This appendix gives a representative set of PITL theorems and derived inference rules together with their proofs. Many are used either directly or indirectly in the completeness proof for PITL with both ﬁnite and inﬁnite time. We have partially organised the material, particularly in Section 8.2, along the lines of some standard modal logic systems.
The PITL theorems and derived rules have a shared index sequence (e.g., BfChopImpChop – BoxChopEqvChop are followed by BfGen rather than DR1). We believe that this convention simpliﬁes locating material in this appendix.
Proof steps can refer to axioms, inference rules, previously deduced theorems, derived inference rules and also the following:
|Our assumption of axiomatic completeness for PITL with just ﬁnite time
|permits any valid implication of the form ﬁnite ⊃ f.