Description: 3 of 3 axioms for propositional calculus due to Lukasiewicz. Copy of luk-3 and pm2.24 , but introduced as an axiom.
One might think that the similar pm2.21 ( -. ph -> ( ph -> ps ) ) is a valid replacement for this axiom. But this is not true, ax-3 is not derivable from this modification.
This can be shown by designing carefully operators -. and -> on a finite set of primitive statements. In propositional logic such statements are T. and F. , but we can assume more and other primitives in our universe of statements. So we denote our primitive statements as phi0 , phi1 and phi2. The actual meaning of the statements are not important in this context, it rather counts how they behave under our operations -. and -> , and which of them we assume to hold unconditionally (phi1, phi2). For our disproving model, I give that information in tabular form below. The interested reader may check by hand, that all possible interpretations of ax-mp , ax-luk1 , ax-luk2 and pm2.21 result in phi1 or phi2, meaning they always hold. But for wl-luk-ax3 we can find a counter example resulting in phi0, not a statement always true.
The verification of a particular set of axioms in a given model is tedious and error prone, so I wrote a computer program, first checking this for me, and second, hunting for a counter example. Here is the result, after 9165 fruitlessly computer generated models:
ax-3 fails for phi2, phi2
number of statements: 3
always true phi1 phi2
Negation is defined as
----------------------------------------------------------------------
-. phi0 | -. phi1 | -. phi2 |
phi1 | phi0 | phi1 |
p->q | q: phi0 | q: phi1 | q: phi2 |
p: phi0 | phi1 | phi1 | phi1 |
p: phi1 | phi0 | phi1 | phi1 |
p: phi2 | phi0 | phi0 | phi0 |
Ref | Expression | ||
---|---|---|---|
Assertion | ax-luk3 | |- ( ph -> ( -. ph -> ps ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | wph | |- ph |
|
1 | 0 | wn | |- -. ph |
2 | wps | |- ps |
|
3 | 1 2 | wi | |- ( -. ph -> ps ) |
4 | 0 3 | wi | |- ( ph -> ( -. ph -> ps ) ) |