IJAHP Essay: Saaty/Pairwise comparisons and their contribution to understanding consciousness 

 

 
 
 

International Journal of the 

Analytic Hierarchy Process 

199 Vol. 8 Issue 1 2016 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v8i1.381 

Pairwise Comparisons and their Contribution to Understanding 

Consciousness 

Thomas L. Saaty 

Distinguished University Professor 

University of Pittsburgh 

Pittsburgh, Pennsylvania, USA  

E-mail: saaty@katz.pitt.edu 

 

Making comparisons of elements with respect to criteria is an ability we are born with as 

are all conscious living things. We derive priorities using informed judgment from 

comparisons of elements in regard to influence: dominance of the importance, preference 

or likelihood with respect to a property they share. Priorities represent a rank order of the 

things that are compared. They also represent proportionality of the numbers that 

represent that order.  

The priorities are derived from pairwise comparison matrices which lead to the 

consideration of eigenvalues, in particular the principal eigenvalue, of a positive and 

occasionally nonnegative matrix. This eigenvalue becomes a useful tool in the 

measurement of inconsistency of judgments. When we use a hierarchic structure to make 

a decision we must synthesize these different priorities through multiplication followed 

by addition in an appropriate manner. 

With network structures, we use the idea of a supermatrix. In the supermatrix, the entries 

are block matrices whose columns are eigenvectors derived from paired comparison 

matrices. The columns of the supermatrix must be stochastic in order for it to converge. 

This is accomplished by weighting each column of block matrices so the columns of 

eignenvectors in the blocks sum to one and we can raise the weighted supermatrix to 

powers to obtain the limiting priorities that represent the answer we seek.  

With a neural structure, we need eigenfuntions to represent the priorities. The firing of 

each neuron can be represented by an eigenfunction, an exponentially damped periodic 

function of period one. There are four fundamental kinds of eigenfunctions that belong to 

the four division algebras: real, complex, quaternionic and octonionic.  Each kind is 

suitable for certain kinds of sensations and thoughts. An article that investigates the 

interpretation of how these eigenfunctions can be used to describe consciousness in the 

neural domain will appear in the Journal of Neural Networks. Beyond that are other 

hypercomplex numbers that in addition to non-commutativity as with quaternions and 

octonions and non-associativity as with octonions are inadequate to characterize them. 

For example, sedonions which are non-commutative and  non-associative have the 

peculiar property that the product of two non-zero elements can be zero as can happen 

with multiplying matrices. 

mailto:saaty@katz.pitt.edu


IJAHP Essay: Saaty/Pairwise comparisons and their contribution to understanding consciousness 

 

 
 
 

International Journal of the 

Analytic Hierarchy Process 

200 Vol. 8 Issue 1 2016 

ISSN 1936-6744 

http://dx.doi.org/10.13033/ijahp.v8i1.381 

A useful contribution of the ANP in applications is that it opens a new vista into 

nonlinear thinking that involves feedback for which language alone is unable to cope, by 

expressing in words and sentences, the consequences of all the influences that work 

together to shape the future. Logical thinking is linear and language itself is linear and 

moves from assumptions to conclusions linearly. The decision framework of benefits, 

opportunities, costs and risks has such complexity in it that logic and language alone 

cannot cope. Kurt Gödel’s incompleteness theorem about undecidable proofs may also be 

questioned in using the comprehensive but incomplete network structures in complex 

decision making. This is because a built in assumption is that the system they represent is 

always open to adding new factors that influence the outcome.