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1 What is Input/Output Logic? David Makinson, King’s College London Leendert van der Torre, CWI Amsterdam

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2 Our Problem Arises in investigation of conditional directives Doubly problematic subject We should take seriously: directives do not carry truth-values directionality of the conditional structure inputs (conditions) need not reappear as outputs (goals) contraposition not in general valid

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3 Methodology Don’t invent yet another non-classical logic – think of new ways of using classical logic All concepts should be understood relative to a given code or base The fundamental question: Given a code, explicitly containing certain directives, what others are implicit in it?

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4 Logic as an Assistant input TRANSFORMATION ENGINE Figure 1 What does the real work? LOGIC prepares (coordinates) output packages

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5 Our Language Propositions: boolean connectives only Conditional directives: pairs (a,x) of boolean formulae Code (base, generating set): set G of such pairs Input/output operation: –(b,y) out(G)given code G, produces conditional directive (b,y) as output –y out(G,b)given code G and a condition b, produces an output y Latter for semantic definitions, former for derivations

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6 Simple-Minded Output

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7 Simple-Minded Output - Example a, b c, d, c (a,x), (b,y), (d,x) (y,z) x, y Cn({x, y})

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8 Simple-Minded Output – Derivational Characterization Strengthening inputSIFrom (a,x) to (b,x) whenever b a Weakening outputWOFrom (a,x) to (a,y) whenever x y Conjoining outputAND From (a,x), (a,y) to (a,x y) TautologyTAUTFrom no premises to (t,t)

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9 Example of derivation (a,x)(b,y) SI SI ( A,x) ( A,y) AND ( A,x x y) WO ( A, (x y) w)

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10 Basic Output

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11 Reusable output

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12 Basic Reusable Output

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13 Derivational Characterizations Disjunctive InputORFrom (a,x), (b,x) to (a b,x) Gives Basic Output out 2 Cumulative Transitivity CTFrom (a,x), (a x,y) to (a,y) Gives Reusable Output out 3 Disjunctive Input + Cumulative Transitivity OR + CTBoth as aboveGives Reusable Basic Output out 4

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14 Authorizing Input to Reappear as Output Semantically: In the definitions, replace set R by R {(x,x): x any formula} Derivationally: Add zero-premise rule: From no premises to (x,x) This gives 4 additional systems: out i + for 1 = 1,2,3,4 Not all distinct! out 2 + = out 4 + = Cn(m(G) A) where m(G) is materialization of G i.e. m(G) = {a x: (a,x) G}

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15 Deriving CT in out 2 + (a,x) (a x, a x) ID (a x,y) SI (a x, x) AND (a x, x (a x)) WO (a x,y) ……… OR (a,y)

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16 Further Developments: Contrary-to-goal inputs Deontic logic: problem of ‘contrary-to-duty’ conditions of obligations. Example: G consists of If main dish is steak, the wine should be red If main dish steak but the wine not red, then wine should be rosé Input: steak, wine not red. Output: red, rosé! Can be tackled in input/output logic by imposition of consistency constraints on the application of the operation

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17 Further Developments: Different Kinds of Permission

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18 Different Kinds of Permission: Example Code contains: Obligation: It is required to fill in an annual income-tax form, if employed. Permission: It is permitted to vote, if 18 years of age or over. Does it follow that it is permitted to vote, if employed? Yes: Nothing in the code forbidding it (negative permission). No: A person may be employed at 17 and not covered by the explicit permission (positive permission - static). Yes: If we were to forbid a person to vote when employed, we would be creating an incoherence in the code (positive permission - dynamic).

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19 Future Developments: Networks of Input/Output Operations Output of one operation feeds in as input to another Family of nodes, linked by an accessibility relation. At each node, a generator set R and an output operation out i. An entry point for the entire net, and an exit point. Input fed into entry point, output collected from exit point. To be developed….

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20 References Authors: David Makinson & Leendert van der Torre Brief outline: What is input/output logic?. Foundations of the Formal Sciences II: Applications of Mathematical Logic in Philosophy and Linguistics, pp Dordrecht: Kluwer, Trends in Logic Series, vol 17, Detailed analysis: Input/output logics. Journal of Philosophical Logic 29 (2000) With consistency contraints: Constraints for input/output logics, Journal of Philosophical Logic 30 (2001) Different kinds of permission: Permission from an input/output perspective. To appear in Journal of Philosophical Logic. From authors:

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