1. Modal logic and databases

2. Terms Object terms Concept terms ↓ t: object denoted by concept t in some context Type designations: o (object) and c (concept)

3. Syntax

4. Semantics

5. Valuation

6. Truth and modal models

7. Relational databases Record: basic unit of information in rdb  Can’t return it directly as answer to query  Each one is a possible world Accessibility (i.e. “proximity” of possible worlds): the S5 logic db field attributes: individual concepts db field entries: individual objects

8. Sample database

9. Axioms

10. The worlds and mappings Validity: true in every world

11. Rigidity t is rigid if it always designates the same object, no matter which world  FWIW: in linguistics, proper names are rigid Rigidity can be relative w/rt subsets of all possible worlds  Databases: functional dependencies (e.g. between attributes)

12. Designation Designation is only possible when interpretation is grounded.

13. A query returns: 2 and 5

14. Another query returns: t

15. Additional relation(ship)s now add...

16. Higher-order relations relation of type <>: PERSON relation of type <>: LOCATION

17. The new (relational) constraint axioms

18. The new instance axioms

19. A sample derivation Prove: Strategy: Prove X Prove ¬X is false

20. A shorthand derivation

21. The overall derivation

22. The strategy Prove ¬X is false

23. Reduce query’ to disjuncts

24. Introduce Axiom 5

25. Apply the shorthand derivation Φ

26. Instantiate with query objects

27. λ-reduce

28. Apply and reduce Axiom 7

29. Apply shorthand rule to 11

30. Apply disjunction rule

31. Instantiate

32. Contradiction! (lhs)

33. Instantiate

34. Contradiction!

35. Another example

36. Attributes and relations

37. Sample query 1 Which items have 2 cylinders?

38. Checking query 1 (for instance 3) check the relevant world(s) with appropriate mappings: and resolve each conjunct...

39. Sample query (2) What choices does a customer have when purchasing a 4-cylinder car?

40. Sample query (3) What features can a customer choose that are available for more than one product?