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.

## No comments:

## Post a Comment