Formal Logic Basics: And, Or, Neither, Nor…

by on September 22nd, 2009

We’ve covered, in an earlier articles, how to deal with the simplest formal logic statement: If X, then Y.  But what happens when our necessary or sufficient factors become more complicated?  Let’s look at a couple of examples, using the idea of a vegetable salad.  The simplest statement and its contrapositive might look like this:

If the salad has lettuce, then it has tomatoes.

If the salad has no tomatoes, then it has no lettuce.

Now let’s add more vegetables (and more complicated logic):

If the salad has lettuce or spinach, then it has tomatoes and peppers.

Here’s an important idea: when you are forming a contrapositive, you already know that the necessary and sufficient factors are switched around and negated.  But now you also have to remember that “and” becomes “or,” and vice versa.  So the statement above becomes:

If the salad has no tomatoes or no peppers, then it has no lettuce and no spinach.

I find it extremely helpful to individually negate each element of the statement; otherwise, it’s easy to get confused.  For instance, if I only negate the first part of the statement above and say to you, “If the salad has no tomatoes or peppers…” you might interpret that as meaning that neither of those vegetables should be in the salad.  But in formal logic terms, it would technically mean that I either want peppers or no tomatoes.  Neither of those ideas, though, is what I mean to say in the contrapositive; the intended meaning is that I want no tomatoes or no peppers.

The pairing of “neither” and “nor” can also cause some consternation.  The easiest way to deal with that is to remember that “neither X nor Y” is the same thing as “no X and no Y.”  The example above can be rephrased as follows:

If the salad has no tomatoes or no peppers, then it has neither lettuce nor spinach.

So if you need to negate a “neither/nor” statement, the “nor” becomes “or” just as it would if the statement said “and.”

What if the sentence isn’t written in the order in which we expect to find the elements?  For instance, how do we interpret a sentence that says:

The salad has cucumbers if it has onions.

Here we can take the word “if” and read the statement that follows it as the sufficient element.  We can turn that sentence into this:

If the salad has onions, then it has cucumbers.

A final issue is the phrase “only if.”  Let’s go back to our vegetable salad, and look at the following sentence:

The salad has carrots only if it has mushrooms.

Here, you can’t interpret “if” as signaling the sufficient element.  “Only if” statements are interpreted differently than regular “if” statements.  The “only if” statement above means the same thing as this:

If the salad has carrots, then it has mushrooms.

These ideas flesh out the basics of formal logic.  To recap, here are the thoughts that you need to remember:

  1. When forming a contrapositive, turn “and” into “or” and vice versa.
  2. “Neither X nor Y” means “No X and no Y.”
  3. “X only if Y” means “If X then Y.”

Just remember that, as with everything on the GMAT, formal logic becomes much easier with practice, so keep working on it!

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