F L i P : Logical Framework in Python

Jonathan Jacky

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Flip is a logical framework written in Python. A logical framework is a library for defining logics and writing applications such as theorem provers. One Flip application is a proof checker for entering and editing proofs in natural deduction style:

Kaye ex. 9.12, ~Ax.P(x) |- Ex.~P(x)  (0)  Comment
~Ax.P(x)                  (1)  Given
|~Ex.~P(x)                (2)  Assumption
||Let x be arbitrary      (3)  New variable for subproof
|||~P(x)                  (4)  Assumption
|||Ex.~P(x)               (5)  E-Introduction (4)
|||F                      (6)  Contradiction (5) (2)
||~~P(x)                  (7)  Reductio Ad Absurdum (4) (6)
||P(x)                    (8)  Not-Elimination (7)
|Ax.P(x)                  (9)  A-Introduction (3) (8)
|F                       (10)  Contradiction (9) (1)
~~Ex.~P(x)               (11)  Reductio Ad Absurdum (2) (10)
Ex.~P(x)                 (12)  Not-Elimination (11)

This is the output from the checker. Here is the proof script. Here is a more Pythonic example.

The checker can use different logics; Flip comes with several. You can add another logic, or add axioms and derived rules, by writing a module in Python. Python is both the object language and the metalanguage. Formulas, inference rules, and entire proofs are Python expressions. Prover commands are Python functions. The Python interpreter itself is the only user interface to the proof checker application. (It is not necessary to know much Python to use the checker.)

Flip was undertaken as a Python programming exercise. It is not intended to compete with industrial-strength theorem provers such as HOL, nor with nicely-designed educational provers such as Jape, etc. That said, the checker is quite capable of working the examples and exercises in university-level textbooks on logic for computer science or mathematics, such as Kaye, Huth and Ryan, and Bornat.

Always demand proof, proof is the elementary courtesy that is anyone's due.     — Valéry

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