Geometry: The Importance of Being Direct
The Importance of Being Direct
The next type of argument is the direct proof, which can be interpreted as a transitive property of implication. Consider these two statements:
- If it is raining, the street is wet.
- If the street is wet, then the street is slippery.
You can eliminate the middleman and draw the conclusion “if it is raining, then the street is slippery” directly. This is the type of reasoning that Sherlock Holmes is famous for. He omits the middle statements and draws what appear to be surprising and clever conclusions as a result.
You can put this into a logical format and analyze it using P's Q's and R's. You haven't analyzed three statements together yet, but it's not much different than analyzing two statements together. You just have a lot more options for the truth value combinations. In fact, when you form a compound statement using three individual statements, your truth table will have 8 rows.
If P is the statement “it is raining”, Q is the statement “the street is wet,” and R is the statement “the street is slippery,” then your premises are P → Q and Q → R. Your conclusion is P → R. Let's construct the truth table for our compound statement ((P → Q) ∧ (Q → R)) → (P → R).
|Truth table for ((P → Q) ∧ (Q → R)) → (P → R)|
|P||Q||R||P → Q||Q → R||((P → Q) ∧ (Q → R))||P → R||((P → Q) ∧ (Q → R)) → (P → R)|
You can determine how many rows a truth table with four statements would require using inductive reasoning. A truth table involving one statement requires two rows. A truth table involving two statements requires four rows. When a truth table involves three statements, use eight rows. Following the pattern of doubling, a truth table involving four statements would need sixteen rows. In fact, a truth table involving n rows requires 2n rows.
Looking at the truth values for ((P → Q) ∧ (Q → R)) → (P → R), you see nothing but Ts. That's a great report card for a compound statement, and as a result, it gets put on the truth table equivalent of the honor roll. It is a tautology, and hence a valid argument. So “implies” has a transitive property.
You can apply this transitive property several times, as you will see in later examples.
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.