The Old Knewton Blog

Thoughts on “Unless Statements” from Alex K.

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KH3A conversation with a good friend this morning got me thinking about LSAT prep when she announced:

“I will move out of my apartment, unless my rent is lowered.”

As any good friend would, I translated her announcement: P unless Q.

Her intended meaning, I thought, probably implied two conditions:

1. If I do not move out of my apartment, then my rent was lowered. (“If not P, then Q.”)
2. If I move out of my apartment, my rent was not lowered. (“If P, then not Q.”)

In everyday usage, we intend most “unless” statements to be translated in this way, as biconditional statements.

For LSAT purposes, however, only the first conditional statement (“if not P, then Q”) is a valid translation of the original statement: “P unless Q.” If she stays in her apartment, then we can be sure that her rent was lowered. The necessary condition, lower rent, must have occurred considering the outcome that she did not move. If she moves out of her apartment, however, I can’t be certain that her rent was not lowered. She could have decided to move for an unrelated reason, like a fire or an infestation of beetles.

Consider this statement in which the implication is more clearly uni-directional:

“You will die, unless you take this antidote.” (P unless Q)

As a conditional, this becomes:

“If you do not die, you have taken this antidote” (“If not P, then Q”).

The necessary condition, taking the antidote, must have occurred considering the outcome, that you do not die. However, the statement cannot be translated as:

“If you die, you have not taken this antidote” (“If P, then not Q”).

There are plenty of ways you can die even after having taken the antidote. The antidote does not necessarily prevent death–we know only that you will certainly die if you do not take it.

Thus, in formal logic, any statement “P unless Q” can be translated into a conditional only as “if not P, then Q” and its contrapositive “if not Q, then P.”

Similarly, “not P unless q” can be translated as “if P, then Q” and its contrapositive “if not Q, then not P.”

“P unless not Q” can be translated as “if not P, then not Q” and its contrapositive “if Q, then P.”

And, “Not P, unless not Q” can be translated as “If P, then not Q” and its contrapositive, “If Q, then not P.”

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Written by Knewton

August 20, 2009 at 3:50 PM

Posted in Uncategorized

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