In solving problems by explicit logical analysis, three principles can be distinguished: the principle of constructing hierarchies, the principle of establishing priorities, and the principle of logical consistency.
1. Structuring Hierarchies - Humans have the ability to perceive things and ideas, to identify them, and to communicate what they observe. For detailed knowledge our minds structure complex reality into its constituent parts, and these in turn into their parts, and so on hierarchically. The number of parts usually ranges between five and nine. By breaking down reality into homogeneous clusters and subdividing these clusters into smaller ones, we can integrate large amounts of information into the structure of a problem and form a more complete picture of the whole system.
2. Setting Priorities - Humans also have the ability to perceive relationships among the things they observe, to compare pairs of similar things against certain criteria, and to discriminate between both members of a pair by judging the intensity of their preference for one over the other. Then they synthesize their judgments─ through imagination or, with the AHP, through a new logical process ─ and gain a better understanding of the whole system.
Relationships represent the relative impact of the elements of a given level on each element of the next higher level. In this context the latter element serves as a criterion and is called a property. The result of this discrimination process is a vector of priority, or of relative importance, of the elements with respect to each property. This pairwise comparison is repeated for all the elements in each level. The final step is to come down the hierarchy by weighing each vector by the priority of its property. This synthesis results in a set of net priority weights for the bottom level. The element with the highest weight is the one that merits the most serious consideration for action, although the others are not ruled out entirely. This principle and the next are fully explained in Chapter 5 of Dr. Saaty’s book.
3. Logical Consistency - The third principle of analytic thought is logical consistency. Humans have the ability to establish relationships among objects or ideas in such a way that they are coherent─ that is, they relate well to each other and their relations exhibit consistency. Consistency means two things. The first is that similar ideas or objects are grouped according to homogeneity and relevance. For example, a grape and a marble can be grouped into a homogeneous set if roundness is the relevant criterion but not if flavor is the criterion. The second meaning of consistency is that the intensities of relations among ideas or objects based on a particular criterion justify each other in some logical way. Thus if sweetness is the criterion and honey is judged to be five times sweeter than sugar, and sugar twice as sweet as molasses, then honey should be taken to be ten times sweeter than molasses. If honey is judged to be only four times sweeter than molasses, then the judgments are inconsistent and the process may have to be repeated if more accurate judgments could be obtained.