# Formal Logic – Necessary & Sufficient?

One of the major concepts to understand on the LSAT is the use of formal logic. Formal logic is found predominantly in the Analytical Reasoning section of the LSAT, but it is also found in the Logical Reasoning section in inference questions and assumption questions.

Formal logic is founded on the understanding of necessary and sufficient assumptions. A necessary assumption is something that MUST be true, whereas a sufficient assumption is something that COULD be true. Formal logic is the use of rules to make deductions.

In the Analytical Reasoning section of the LSAT you want to diagram formal logic and necessary/sufficient problems. For If—then statements you want to diagram them as:

If A then B:         A —–> B

The left side of the arrow (A) is the sufficient condition, and then right side of the arrow is the necessary condition (B). Thus, it is possible for A to happen (although it doesn’t have to happen), but if A does happen then B MUST happen as well.

When you have a conditional statement you can also write the if—–then statement’s contra-positive. A contra-positive is just a further deduction that can be made from a conditional statement. To do this you will negate both terms and then flip them to the other side of the arrow.

If A then B:      ~B —-> ~A

Other Examples:

1) If not X then Y:      ~X —–> Y     OR     ~Y —–> X

2) if not S then not T:    ~S —-> ~T    OR    T —-> S

Another common type of formal logic seen on the LSAT is the “only if” statement.  For example, A only if B. Only if means that if A does happen then B must also happen. That means that:

A only if B can be rewritten as: if A then B:     A—–> B

Once again the contra-positive would be ~B —-> ~A

Another common type of formal logic seen on the LSAT is the “if and only if” and “if but only if” statements. “If and only If” is a bi-conditional logical connective between statements. This means that the truth of one of these statements requires the reverse to also be true. For example:

A if but only if B can be written as: if A then B AND if B then A:      A <—–> B

The contra-positive would be: ~A <—–> ~B

Also, the phrase “if and only if” means the same thing as “if but only if” in terms of formal logic and would be written the same way as the above example.

***”~” means NOT ***