Independence results like this are often established by semantics.
Common Mistakes:
- Few students forgot to show that MP preserves ‘truths’, thus they didn’t state that all theorems are ‘true’.
- Checking all possible valuations of the axiom schemata is crucial in the proof of independence.
Challenge: Prove that Peirce’s Law A\supset B\supset A\supset A cannot be deduced by using only axiom schemata A\supset .B\supset A and [A\supset .B\supset C]\supset .A\supset B\supset .A\supset C.
Hint: design a three-valued truth assignment and show that any consequence deduced from the axioms will always take value T and Peirce’s Law won’t. You may consider the following table and try to figure out what values x, y and z should take respectively.
| A | B | A\supset B |
|---|---|---|
| T | T | T |
| T | M | M |
| T | F | F |
| M | T | x |
| M | M | y |
| M | F | z |
| F | T | T |
| F | M | T |
| F | F | T |