# DeMorgan's Laws Analysis in Ferguson's Bilateral System

## Ferguson's Exact Statement (p. 13)

"The reader can confirm that the bilateral approach in fact improves on the presentation for wK inasmuch as DeMorgan's laws are reestablished; as S*_fde and AC are our actual targets, this should relieve concerns about their failure in wK."

## Key Observation

Ferguson states: "One further observation is required, establishing that **V₃² is in fact closed** under the bilateral interpretation of the restricted quantifiers."

This is followed by **Lemma 2**: "V₃²—the collection of S*_fde truth values—is closed under the above interpretation of the restricted quantifiers."

## Critical Insight

Ferguson seems to be saying that DeMorgan's laws are "reestablished" in a specific sense:
1. The set V₃² is **closed** under the quantifier operations
2. This closure property is what "improves" on wK

## What "Reestablished" Might Mean

Looking at Definition 15, the bilateral quantifiers use:
- For ∃: Verification uses ∃̇, Falsification uses ∀̇
- For ∀: Verification uses ∀̇, Falsification uses ∃̇

This **duality** between ∃ and ∀ in the verification/falsification coordinates might be what Ferguson means by DeMorgan's laws being "reestablished."

## The Confusion

Ferguson might mean that DeMorgan's laws hold at the **semantic level** for the bilateral truth values, not necessarily at the **syntactic level** for formulas.

Consider:
- In bilateral semantics, negation swaps coordinates: I(~φ) = ⟨I₁(φ), I₀(φ)⟩
- The quantifiers have built-in duality between ∃̇ and ∀̇
- This might restore DeMorgan-like relationships at the semantic level

## Our Implementation Issue

Our tests check syntactic DeMorgan's laws:
```
~(P(a) & Q(a)) ⊢ (~P(a) | ~Q(a))  # Fails
```

But Ferguson might mean semantic DeMorgan's laws on the bilateral pairs:
```
⟨u,v⟩ for ~(P & Q) relates properly to ⟨u',v'⟩ for (~P | ~Q)
```

## The Gap

The statement "V₃² is in fact closed under the bilateral interpretation" suggests that the restoration of DeMorgan's laws is about:
1. **Closure properties** of the truth value space
2. **Semantic relationships** between operations
3. Not necessarily **syntactic validity** of inferences

## Hypothesis

Ferguson's claim about DeMorgan's laws being "reestablished" might mean:
1. The bilateral semantic space has the right algebraic properties
2. The duality between ∃ and ∀ is preserved in the bilateral setting
3. But this doesn't automatically make syntactic DeMorgan inferences valid

## Test Needed

We should test whether the semantic values satisfy DeMorgan's laws:
- Given bilateral values for P and Q
- Compute ~(P & Q) using bilateral semantics
- Compute (~P | ~Q) using bilateral semantics  
- Check if they're equal

This is different from checking if the syntactic inference is valid in the tableau system.

## Conclusion

The discrepancy might not be a bug but a misunderstanding of what Ferguson means by "DeMorgan's laws are reestablished." He might be referring to:
1. Algebraic properties of the bilateral semantic space
2. Duality preservation in quantifier operations
3. Not syntactic validity of DeMorgan inferences

This needs further investigation at the semantic level rather than the proof-theoretic level.