Truth tables

AND truth table

The conjunction and produces the following set of possible outcomes:

P	Q	P AND Q
TRUE	TRUE	TRUE
TRUE	FALSE	FALSE
FALSE	TRUE	FALSE
FALSE	FALSE	FALSE

As you can see from the truth table, it is only if both conditions are true that the conjunction will equate to true. If one or other or both of the conditions in the conjunction are false, then the conjunction equates to false.

Also notice that when conditions are connected by AND that a single condition being false is sufficient for the conclusion to be false, but a single condition being true is not sufficient for the conclusion to be true (you would need to know the value of the other conditions).

In this way, the conjunction itself has its own truth value which is distinct from each of the conditions contained within (ie one of the conditions may be true, but the value of the conjunction is false).

OR truth table

The disjunction or produces the following set of possible outcomes:

P	Q	P OR Q
TRUE	TRUE	TRUE
TRUE	FALSE	TRUE
FALSE	TRUE	TRUE
FALSE	FALSE	FALSE

As you can see, if one or other of the conditions is true, the disjunction will equate to true. It is only if both conditions are false that the entire collection equates to false.

Also notice that when conditions are connected by OR that a single condition being true is sufficient for the conclusion to be true, but a single condition being false is not sufficient for the conclusion to be false (you would need to know the value of the other conditions).

A disjunction also has its own truth value distinct from its conditions (ie one of the conditions may be false, but the value of the disjunction is true).

Uncertain truth tables

The following truth tables show how uncertainty works:

P	Q	P AND Q
TRUE	UNCERTAIN	UNCERTAIN
UNCERTAIN	TRUE	UNCERTAIN
FALSE	UNCERTAIN	FALSE
UNCERTAIN	FALSE	FALSE
UNCERTAIN	UNCERTAIN	UNCERTAIN

P	Q	P OR Q
TRUE	UNCERTAIN	TRUE
UNCERTAIN	TRUE	TRUE
FALSE	UNCERTAIN	UNCERTAIN
UNCERTAIN	FALSE	UNCERTAIN
UNCERTAIN	UNCERTAIN	UNCERTAIN

The uncertain operator causes the condition to return true only if its value is uncertain. A condition using the uncertain operator returns false if the underlying value is not uncertain.