Implication-matrix Model: Difference between revisions
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An '''Implication Matrix Model''' includes a reflexive, binary matrix Ψ and an association … where Bl represents | |||
An | "implies." | ||
matrix Ψ and an association … where Bl represents | |||
"implies." More specifically, if Ψij = 1, the truth of hj | More specifically, if Ψij = 1, the truth of hj follows from the truth of vi. | ||
follows from the truth of vi | |||
follows from the falsity of hj. If truth of hj is represented | Also, the falsity of v1 necessarily follows from the falsity of hj. | ||
by the equation hj = 1, then falsity is represented both by | |||
If truth of hj is represented by the equation hj = 1, then falsity is represented both by | |||
hj = 0 and hj = 1. | hj = 0 and hj = 1. | ||
From a structural point of view, | From a structural point of view, | ||
Ψij = 1 means that there is a digraph path oriented from | Ψij = 1 means that there is a digraph path oriented from v1 to hj in an implication digraph. | ||
v1 to hj in an implication digraph. | |||
It follows that, if Ψ(V, H) represents an | It follows that, if Ψ(V, H) represents an [[Implication Matrix]] indexed by the ordered sets V and H, the transpose matrix Ψ(H,V) will be an [[Imlplication Matrix]] indexed by H and V. | ||
matrix Ψ(H,V) will be an | |||
by H and V. | |||
If the index pair of an | If the index pair of an [[Imlplication Matrix]] is (V, V), the matrix is a [[Self-implication Matrix]]. | ||
matrix is a | |||
If V and H have no elements in common, the matrix is a | If V and H have no elements in common, the matrix is a [[Cross-implication Matrix]]. | ||
All other implication matrices are hybrid. | All other implication matrices are hybrid. | ||
An Implication Matrix Model is not necessarily complete. | An Implication Matrix Model is not necessarily complete. | ||
Likewise, because of the transitivity of the Implication | If it is complete, then the Implication Matrix Ψ is also a [[Reachability Matrix]], because of the transitivity of the implication relation, i.e., Ψ2 = Ψ and Ψ + I = Ψ, where I is the [[Identity Matrix]]. | ||
relation, any power of an Implication Matrix is an Implication Matrix. | |||
The Boolean sum of any two Implication Matrices with the same index pairs is clearly an [[Implication Matrix]]. | |||
Likewise, because of the transitivity of the Implication relation, any power of an [[Implication Matrix]] is an [[Implication Matrix]]. |
Revision as of 16:53, 6 January 2022
An Implication Matrix Model includes a reflexive, binary matrix Ψ and an association … where Bl represents "implies."
More specifically, if Ψij = 1, the truth of hj follows from the truth of vi.
Also, the falsity of v1 necessarily follows from the falsity of hj.
If truth of hj is represented by the equation hj = 1, then falsity is represented both by
hj = 0 and hj = 1.
From a structural point of view, Ψij = 1 means that there is a digraph path oriented from v1 to hj in an implication digraph.
It follows that, if Ψ(V, H) represents an Implication Matrix indexed by the ordered sets V and H, the transpose matrix Ψ(H,V) will be an Imlplication Matrix indexed by H and V.
If the index pair of an Imlplication Matrix is (V, V), the matrix is a Self-implication Matrix.
If V and H have no elements in common, the matrix is a Cross-implication Matrix.
All other implication matrices are hybrid.
An Implication Matrix Model is not necessarily complete.
If it is complete, then the Implication Matrix Ψ is also a Reachability Matrix, because of the transitivity of the implication relation, i.e., Ψ2 = Ψ and Ψ + I = Ψ, where I is the Identity Matrix.
The Boolean sum of any two Implication Matrices with the same index pairs is clearly an Implication Matrix.
Likewise, because of the transitivity of the Implication relation, any power of an Implication Matrix is an Implication Matrix.