1Introduction

It is the mark of an instructed mind to rest satisfied with that degree of precision which the nature of the subject admits, and not to seek exactness where only an approximation of the truth is possible.

Aristotle, 384–322 B.C.Ancient Greek philosopher

Precision is not truth.

Henri E. B. Matisse, 1869–1954Impressionist painter

All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestrial life but only to an imagined celestial existence.

Bertrand Russell, 1923British philosopher and Nobel Laureate

We must exploit our tolerance for imprecision.

Lotfi Zadeh, 1973Professor, Systems Engineering, UC–Berkeley

The preceding quotes, all of them legendary, have a common thread. That thread represents the relationship between precision and uncertainty. The more uncertainty in a problem, the less precise we can be in our understanding of that problem. It is ironic that the oldest quote is attributed to the philosopher who is credited with the establishment of Western logic—a binary logic that admits only the opposites of true and false, a logic that does not admit degrees of truth in between these two extremes. In other words, Aristotelian logic does not admit imprecision in truth. However, Aristotle’s quote is so appropriate today; it is a quote that admits uncertainty. It is an admonishment that we should heed; we should balance the precision we seek with the uncertainty that exists. ...

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