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.

Reusability (out₃): Output can feed back as input: if (a,b) and (b,c) are generators, input a yields c.

Throughput (out⁺): A separate property — the identity rule, input included in output. Without it a ∉ out(G, a), since a norm's body is not itself something that ought to be.

Permissions: Negative: not prohibited (¬x ∉ out(G, a)). 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
out₁	—	Simple-minded output	Cn(G(Cn(A)))
out₂	—	Basic output	Intersection over complete extensions
out₃	—	Reusable output	Output may feed back as input
out₄	—	Basic reusable output	out₂ and out₃ combined
out⁺	—	Throughput	Output operation with identity
SI	—	Strengthening the input	Rule characterizing out₁
WO	—	Weakening the output	Rule characterizing out₁
AND	—	Conjunction of outputs	Rule characterizing out₁
OR	—	Disjunction of inputs	Rule distinguishing out₂
CT	—	Cumulative transitivity	Rule distinguishing out₃
ID	—	Identity	Rule giving throughput
V	—	Complete set	Maximal consistent extension of the input
⁺	U+207A	Superscript plus	Marks the throughput variants
₁	U+2081	Subscript one	Indexes the output operations
∈	U+2208	Membership	(a, x) ∈ G
∉	U+2209	Non-membership	a ∉ out(G, a) without throughput
∩	U+2229	Intersection	Defines out₂ and out₃
⊆	U+2286	Inclusion	A ⊆ V in the definition of out₂
⊇	U+2287	Reverse inclusion	B ⊇ G(B), the fixed-point condition
¬	U+00AC	Negation	Negative permission

Metatheory. The central results of Makinson and van der Torre (2000) are representation theorems: each semantically defined output operation coincides exactly with a derivation system built from a short list of rules. out₁ is generated by SI, WO and AND; out₂ adds OR; out₃ adds CT to the first three; out₄ has all five. The throughput variants out_i⁺ are obtained by adding ID, and without it a ∉ out(G, {a}), since the body of a norm is not itself something that ought to hold. None of this is an entailment relation between norms — out is an operation on sets of formulas — which is how the framework sidesteps Jørgensen's dilemma: norms carry no truth values, so no consequence relation among them is required. Makinson and van der Torre (2001) added constraints, taking the maximal subsets of G whose output is consistent with a given requirement and reading off the intersection or the union of their outputs, which yields sceptical and credulous variants; their 2003 paper separates negative from positive permission on the same machinery.

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

Limitations. Detachment in the unconstrained operations is unconditional, so a norm whose body holds delivers its head whatever else is the case; contrary-to-duty cases are reached only through the constrained versions, and those proceed by discarding norms rather than ranking them. Nothing in the framework selects among the maximal consistent subfamilies: the sceptical output can be empty where a duty plainly remains, and the credulous union of outputs need not itself be consistent. Reusability makes out₃ chain norms through intermediate heads even when the intermediate was reached by violating something. The generator set is fixed, so norm creation, derogation and priority between sources of authority all lie outside the theory, as do time, agency and degrees of confidence. Concurrent obligations aggregate by conjunction only, which is why reparation chains need a separate operator.