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Logical Connectives

While no training in formal logic is required for the LSAT, essentially it is a logic test. So, some knowledge of formal logic will give you a definite advantage.
 
To begin, consider the seemingly innocuous connective “if..., then....” Its meaning has perplexed both the philosopher and the layman through the ages.
 
The statement “if A, then B” means by definition “if A is true, then B must be true as well,” and nothing more.

For example, we know from experience that if it is raining, then it is cloudy. So, if we see rain falling past the window, we can validly conclude that it is cloudy outside.


There are three statements that can be derived from the implication “if A, then B”; two are invalid, and one is valid.


From “if A, then B” you cannot conclude “if B, then A.” For example, if it is cloudy, you cannot conclude that it is raining. From experience, this example is obviously true; it seems silly that anyone could commit such an error. However, when the implication is unfamiliar to us, this fallacy can be tempting.

Another, and not as obvious, fallacy derived from “if A, then B” is to conclude “if not A, then not B.” Again, consider the weather example. If it is not raining, you cannot conclude that it is not cloudy—it may still be overcast. This fallacy is popular with students.

Finally, there is one statement that is logically equivalent to “if A, then B.” Namely, “if not B, then not A.” This is called the contrapositive, and it is very important.
 
If there is a key to performing well on the LSAT, it is the contrapositive.

To show the contrapositive’s validity, we once again appeal to our weather example. If it is not cloudy, then from experience we know that it cannot possibly be raining.
We now know two things about the implication “if A, then B”:
  1. If A is true, then B must be true.
  2. If B is false, then A must be false.
If you assume no more than these two facts about an implication, then you will not fall for the fallacies that trap many students.
 
We often need to rephrase a statement when it’s worded in a way that obscures the information it contains. The following formulas are very useful for rewording and simplifying the conditions of a game.
 
On the LSAT, as in everyday speech, two negatives make a positive—they cancel each other out.
 
not(not A) = A
 
Example

“It is not the case that John did not pass the LSAT”

means the same thing as

“John did pass the LSAT.” 

The statement “if A, then B; and if B, then A” is logically equivalent to “A if and only if B.” Think of “if and only if” as an equal sign: if one side is true, then the other side must be true, and if one side is false, then the other side must be false.
 

 
(If A, then B; and if B, then A) = (A if and only if B)
 

A if and only if B

A

B

True

True

False

False

 
Example

“If it is sunny, then Biff is at the beach; and if Biff is at the beach, then it is sunny”

is logically equivalent to

“It is sunny if and only if Biff is at the beach.”

“A only if B” means that when A occurs, B must also occur. That is, “if A, then B.”

 
 
A only if B = if A, then B
 
Example

“John will do well on the LSAT only if he studies hard”

is logically equivalent to

“If John did well on the LSAT, then he studied hard.”

(Note: Students often wrongly interpret this statement to mean “if John studies hard, then he will do well on the LSAT.” There is no such guarantee. The only guarantee is that if he does not study hard, then he will not do well.)

The statement “A unless B” means that A is true in all cases, except when B is true. In other words if B is false, then A must be true. That is, if not B, then A.
 

 
A unless B = if not B, then A
 
Example

“John did well on the LSAT unless he partied the night before”

is logically equivalent to

“If John did not party the night before, then he did well on the LSAT.”

The two statements “if A, then B” and “if B, then C” can be combined to give “if A, then C.” This is called the transitive property.
 

 
(“if A, then B” and “if B, then C”) = (“if A, then C”)
 
Example

From the two statements

“if John did well on the LSAT, then he studied hard” and “if John studied hard, then he did not party the night before the test”

you can conclude that

“if John did well on the LSAT, then he did not party the night before the test.”   


These fundamental principles of logic are never violated in either the games or the arguments. *Hence, by using the above logical connectives, you can safely reword any statement on the LSAT.




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