Brief Notes On QL

Axioms originally formulated within: A New Axiomatization of Unified Quantum Logic by M. Pavicic.

WFF for Zero Order QL (ZQL)

1. Lexicon:

i. Logical Connectives: ¬, →, ∨, ∧

ii. Propositions: p, q, r, ...

2. Grammar:

i. If p is a Proposition in ZQL, p is a WFF.

ii. If p is a WFF, so is ¬p.

iii. If p, q are WFF, so is p → q.

iv. If p, q are WFF, so is p v q.

v. If p, q are WFF, so is p ∧ q.

vi. There are no other WFF of ZQL.

Inference System for Zero Order QL

3. Axioms:

[A1] A → A

[A2] A → ¬¬A ∧ ¬¬A → A

[A3] A → A ∨ B

[A4] B → A ∨ B

[A5] B → A ∨ ¬A

4. Rules of Inference (here | denotes turnstile):

[R1] (A → B), (B → C) | A → C

[R2] (A → B) | (¬B → ¬A)

[R3] (A → C), (B → C) | (A v B → C)

[R4a] (B v ¬B) → A | A

[R4b] A | (B v ¬B) → A

Kripke Frames

See notes on page 9 for a Modal Logic semantics for ZQL.