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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 Addition

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

Disjunctive Addition adds any statement, true or false, to a given true statement.

Let's consider a statement "The Moon revolves around the Earth". We know that this statement is true -- it is a proven fact. Now that we are given this true statement, we can add any other statement to it by applying Disjunctive Addition. This is how it is done formally:

p: "The Moon revolves around the Earth."
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p v q: "The Moon revolves around the Earth or the Earth is larger than the Moon."

The given statement p is above the line of dashes, and the new expression p v q formed by applying Disjunctive Addition is below the line.

We can also add a statement known to be false to a given true statement:

p: "The Moon revolves around the Earth."
----------
p v q: "The Moon revolves around the Earth or the Earth is smaller than the Moon."

This is possible because, by definition, Disjunctive Addition can add any statement, true or false, to a given true statement.

This is not the extent of the application of Disjunctive Addition. We can add absolutely any statement to a given true statement, even if there does not seem to be a connection between the statements.

p: "The Moon revolves around the Earth."
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p v q: "The Moon revolves around the Earth or smoking causes lung cancer."

This is possible because of the inherent property of the OR operator: A disjunction is true if at least one of its statements is true. Therefore, p v q "The Moon revolves around the Earth or smoking causes lung cancer" is a true statement because we know for a fact that its first part is true -- the moon does revolve around the earth. Knowing that, we don't have to worry about the second part ("smoking causes lung cancer") -- regardless of whether it is true or not, it is not going to affect the entire statement p v q. In this case, the second part happens to be true -- smoking does cause lung cancer, but we might as well have picked any other, true or false, statement.

 

Other examples of Disjunctive Addition

A: "The water is cold."
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A v B: "The water is cold or the day is hot."

 

X: "The painting is extraordinary."
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X v ~Y: "The painting is extraordinary or the artist is not talented."

 

Links to Relevant Problems

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

2-step problem