LOGIC MAIN PAGE

LOGICAL OPERATORS

Sample sentences
AND operator
IF/THEN operator
NOT operator
OR operator
XOR operator

RULES OF LOGIC

Chain Rule
Contrapositive
DeMorgan's Law
Disjunctive Inference
Disjunctive Infer. (XOR)
Double Negation
Modus Ponens
Modus Tollens
Mutual Exclusion
Simplification

VALIDITY PROOFS

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Mutual Exclusion

Mutual Exclusion is a rule of inference pertaining to the XOR operator.

Mutual Exclusion states that in an exclusive disjunction, if one of the exclusive disjuncts is true, then the other has to be false.

Imagine we are given an exclusive disjunction: "Alice married Alex or Paul." We know that this is an exclusive disjunction as opposed to a regular disjunction because Alice could not have married both Alex and Paul. Let's recognize two distinct statements in this exclusive disjunction: 1) Alice married Alex, and 2) Alice married Paul. Now, if we are told that Alice married Paul, we have no choice but to conclude that she did not marry Alex. Formally, it looks like this:

p XOR q: "Alice married Alex or Paul."
q: "Alice married Paul."
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~p: "Alice did not marry Alex."

The given exclusive disjunction p XOR q and the expression q are above the line of dashes, and the conclusion ~p obtained by applying the Mutual Exclusion rule is below the line.

Other examples of Mutual Exclusion

A XOR B: "The client is in Austria or Germany."
B: "The client is in Germany."
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~A: "The client is not in Austria."

X XOR Y: "The cafeteria food is acceptable or appalling."
X: "The cafeteria food is acceptable."
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~Y: "The cafeteria food is not appalling."

These are links to validity proof problems whose solutions contain Mutual Exclusion.