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VALIDITY PROOFS

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."
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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