bucephalus.org
Algebraic properties of hyperdigital logic
online since July 2007
Abstract
Having defined the syntax and semantics of hyperdigital logic
(see [1]),
this paper investigates the basic properties of the quasiorder relation on hyperpropositional formulas of a given carrier set A and degree k.
One set of properties immediately derives from the fact that these formula algebras are quasiboolean algebras. We for example have
 the idempotency of conjunctions:
 the associativity of disjunctions:
 the law of double negation:
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 duality of box and diamond:
But not all phenomena fit into the perspective of traditional logical systems:
 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
always implies
.
But this is no longer true in hyperpropositional logic, where for each degree k there are two formulas with
and
.
From a deeper point of understanding of course, these "surprising" phenomena turn into perfectly normal behaviour.
The text
The full text is currently under construction.
References
[1]
Hyperdigital logic: Summarized overview of its syntax and semantics