Noddy Logic

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.