What is Input/Output Logic?

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

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

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 givencode or base • The fundamental question: Given a code, explicitly containing certain directives, what others are implicit in it?

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

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

6. Simple-Minded Output

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

8. Simple-Minded Output – Derivational Characterization

9. Example of derivation (a,x) (b,y)   SI  SI (A,x) (A,y) ---------------------------------------------- AND (A,x xy)   WO (A, (xy)w)

10. Basic Output

11. Reusable output

12. Basic Reusable Output

13. Derivational Characterizations

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: outi+ for 1 = 1,2,3,4 Not all distinct! • out2+ = out4+ = Cn(m(G)A) where m(G) is materialization of G i.e. m(G) = {ax: (a,x)  G}

15. Deriving CT in out2+ (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)

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

17. Further Developments:Different Kinds of Permission

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).

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 outi. • An entry point for the entire net, and an exit point. • Input fed into entry point, output collected from exit point. • To be developed….

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, pp163-174. Dordrecht: Kluwer, Trends in Logic Series, vol 17, 2003. • Detailed analysis: Input/output logics. Journal ofPhilosophical Logic 29 (2000) 383-408. • With consistency contraints: Constraints for input/output logics, Journal ofPhilosophical Logic 30 (2001) 155-185. • Different kinds of permission: Permission from an input/output perspective. To appear in Journal ofPhilosophical Logic. From authors: makinson@dcs.kcl.ac.uk