Post Logic

Origin. Emil Post (1921). First systematic many-valued logics. n-valued truth tables. Generalized negation cycles. Functional completeness results. Foundation for multi-valued computation.

Models. Truth values: 0, 1, ..., n-1. Cyclic negation: N(i) = i+1 mod n. Designated values: typically {n-1} or {k, k+1, ..., n-1}. All truth functions definable.

Formalism.

Truth values: V_n = {0, 1, 2, ..., n-1} Ordered: 0 < 1 < ... < n-1

Post negation (cyclic): N(v) = (v + 1) mod n N^n(v) = v (returns after n applications)

Post disjunction: v₁ ∨ v₂ = max(v₁, v₂)

Post conjunction: v₁ ∧ v₂ = min(v₁, v₂)

Jᵢ operator: Jᵢ(v) = n-1 if v = i, else 0 "Exactly value i" detector.

Functional completeness: {N, max, J₀} generates all n-valued functions. Any truth function expressible.

Post implication: v₁ → v₂ = min(n-1, n-1-v₁+v₂)

Designated values: Typically D = {n-1} (only top value). Valid: all arguments yield designated outputs.

Symbols.

Symbol	Unicode	Name	Meaning
N	—	Negation	Cyclic successor
Jᵢ	—	J operator	Value detector
∨	U+2228	Disjunction	Maximum
∧	U+2227	Conjunction	Minimum
Vₙ	—	Values	n-element set

Metatheory. Functional completeness. Decidable validity. Finite matrices. Post algebras. Relationship to multi-valued switching. Generalization of Boolean algebra.

Applies to. Multi-valued circuit design. Switching theory. Theoretical computer science. Mathematical logic foundations. Finite-valued semantics.

Limitations. Cyclic negation unnatural for some interpretations. Many systems since Post. Limited philosophical motivation. Engineering focus.