Transitive Relations
Mathematical Definition
A binary relation R defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.
ISM is based on Transitive Logic
ISM is based on the mathematical concept of Transitive Logic. This concept is used to reduce the number of comparisons (and thus time) needed to construct the MAP. Transitive Logic is also used to simplify the topology of the visual model. Transitive Logic states that for any 3 elements (A, B, C) with a given relation (�) when:
A has the relation to B, (written A*B)
And B has the relation to C, (written B*C)
Then A has the relation to C, (written A*C or A*B*C)
For example:
If travel by car is faster than travel by bicycle
And travel by bicycle is faster than travel by walking
Then travel by car is faster than travel by walking
Applying transitive logic also means that for any 3 elements (A, B, D) with a given relation, when:
A has the relation to
And A does not have the relation to D
Then B does not have the relation to D
This means that B cannot have the relation to D or, using transitive logic, A would have to have the relation to D also
For example:
If travel by car is faster than travel by bicycle
And travel by car is not faster than travel by airplane
Then travel by bicycle is not faster than travel by airplane
ISM Software uses this type of logic to infer some of the relationships based on the answers given by the participants of a Structured Democratic Dialogue. The number of inferred answers in an ISM session varies with the situation, but it is often in the order of 70%. This represents considerable time-saving.
Examples of Transitive Relations
| R | Explanation |
|---|---|
| precedes | if A precedes B, and B precedes C, necessarily A precedes C. This would be true whatever the nature of A, B, and C,although the ideas can be made more precise by thinkingof A, B, and C as events occurring at specific times. |
| includes | if A includes B, and B includes C, necessarily A includes |
| included in | |
| is less than | |
| is greater than | |
| supports | |
| implies | |
| causes |
The generic phrase is subordinate to can be used to represent any of the above relations.
Transitive relations are quite common. However, one shoud be careful not to grant transitivity when it is not present.
For ecxample, the contextual relation "is preferred to" is not transitive.
If a person says "blue is preferred to red" and "red is preferred to yellow," it still may be that the person will say "yellow is preferred to blue," hence, transitivity is violated.
are next-door neighbours is not but is neighbour of is!
Such contextual relations can be referred to as "transitive preference" and "intransitive preference."