Re: looking for a predicate hierarchy

"Dmitry A. Kazakov" <mailbox@xxxxxxxxxxxxxxxxx> wrote in
news:12cnousl5msxh.1anmyqm356hwb$.dlg@xxxxxxxxxx:


(in logic "uncertain" is usually denoted as _|_, flipped T)

In what logic ?


- True and False are (exclusive) specialization of TrueFalse by
inheritance. - Specialisation can be used to perform state complement
(logical NOT) because most specific specialisation are selected
first.

With these observations, it is easy to build logical tables with the
3 states by writing 4 specialisations (instead of 9).

Hmm, it should be 8.

= 2^3

2 = the depth of the type hierarchy
3 = the number of parameters within the same hierarchy (2 arguments +
the result)

When Boolean and TriState are related types and both arguments and the
result of AND are covariant then the dispatching table of AND will
have 8 slots:

AND : Boolean x Boolean -> Boolean
AND : Boolean x Boolean -> TriState
AND : Boolean x TriState -> Boolean
AND : Boolean x TriState -> TriState
AND : TriState x Boolean -> Boolean
AND : TriState x Boolean -> TriState
AND : TriState x TriState -> Boolean
AND : TriState x TriState -> TriState

Taking into account commutativity of AND, the number of distinct
signatures can be reduced to 6:

AND : Boolean x Boolean -> Boolean
AND : Boolean x Boolean -> TriState
AND : Boolean x TriState -> Boolean
== AND : TriState x Boolean -> Boolean
AND : Boolean x TriState -> TriState
== AND : TriState x Boolean -> TriState
AND : TriState x TriState -> Boolean
AND : TriState x TriState -> TriState

Because TriState is a generalization of Boolean (or equivalently
Boolean is a specialization of TriState)

AND : Boolean x Boolean -> Boolean
<= AND : Boolean x Boolean -> TriState
AND : Boolean x TriState -> Boolean
== AND : TriState x Boolean -> Boolean
AND : Boolean x TriState -> TriState
== AND : TriState x Boolean -> TriState
AND : TriState x TriState -> TriState
=> AND : TriState x TriState -> Boolean

Without implication, your three-valued logic is not fully specified.

.

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