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is a formula.A\/B

**Alternative Forms** A disjunction of `A` and `B` may also
appear in other syntactic forms, e.g. as

`A`+`B`,`A`|`B`,`A``||`

`B`;- "
`A`or`B`"; `or(`

.`A`,`B`)

The last form is the input syntax for the Logic Evaluator.

ABA\/Bfalsefalsefalsefalsetruetruetruefalsetruetruetruetrue

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

The Java expressionpublic 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 From Definition *Semantics of Disjunction*, we can deduce the
following properties of disjunctions.

Disjunction is alsoA\/BiffB\/A.

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

The disjunctionA\/Biff ~(~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,

for conjunctions ofA_{0}\/A_{1}\/ ... \/A_{n-1}

```
````or(``A`_{0}, `A`_{1}, ..., `A`_{n-1})

of an arbitrary number of formulas.
**Exclusive Disjunction** Finally we give the definition of the
"exclusive or" connective mentioned above.

(is a formula, theAxorB)

In other words,

ABAxorBfalsefalsefalsefalsetruetruetruefalsetruetruetruefalse

We then have the following relationship between both kinds of disjunctions.

(AxorB) 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`).

Author: Wolfgang Schreiner

Last Modification: October 4, 1999