The ability to logically negate a statement—whether conditional, causal, etc.—is critical to your success on the LSAT. It comes up most commonly in the Logical Reasoning section of the test, although any question stem using the word “EXCEPT” (always capitalized) will require you to logically negate that stem. The list does not stop here: every time you apply the contrapositive of a conditional statement, you will need to reverse and negate the two conditions that constitute that statement (this is relevant to Must Be True, Justify, and Parallel Reasoning questions mostly, but can also be critical in other sections of the test). Negating statements is also useful in Assumption questions, because proving the correct answer choice requires application of the Assumption Negation Technique: the correct answer choice, when logically negated, must weaken the conclusion of the argument. And, of course, the ability to understand the logical opposite of a conditional statement will be directly relevant to many Cannot Be True questions, where the correct answer choice is the one that can be disproven using the information contained in the stimulus.
As described in The PowerScore Logical Reasoning Bible (and in Lesson 2 of our Full Length LSAT course), negating a statement consists of creating the logical opposite of that statement, i.e. a statement that denies the truth of the original. The logical opposite of Jon is wise is Jon is not wise, where the affirmative and negative counterparts of the oppositional construct are mutually exhaustive as well as mutually exclusive. The same is true of quantified expressions. For instance, the logical opposite of I gained more than 20 points on my latest practice test is I did not gain more than 20 points on my latest practice test. How many points did you actually gain? The only possibility precluded is the possibility of gaining more than 20 points. You could have gained exactly 20 points, or maybe you did not gain any. In quantity constructs, all/not all, some/none, most/not most are logically proper oppositional constructs. By contrast, a construct such as All pleasure is good/No pleasure is good does not contain a logical opposition, because these opposites do not mutually exhaust the domain.
Formulating logical opposites may seem mechanistic, and in many ways it is. Nevertheless, it is important to understand the implications of your oppositional construct. Take, for instance, the logical opposite of I believe that stealing is moral. The correct form of the logical opposite (I do not believe that stealing is moral) amounts to a negation of the embedded clause (I believe that stealing is not moral). Similarly, when logically negating a statement such as It is rational not to eat food that is poisonous, consider the implications of the double negative construct (It is not rational not to eat food that is poisonous) and immediately simplify it: It is rational to eat food that is poisonous.
Negative statements are generally less informative than affirmatives; they are more complex and harder to process. Simplification is key!
What happens when you need to negate a conditional statement? In that case, you need to show that the sufficient condition can occur even if the necessary condition does not occur, i.e. that the necessary condition is not, in fact, necessary. So, the logical opposite of “All that glitters is gold” is simply “All that glitters is not gold.” To put it another way, some things may glitter even if they are not gold, because not everything that glitters is gold.
Quite often, the logical opposite of a conditional statement is formed using the phrase “even if.” The logical opposite of the statement Unless you practice, you will not succeed is You can succeed even if you do not practice. This is because the original statement posits practicing as a necessary condition for success (Succeed –> Practice). To contradict this statement, you need to show that practice is not a necessary condition for success. Note that even if is not, by itself, an indicator of a sufficient condition: it merely states that the lack of practice does not prevent one from succeeding, not that the lack of practice somehow ensures success.
Now, consider what happens when negating compound sentences, such as conjunctions of the form X and Y and disjunctions of the form X or Y (or both).
To negate a conjunction such as I love you and you love me requires showing that the two clauses cannot be both true: it is simply not the case that we both love each other. Using De Morgan’s laws for interrelating conjunctions and disjunctions, this would mean one of three things: either I don’t love you, or else you don’t love me, or else neither of us loves the other. The either/or construction does not preclude the possibility of both propositions being true, unless specifically told otherwise (e.g. either X or Y, but not both).
To negate a disjunction such as Jon is either a doctor or a lawyer requires showing that Jon is neither a doctor nor a lawyer (i.e. that Jon is not a doctor and that Jon is not a lawyer). At the most basic level, negating a conjunction (and) requires the use of or, whereas negating a disjunction (or) requires the use of and.
If you wish to practice sentential negation of propositions that are both conditional and compound, try these out:
- Only those who are both romantic and cautious can truly fall in love.
- You are not a realist unless you either believe in miracles or in yourself.
- You are neither a realist nor a humanist if you believe in miracles.
- It is rational not to believe in miracles, unless you are either romantic or idealistic.
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Image courtesy of Sean MacEntee