Re: An Affine Logic Example: Łukasiewicz Logic

On 12/21/2024 02:20 PM, Mild Shock wrote:

Hi,
>
An example of an affine Logic, is this 3-valued
Logic with the following implication truth table:
>
     F    U    T
F    T    T    T
U    U    T    T
T    F    U    T
>
It satisfies modus ponens:
>
/* Implication Elimination */
?- tauto((X & (X->Y) => Y)).
true.
>
It satisfies the types of combinators BCK:
>
/* K Combinator */
?- tauto((X -> Y -> X)).
true.
>
/* B Combinator */
?- tauto(((Y -> Z) -> ((X -> Y) -> (X -> Z)))).
true.
>
/* C Combinator */
?- tauto(((X -> (Y -> Z)) -> (Y -> (X -> Z)))).
true.
>
And surprise surprise, it doesn't satisfy contraction,
the formula that Julio doubted that it is unprovable:
>
?- tauto(((X -> (X -> Y)) -> (X -> Y))).
false.
>
Bye
>

quasi-modal
How about instead
B both
N neither
X don't care
? don't know
T true
F false
It depends on propositions fulfilling question words,
all of them.
That you have "material implication"
is not necessarily anybody else's problem.
I.e., nobody needs "the quasi-modal", at all,
except to make broken logics like those.