dedekind::category::Chain< T > Struct Template Reference

Chain<T>: the finite Kleene chain over an integral carrier T, the full-T-range many-valued logic. Boole (2 grades) and Kleene (3) are the small distinguished members; Chain<int> spans the implementation's full int range. More...

Collaboration diagram for dedekind::category::Chain< T >:

Public Types
using	Ω = T
using	JoinOp = Sup
	The op-types OR / AND / RFL are: join / meet are max / min (the numeric order is the truth order), i.e. the value-returning Sup / Inf from :mereology; ¬ is logic_complement<Chain<T>>. Chain<T> reaches IsDistributiveLattice through these.
using	MeetOp = Inf
using	RflOp = logic_complement<Chain<T>>

Static Public Member Functions
static constexpr T	AND (T a, T b)
	Meet ∧ = numeric minimum (the numeric order is the truth order).
static constexpr T	OR (T a, T b)
	Join ∨ = numeric maximum.
static constexpr T	RFL (T a)
	¬a = ~a (bitwise NOT): the order-reversing involution swapping the poles (⊥ ↔ ⊤). Overflow-free (no arithmetic).

Static Public Attributes
static constexpr T	True = std::numeric_limits<T>::max()
	The top pole ⊤ = the carrier maximum.
static constexpr T	False = std::numeric_limits<T>::min()
	The bottom pole ⊥ = the carrier minimum.

Detailed Description

template<std::integral T>
requires (std::same_as<T, std::remove_cv_t<T>> && !std::same_as<T, bool>)
struct dedekind::category::Chain< T >

Chain<T>: the finite Kleene chain over an integral carrier T, the full-T-range many-valued logic. Boole (2 grades) and Kleene (3) are the small distinguished members; Chain<int> spans the implementation's full int range.

logic__Species_3

Meet AND = min, join OR = max (the numeric order is the truth order); reflection RFL = ~a (bitwise NOT), the order-reversing involution swapping ⊥ = numeric_limits::min and ⊤ = numeric_limits::max. A bounded De Morgan chain, hence Kleene (chain implies normality), not Boolean: only the two poles are complemented.

Signed and unsigned integrals both work (unsigned poles 0 / UMAX; signed INT_MIN / INT_MAX, where ~x = -1-x on two's complement). bool is excluded (its ~ promotes, breaking RFL: Boole is the 2-chain). Floating point is excluded by std::integral: totality needs min / max / ¬ everywhere, and NaN breaks the total order.

The bounds are the carrier's own min / max, so the value set is the chain. A sub-interval like [0,100] (a "Percentage" confidence, ¬-fixed at 50) is not sound here: Ω would still be all of T, so True would not be the top and ¬ could leave the interval. A genuinely bounded [Lo,Hi] chain needs a carrier that enforces the range: see Percent / Percentage below (#906). The species names the ops (min / max / ¬); raw T keeps its own boolean &&, so the concepts gate the species' named ops (register- agnostic). See #901.

Member Typedef Documentation

JoinOp

template<std::integral T>

using dedekind::category::Chain< T >::JoinOp = Sup

The op-types OR / AND / RFL are: join / meet are max / min (the numeric order is the truth order), i.e. the value-returning Sup / Inf from :mereology; ¬ is logic_complement<Chain<T>>. Chain<T> reaches IsDistributiveLattice through these.

(#923)

Note

#935 makes no boundedness claim about Chain<T> either way: the machine-int-vs-ℤ question (register the numeric_limits poles as Sup / Inf identities to reach IsBoundedLattice, matching IsBoundedDeMorganChain<Chain<int>>, vs treat the carrier as unbounded ℤ) is a deliberate design decision left out of scope: FIXME(#941).

---

The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/category/logic.cppm