Are the following lambda-calculus definitions axiomatic?
- true: $\lambda xy.x$
- false: $\lambda xy.y$
Is the definition truly arbitrary? In my impression, it looks like we could just swap the definitions for true and false around. Would an alternative definition such as the following one be ok:
- true: $\lambda xy.y$
- false: $\lambda xy.x$
Is it possible that "true" is just an arbitrary choice, while "false" then needs to be the opposite of "true" in a certain way?
Would the following definition by outright wrong:
- true: $\lambda xy.xz$
- false: $\lambda xy.yz$
If more than one definition for true and false is possible, how could we describe the class of "consistent" definitions for true and false?