Assumptions Part 3: Logical Opposites, Quantifiers, and Hard-to-Negate Statements

by on March 8th, 2012

We’ve spent a fair bit of time defining necessary assumptions and talking about strategies to determine necessity, so I’m not going to dwell on these subjects. If you’re just joining us, check out the first two parts in this series here and here.

What we’ll do instead is take a closer look at how to negate statements, particularly statements with quantifiers. In order to do so, we’ll need to define two terms that will sometimes overlap and won’t at other times. The first of these is the “logical opposite.” The logical opposite of a statement is its negated version. It is what we’re always looking for when we negate an assumption. The second of these terms is the “polar opposite.” Polar opposites are, as the name implies, more extreme than logical opposites.

Let’s take a look at an example.

The economy shed more than 500,000 jobs during the present recession.

The properly negated form of this statement (i.e. the logical opposite) is:

The economy did not shed more than 500,000 jobs during the present recession.

Here’s where it gets tricky. How many ways are there for the above statement to be true (i.e. what are the possibilities)? Think about this for a few seconds and read on.

Possibility 1: The economy shed less than 500,000 jobs during the present recession.

This one was obvious, but there’s one more way for the logical opposite to be true.

Possibility 2: The economy shed exactly 500,000 jobs during the present recession.

Confused? Take a look at the logical opposite again. The only possibility precluded is the possibility of the economy having shed MORE than 500,000 jobs during the present recession.

This brings us to the polar opposite. As the name implies, the polar opposite is the most extreme possibility, i.e. “The economy shed less than 500,000 jobs during the present recession.”

Let’s take a look at another interesting case.

Most of the people on this boat are deluded.

On the GMAT, “most” means greater than fifty percent, so the logic required here is identical to the previous case. The logical opposite is therefore:

It is not true that most of the people on this boat are deluded.

This encompasses two possibilities:

Possibility 1: Less than half of the people on this boat are deluded or, in other words, most of the people on this boat are NOT deluded (the polar opposite).

and

Possibility 2: Exactly half of the people on this boat are deluded.

In other cases, there is no distinction between the logical opposite and the polar opposite. Consider the following statement, for example.

Some bulldogs are more ferocious than pitbulls.

“Some” is an ambiguous term that is often misinterpreted by many to mean at least one, but less than half. This is INCORRECT. Saying some bulldogs are more ferocious than pitbulls DOES NOT imply that some bulldogs are not more ferocious than pitbulls. The word “some” can actually encompass a wide range of possibilities, anywhere from one to all.

So the logical opposite of “anywhere from one to all” must incontrovertibly be “none” or “zero.” The logical opposite of the statement above therefore becomes

No bulldogs are more ferocious than pitbulls.

Since we have just the one possibility here, there is no distinction between the logical opposite and the polar opposite. In fact, the very notion of a polar opposite really only applies when the logical opposite comprises more than one possibility.

Finally, let’s take a look at a particularly pesky statement.

No crusader is unaware of the fact that his cause might merely be a poorly founded figment of his imagination that will not stand up to scrutiny.

How exactly do we negate this beast? Do we get rid of the first “No”? Do we say he is “aware of the fact…”? Do we say “…his cause cannot be a poorly founded..”? Does it become “…will stand up to scrutiny”? Should we do all of the above, or some combination thereof?

There’s actually a pretty easy way to approach a statement like this. Just preface it with the words “It is not true that…”.

Like so:

It is not true that no crusader is unaware of the fact that his cause might merely be a poorly founded figment of his imagination that will not stand up to scrutiny.

Easy enough? Here comes the hard part. What exactly does that sentence mean?

Hopefully writing in that form helps you think about what is required for the statement not to be true. There are two key ideas here that take precedence over everything else in the sentence. The first is the awareness of the crusader. What it is he is aware or unaware of is immaterial. All we’re interested in is whether he is aware or not. So for all practical purposes you can cross the rest of the sentence out, making your life that much easier.

It is not true that no crusader is unaware…

But there’s another, even more important idea that takes precedence here. If you’re thinking “quantifier,” you’re absolutely correct. We actually don’t even care about whether the crusader is aware or not, since the real issue at hand is how many crusaders are unaware, not whether any individual crusader is aware/unaware. So we can trim this further to end up with the following:

It is not true that no crusader is…

What is the logical opposite of “none”? Careful here. Hopefully you applied your newly acquired knowledge of the distinction between logical and polar opposites. It is NOT “all.” Well, not entirely, anyway. We have to account for all possibilities between “one” and “all.” We can represent that as follows.

At least one crusader is…

Now, you can add back the rest of that mess.

At least one crusader is unaware of the fact that his cause might merely be a poorly founded figment of his imagination that will not stand up to scrutiny.

Note that we could have replaced “at least one” with “some,” and the logic of the statement would have been identical.

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