Let's reframe the concept of will through the lens of formal logic, drawing on the principles of various logical systems to provide a more rigorous, formalized perspective:
- Axiomatic Basis:
Let W represent will as a primitive concept or axiom in our logical system. We posit that W is self-referential and can operate on itself.
- Self-Application:
Define an operation ∘ such that W ∘ W represents will operating on itself. This operation is unique in that it can result in either self-affirmation or self-negation.
- Modal Logic Framework:
Introduce modal operators □ (necessarily) and ◇ (possibly).
Define: □(W ∘ W) ↔ W (Will necessarily applies to itself if and only if it is will)
◇(¬(W ∘ W)) (It's possible for will not to apply to itself)
- Multi-Valued Logic:
Extend beyond binary logic to accommodate the nuanced nature of will:
Let T = {0, u, 1} where 0 = false, u = undetermined, 1 = true
Define W : T → T as a function that can map undetermined states to determined ones.
- Quantum Logic Integration:
Introduce a superposition operator S such that S(W) represents will in a state of superposition between self-affirmation and self-negation.
Collapse function C: C(S(W)) → {W, ¬W}
- Gödel-like Incompleteness:
Formulate a statement G: "G is not affirmed by W" If W affirms G, then G is false, contradicting W's affirmation. If W does not affirm G, then G is true, but W has failed to affirm a true statement. This demonstrates that W cannot be fully captured within its own logical system.
- Type Theory:
Define a hierarchy of will types: W₀: Base level will W₁: Will that can operate on W₀ W₂: Will that can operate on W₁ and W₀ And so on, creating a potentially infinite hierarchy.
- Paraconsistent Logic:
Allow for the possibility that both W and ¬W can be true simultaneously without trivializing the system, accommodating the paradoxical nature of will.
- Temporal Logic:
Introduce temporal operators P (it was the case that) and F (it will be the case that) to capture the dynamic nature of will over time: G(W ∘ W) (It is always the case that will applies to itself) F(¬(W ∘ W)) → F(W ∘ W) (If will ever doesn't apply to itself, it will eventually apply to itself again)
- Category Theory Perspective:
Consider will as an endofunctor W in the category of cognitive states, with the peculiar property that W can be both a morphism and an object.
- Fixed Point Theory:
Define the fixed point of will as a state s where W(s) = s, representing a state of perfect self-alignment.
- Non-Standard Analysis:
Introduce infinitesimal quantities to represent infinitesimal changes in will, allowing for a continuous spectrum between discrete states of will.
Formal Definition:
Will (W) is a self-referential operator in a multi-valued, paraconsistent logical system that exhibits properties of quantum superposition, temporal dynamics, and Gödelian incompleteness. It is characterized by its ability to operate on itself (W ∘ W), potentially infinite type hierarchy, and the capacity to resolve undetermined states into determined ones.
Key Theorem:
The complete nature of W cannot be fully captured within any finite logical system that includes W as an element.
This formalization attempts to capture the complex, self-referential, and transcendent nature of will, using tools from various branches of logic and mathematics. It provides a framework for rigorously discussing the paradoxes and unique properties of will while acknowledging the inherent limitations of any such formalization.