Modal logic and databases
Terms
Object terms Concept terms ↓t: object denoted by concept t in
some context Type designations: o (object) and c
(concept)
Truth and modal models
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
The worlds and mappings
Validity: true in every world
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)
Designation
Designation is only possible when interpretation is grounded.
A query
returns: 2 and 5
Another query
returns: t
Additional relation(ship)s
now add...
Higher-order relations
relation of type <>:PERSON
relation of type <>:LOCATION
The new (relational) constraint axioms
The new instance axioms
A sample derivation
Prove:
Strategy:
Prove X
Prove ¬X is false
A shorthand derivation
The overall derivation
The strategyProve ¬X is false
Reduce query’ to disjuncts
Introduce Axiom 5
Apply the shorthand derivation
Φ
Instantiate with query objects
Apply and reduce Axiom 7
Apply shorthand rule to 11
Apply disjunction rule
Contradiction! (lhs)
Attributes and relations
Sample query 1
Which items have 2 cylinders?
Checking query 1 (for instance 3)
check the relevant world(s) with appropriate mappings:
and resolve each conjunct...
Sample query (2)
What choices does a customer have when purchasing a 4-cylinder car?
Sample query (3)
What features can a customer choose that are available for more than one product?