Defining Logics
Rowan Davies & Frank Pfenning
"A judgmental reconstruction of modal logic"

Denotational Approach
  m \in F1 ^ F2, m \in F1 -> F2 (m semantics)

Hilbert Style
  P says F, P1 speaksFor P2, F1 -> F2, F1 ^ F2

Logics: Natural Deduction
  1. Define basic Judgment
         |- F true
  2. Define hypothetical judgment
       F1 true, F2 true, ..., Fk true |- F true
       \gamma |- F true

       x1:F1 true, x2:F2 true, ..., xk true Fk true |- F true
       \gamma |- F

       a) Weakending Property
	  If \gamma |- F, then \gamma,\gamma' |- F
       b) Exchange Property
	  If \gamma x1 F1, x2, F2, \gamma2 |- F
	  then \gamma1 x2 F2 \gamma F1 |- F
       c) Contraction Property
	  If \gamma1 x:F1, y:F1 |- F
	  then \gamma1 z:F1 |= F
       d) Substitution Property
	  If \gamma |- F and \gamma,F |- F'
	  then \gamma |- F'
  3. Define some connectives for the logic
       a) introduction rules
       b) elimination rules

       - local soundness
	 elimination rules not too strong to the intro rules
	 (all proofs in which an elmination rule directly
	  follows an introduction, can be rebuild to a proof
	  that does not have elimination )

       - local completeness
	 elimination rules are strong enough 
	 (given any proof of a connective, we can apply the
	  eliminatation forms to extract enough info to
	  reapply the intro forms and come up with a proof
	  of the original judgment)
