Consider a logic where "false" facts f add up
to F when multiple errors occur.
Introduce a "noddy" n joker that can be played
to turn a single f into a "true" t.
Show the truth tables for:
AND OR NOT IMPLIES
A & B A | B ! A A ⇒ B
t t t t T t f t t t t
f f t f t t t f f t t
t f f t t f t f f
f F f f f f f t f
n n t n t t F n n t t
t n n t t n t n n
n t f n n f n F f
f t n f n n f t n
n N n n f n n N n
F F t F t t n F F t t
t F F
F F f F F f F N f
f t F
F F n F N n F F n
n F F
F F F F F F F N F
Where F = f&f ≠ f is kept,
while we substitute:
T := t directly and N := n finally.
Both logic operations & and |
are commutative, so that: A&B = B&A
but only OR |
is associative: A|(B|C) = (A|B)|C
There's a simple arithmetical trick behind our truth tables:
We gave "true" the value t=0
and by putting f=1 as one "false"
we allow AND & to add facts A+B
to larger natural numbers F>1
which are deemed "falser".
The new truth value "noddy" n=-1
acts as our negative unit, so f&n = t.
To OR | was calculated by multiplication A*B
and the negation !A evaluated as (A|n)&f
equals 1 - A.
Now the conundrum F & n = 2-1 | 3-1 = f | F
is solved by 1*2 = 2 = F.
We defined implication A⇒B by (!A)|B = (1-A)*B
and alternatively as B - A*B = (n|A|B)&B
where you may choose to leave conflicting cases "undefined" ?
rather than resolving every operation as above.
