Implication-matrix Model: Difference between revisions

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IMPLICATION-MATRIX MODEL
An '''Implication Matrix Model''' includes a reflexive, binary matrix Ψ and an association … where Bl represents
An implication Matrix model includes a reflexive, binary
"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; also the falsity of v1 necessarily
 
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 implication
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 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 implication matrix is (V, V), the
If the index pair of an [[Imlplication Matrix]] is (V, V), the matrix is a [[Self-implication Matrix]].  
matrix is a self-implication matrix.  


If V and H have no elements in common, the matrix is a cross-implication matrix.  
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.
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  
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.