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A \/ Bis a formula.
Alternative Forms A disjunction of A and B may also appear in other syntactic forms, e.g. as
|| B;
or(A, B).
The last form is the input syntax for the Logic Evaluator.
A B A \/ B false false false false true true true false true true true true
In other words, A \/ B is false if and only if both A and B are false.
Since A \/ B is also true if both A and B are true, it must not be read as "either A or B" (which would indicate otherwise). Such an "exclusive or" connective needs an extra definition (which is given at the end of this section).
Operational Interpretation In the Logic Evaluator, a disjunction is represented by an object of the Java type
public final class Or implements Formula
{
private Formula formula0;
private Formula formula1;
public Or(Formula _formula0, Formula _formula1)
{
formula0 = _formula0;
formula1 = _formula1;
}
public boolean eval() throws EvalException
{
if (formula0.eval())
return true;
else
if (formula1.eval())
return true;
else
return false;
}
}
(new Or(A, B)).eval() computes the
truth value of A \/ B. As one can see, if A evaluates
to true, the result is immediately true, i.e., the truth
value of B does not matter any more. Only if A evaluates to
false, also B is evaluated; the result is false only
if both formulas are false.
From Definition Semantics of Disjunction, we can deduce the following properties of disjunctions.
A \/ B iff B \/ A.Disjunction is also associative, i.e., for all formulas A, B, and C, we have
A \/ (B \/ C) iff (A \/ B) \/ C.
We have an important duality between conjunctions and disjunctions expressed by the following law.
~(A /\ B) iff ~A \/ ~B, ~(A \/ B) iff ~A /\ ~B.
Consequence From above laws, we have
A \/ B iff ~(~A /\ ~B).The disjunction A \/ B is therefore frequently defined just as a syntactic abbreviation of ~(~A /\ ~B).
Convention Because of associativity, it does not matter in which particular way parentheses are placed in nestings of disjunctive formulas. We will therefore write A \/ B \/ C instead of A \/ (B \/ C) respectively (A \/ B) \/ C and, in general,
A0 \/ A1 \/ ... \/ An-1for conjunctions of n formulas (for every n). Also the Logic Evaluator allows disjunctions
or(A0, A1, ..., An-1)
of an arbitrary number of formulas.
Exclusive Disjunction Finally we give the definition of the "exclusive or" connective mentioned above.
(A xor B)is a formula, the exclusive disjunction (Ausschließende Oder-Verknüpfung) of A and B, The meaning of this formula is defined by the following truth table:
In other words, A xor B is true, if and only if exactly one of A or B is true.
A B A xor B false false false false true true true false true true true false
We then have the following relationship between both kinds of disjunctions.
(A xor B) iff (A /\ ~B) \/ (~A /\ B).
Because of this law, the formula (A xor B) is frequently just defined as a syntactic abbreviation for (A /\ ~B) \/ (~A /\ B).