The Moral Law and The Good in Temporal Modal Logic with Propositional Quantifiers
The Moral Law is fulfilled (in a possible world w at a time t) iff (if and only if) everything that ought to be the case is the case (in w at t), and The Good (or The Highest Possible Good) is realised in a possible world w' at a time t' iff w' is deontically accessible from w at t. In this paper, I will introduce a set of temporal alethic deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.