Algebraic properties of hyper-digital logic
online since July 2007
Having defined the syntax and semantics of hyper-digital logic
this paper investigates the basic properties of the quasi-order relation on hyper-propositional formulas of a given carrier set A and degree k.
One set of properties immediately derives from the fact that these formula algebras are quasi-boolean algebras. We for example have
Another set of properties involves the "box" and "diamond" operators and many results are similar to axioms and theorems of modal logic, such as
- the idempotency of conjunctions:
- the associativity of disjunctions:
- the law of double negation:
But not all phenomena fit into the perspective of traditional logical systems:
- the duality of box and diamond:
From a deeper point of understanding of course, these "surprising" phenomena turn into perfectly normal behaviour.
- If say
is the carrier set then
Let S and T be two carrier sets, S a subset of T.
In traditional propositional logic, we can always increase the carrier set at any time in the sense that
But this is no longer true in hyper-propositional logic, where for each degree k there are two formulas with
The full text is currently under construction.
Hyper-digital logic: Summarized overview of its syntax and semantics