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LOGIC MAIN PAGE

LOGICAL OPERATORS

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

RULES OF LOGIC

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

VALIDITY PROOFS

2-step
3-step
4-step
5-step or more
Bad Argument

Disjunctive Inference with XOR

Disjunctive Inference with XOR is a rule of inference pertaining to the XOR operator.

Disjunctive Inference with XOR states that in an exclusive disjunction, if one of the exclusive disjuncts is false, then the other has to be true. This is what is commonly referred to as the process of elimination.

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 did not marry Alex, we have no choice but to conclude that she married Paul. Formally, it looks like this:

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

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

 

Other examples of Disjunctive Inference with XOR

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


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

 

Links to Relevant Problems

These are links to validity proof problems whose solutions contain Disjunctive Inference with XOR.

2-step problem