises below are true. What conclusions can be drawn?

  1. If it is raining, the baseball game will be cancelled.
  2. It is raining.

Solution: If you accept these statements to be true, then you can logically conclude that the baseball game will be cancelled. You can examine this type of argument easier if you switch over to P's and Q's. Let P be the statement “it is raining” and Q be the statement “the baseball game will be cancelled.” Then your first statement is P → Q and the second statement is P. The conclusion is that the baseball game will be cancelled, which is Q. So you are assuming two things: P → Q and P, and claiming that Q follows. The compound statement for assuming P → Q and P, and concluding Q can be written ((P → Q) ∧ P) → Q. Now that's a complicated compound statement! Let's build a truth table for ((P → Q) ∧ P) → Q and see what you get.

Truth table for ((P → Q) ∧ P) → Q
P Q P → Q (P → Q) ∧ P ((P → Q) ∧ P) → Q
T T T T T
T F F T T
F T T T T
F F T T T

Notice that ((P → Q) ∧ P) → Q is a tautology, and hence is a valid argument. So you can logically conclude that the baseball game will be cancelled in this situation.

Let's look at another example.

Example 2: Assume that the two premises below are true. What conclusions can be drawn?

  1. If it is raining, the baseball game will be cancelled.
  2. The baseball game is cancelled.

Solution: It is tempting to conclude that it is raining. But the game could have been cancelled for another reason; maybe one team didn't have enough players. You can examine this scenario using formal logic as well. As you did before, let P be the statement “it is raining” and Q be the statement “the baseball game will be cancelled.” Then the first statement is P → Q and the second statement is Q. The conclusion is that it is raining, which is just P. Again we are assuming two things: P → Q and Q, and claiming that P follows. The compound statement for assuming P → Q and Q, and concluding P can be written ((P → Q) ∧ Q) → P. Let's build the truth table for this compound statement and see what we get.

Truth table for ((P → Q) ∧ Q) → P
P Q P → Q (P → Q) ∧ Q ((P → Q) ∧ Q) → P
T T T T T
T F F F T
F T T T F
F F T F T

Because one of the truth values for ((P → Q) ∧ Q) → P is false, ((P → Q) ∧ Q) → P is not a tautology, and you cannot draw any conclusions about P (whether or not it is raining). It's nice to see that our formal logic confirms what our intuitive logic led us to believe.

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Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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