Interesting. I am rather sceptical about the likelihood that this will lead to concrete, useful results. But the discussion and explanation from the author in the site's comments convinced me that it's a question worth exploring and she's doing it with an open mind and full understanding of the concepts she plans to challenge.
There's ample precedence in math and logic for ideas that seemed foolish at first glance but turned out to lead to fruitful results. My favorite example is imaginary numbers: just ad something that in your previous system is impossible to its axioms, and it turns out to be not only logically consistent but also eventually to have practical applications.
An interesting approach to this might be a language which has first-class constructs for probabilistic programming techniques - something like ProbLog, for example http://dtai.cs.kuleuven.be/problog/
There's ample precedence in math and logic for ideas that seemed foolish at first glance but turned out to lead to fruitful results. My favorite example is imaginary numbers: just ad something that in your previous system is impossible to its axioms, and it turns out to be not only logically consistent but also eventually to have practical applications.