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


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


5-step or more
Bad Argument

Disjunctive Inference

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

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

Imagine we are given a disjunction: "The patient has an adjustment disorder or he has depression." Let's recognize two distinct statements in this disjunction: 1) The patient has an adjustment disorder, and 2) The patient has depression. Now, if we are told that the patient does not have an adjustment disorder, we have no choice but to conclude that the patient suffers from depression. Formally, it looks like this:

p v q: "The patient has an adjustment disorder or he has depression."
~p: "The patient does not have an adjustment disorder."
q: "The patient has depression."

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


Other examples of Disjunctive Inference

A v B: "The student passed the quiz or she passed the exam."
~B: "The student failed the exam."
A: "The student passed the quiz."

X v Y: "The juice is cold or the tea is hot."
~Y: "The tea is not hot."
X: "The juice is cold."


Links to Relevant Problems

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

2-step problem (A)
2-step problem (B)
4-step problem
5-step problem