|
P |
Q | P and Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Logical Operators
A logical operator joins two propositions to form a new, complex, proposition. (If you have not read about propositions, you should do so now. Follow the Next links until you return to this page.) The truth value of the new proposition is determined by the truth values of the two propositions being joined and by the operator that joins them.
Any two propositions P and Q can be conjoined, producing the new, complex, proposition:
P and Q
The proposition P and Q is true if and only if both P and Q are true. It is false otherwise.
Examples:
Any two propositions P and Q can be disjoined, producing the new, complex, proposition:
P or Q
The proposition P and Q is true if and only if either P or Q are true. It is false only if both P and Q are false.
Examples:
|
P |
Q | P or Q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Any two propositions P and Q can be joined by a conditional operator, producing the new, complex, proposition:
If P then Q
The proposition If P then Q is true if and only if either P is false or Q is true. It is false only when P is true and Q is false.
Examples:
|
P |
Q | If P then Q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Proof: A special conditional occurs if we flip the P and Q around: we get if Q then P, which is the same as saying P only if Q.
Any proposition P can be converted into its negation with a negation operator, producing the new, complex, proposition:
Not P
The proposition Not P is true if and only if P is false. It is false only if P is true. The truth table for Not P is as follows:
Examples:
|
P |
Q | Not P |
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | T |
Proof: It does not matter whether Q is true or false. Every time, Not P is true if P is false, and false if P is true.
Any two propositions P and Q can be joined with the biconditional operator, producing the new, complex, proposition:
P if and only if Q
The proposition P if and only if Q is true if and only if both P and Q are true, or if both P and Q are false. It is false only when one of them is true and the other false.
Examples:
|
P |
Q | Not P if and only if Q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Proof: P if and only if Q is the same as:
( If P then Q ) and ( P only if Q ). This is like saying:
( If P then Q ) and ( If Q then P ).
</dl>
The biconditional is a complex operator, built out of simpler operators. Think of it this way:
<dl>The if and only if operator plays a special role in definitions. When we say P if and only if q, we are saying that P says the same thing as Q.