| Description: Define disjunction
(logical 'or'). Definition of [Margaris] p.
49. When
the left operand, right operand, or both are true, the result is true;
when both sides are false, the result is false. For example, it is true
that (2 = 3 ∨ 4 = 4) (see
ex-or in set.mm). After we define the
constant true ⊤ (df-tru 1319) and the constant false ⊥
(df-fal 1320), we will be able to prove these truth table
values:
(( ⊤ ∨ ⊤ ) ↔
⊤ ) (truortru 1340), (( ⊤
∨ ⊥ ) ↔ ⊤ )
(truorfal 1341), (( ⊥
∨ ⊤ ) ↔ ⊤ ) (falortru 1342), and
(( ⊥ ∨ ⊥ ) ↔
⊥ ) (falorfal 1343).
This is our first use of the biconditional connective in a definition; we
use the biconditional connective in place of the traditional
"<=def=>",
which means the same thing, except that we can manipulate the
biconditional connective directly in proofs rather than having to rely on
an informal definition substitution rule. Note that if we mechanically
substitute (¬ φ →
ψ) for (φ ∨ ψ), we end up with an
instance of previously proved theorem biid 227.
This is the justification
for the definition, along with the fact that it introduces a new symbol
∨. Contrast with ∧ (df-an 360), → (wi 4), ⊼
(df-nan 1288), and ⊻ (df-xor 1305) . (Contributed by NM,
5-Aug-1993.) |
