Input-Output Logic

Origin. Makinson and van der Torre introduced input/output logic (2000). Logic for conditional norms and directives. Not truth-functional: norms don't have truth values. Input: factual situation. Output: what ought to be. Foundation for normative reasoning.

Models. Norms as input-output pairs. Norm (a, x): given input a, output x. Not "if a then x is true" but "given a, x is obligatory." Generators: set of basic norms. Derive outputs from inputs via operations. Different operations yield different normative systems.

Formalism.

Norm: (a, x) ∈ G where G is set of generators. a: body/antecedent, x: head/consequent.

Input-output operations: Given input A (set of formulas), compute output.

out₁(G, A) = Cn(G(Cn(A))) Simple-minded: apply norms to logical closure of input.

out₂(G, A) = ∩{Cn(G(V)) : A ⊆ V, V complete} Basic: intersect outputs over all extensions.

out₃(G, A) = ∩{Cn(G(B)) : A ⊆ B = Cn(B) ⊇ G(B)} Reusable: output can feed back as input.

out₄: combines out₂ and out₃.

Throughput (reusable output): Output can reinforce itself: if (a,b) and (b,c) then input a yields c.

Permissions: Negative: not prohibited (¬out(G, a) contains x). Positive: explicitly permitted. Static vs dynamic permissions.

Constraints: Constrained output: outputs must be consistent with constraints.

Symbols.

Symbol	Unicode	Name	Meaning
(a, x)	—	Norm	Conditional directive
G	—	Generators	Basic norms
out	—	Output	Derivation operation
Cn	—	Closure	Logical consequence
A	—	Input	Given facts
⊗	U+2297	Aggregation	Combining norms

Metatheory. Different out operations have different properties. out₁ satisfies SI (strengthening input), WO (weakening output), AND. out₃ adds CT (cumulative transitivity). Complexity: depends on base logic. No truth values for norms — avoids Jørgensen's dilemma.

Applies to. Legal reasoning (statutes as norms). Ethics (moral rules). AI norm-governed agents. Organizational rules. Contracts. Default rules. Policy specification.

Limitations. Many variant systems — no consensus. Permissions complex to handle. Contrary-to-duty puzzles partially addressed. No probabilities. Norm change not modeled. Abstract — implementation not direct.