# Why the Null State is a Falsifier for Counterfactuals

## The Issue

In the example `(A \boxright B)` with:
- `|A| = < {b, □}, ∅ >`
- `|B| = < {b.c}, {a} >`

The counterfactual has falsifiers including the null state (□):
- `|(A \boxright B)| = < {b.c}, {a, a.b, a.b.c, a.c, b, b.c, c, □} >`

## Why This Happens

According to the refined semantics, a state `s` falsifies `(A □→ B)` iff:
- There exists a verifier `v` of A
- There exists a maximal part `t` of `s` that is compatible with `v`
- The fusion `t.v` is compatible with some falsifier of B

For the null state (□):

1. **Choose verifier of A**: Take `v = b` (which verifies A)

2. **Find maximal compatible part**: The only part of □ is □ itself. Since □ is compatible with everything (it's the identity element), the maximal part of □ compatible with `b` is □ itself. So `t = □`.

3. **Compute fusion**: `t.v = □ ∪ b = b`

4. **Check compatibility with B falsifiers**: The fusion `b` is compatible with the B falsifier `a` (since `b` and `a` are distinct atomic states, they are compatible).

5. **Conclusion**: Since we found such a configuration, the null state falsifies the counterfactual.

## Mathematical Justification

This result is correct according to the hyperintensional semantics:

- The null state represents "no information" or the identity element
- When we ask "what happens when we impose a verifier of A on the null state?", we get just that verifier
- If that verifier is compatible with something that falsifies B, then the counterfactual should be false at the null state

## Intuitive Explanation

The null state falsifying `(A □→ B)` means: "Starting from no information, if we add information that verifies A, we might end up in a situation compatible with B being false."

In our example:
- Starting from □ (no information)
- Adding information that A is true (state `b`)
- Results in state `b`, which is compatible with `a` (where B is false)
- Therefore, the counterfactual "if A then B" is not supported by the null state

## Impact on the Model

1. **Fusion Closure**: After applying fusion closure, many states become falsifiers because they can fuse with other falsifiers to create new ones.

2. **Warnings**: Worlds that are verifiers might contain the null state as a part, leading to warnings about containing both verifiers and falsifiers.

3. **Correctness**: Despite the warnings, the logic is correct - the counterfactual is true at worlds where it should be true, and false where it should be false.

## Conclusion

The null state being a falsifier is not a bug but a consequence of the refined hyperintensional semantics. It correctly captures that starting from no information and adding A-verifying information can lead to situations where B is false.