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

2-step
3-step
4-step
5-step or more

OR Operator

The logical operator OR (symbolized v) is called the disjunction.

It is used to join two statements to assert that at least one and possibly both of the statements are true. OR is always inclusive. Here is an example:

Little League coaches know the rules for baseball or soccer.

It is essential not to confuse the inclusive OR, described here, with the exclusive OR, called XOR. In the example given above, there are three possibilities: The Little League coaches know the rules for baseball only, for soccer only, or for both baseball and soccer. The inclusive OR is so called because as you can see, it includes every possible combination of the two statements (one, the other, or both).

In formal logic, we can replace the two statements with the letters p and q (we can use any other letters that we want to). So, let the statement "Little League coaches know the rules for baseball" be p and let the statement "Little League coaches know the rules for soccer" be q.

Now we can rewrite our example in the formal way:

p v q: "Little League coaches know the rules for baseball or soccer."

This is a logical expression. In this expression, p and q are called disjuncts.

Truth table for OR

A truth table lists truth values (that is, either true or false) for every possible combination of component values in a logical expression.

In order to build a truth table for OR, we need to consider the simplest example of a disjunction:

p v q: "Little League coaches know the rules for baseball or soccer."

In formal logic, p and q (called disjuncts) can take on two possible values: true or false. How many combinations of p and q can we come up with? The answer is four. Let us list all the possible combinations:

1. p is true and q is true
2. p is true and q is false
3. p is false and q is true
4. p is false and q is false

We know that a disjunction is true when at least one and possibly both disjuncts are true. There are three cases in which this is valid -- it's the first, second, and third cases. Now we can build a truth table. We are using T for true and F for false.

 p q p v q T T T T F T F T T F F F

As you read the table, you will notice that when p is true and q is true, the entire expression is true. When p is true and q is false, the entire expression is also true, and so on.

Commutativity of Disjunction

We can write p v q or q v p -- these two expressions are equivalent. It's the same as the phrase "She visited Chile or she visited Peru" is completely equivalent to "She visited Peru or she visited Chile."

Rules of Disjunction

These are links to specific rules pertinent to the OR operator.