Devlin's Angle

June 1996

Laws of Thought

By the time they graduate, most mathematics students have heard of Zeno, he of the paradoxes of motion: Achilles and the tortoise, the arrow paradox, and so forth. Less well known is the fact that there was not one but two highly influential Zenos in the world of ancient Greek mathematics.

The better known Zeno, of Zeno's paradoxes, was Zeno of Elea, in Magna Graecia, who lived about 450 B.C. That Zeno was a student of Parmenides, the founder of the Eleatic school of philosophy.

The lesser known Zeno, Zeno of Citium, lived some 150 years later. The legacy of this Zeno is present day computer science.

Well, that's making a bit of a leap. What Zeno of Citium actually did was found the Stoic school of logic. Though modern mathematical logic is popularly credited as having its beginnings in the syllogistic logic of Aristotle, most of the fundamental notions of contemporary propositional logic began not with Aristotle but with Zeno and the Stoics.

As in modern propositional logic, the patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of what those propositions say.

The Stoics examined a number of ways in which two propositions can be combined to give a third, more complicated proposition.

For instance, one, simple way that two propositions may be joined to form a single, new proposition is by means of the connecting word (or 'connective') "and". Present-day logicians bring out the abstract pattern of connectives such as "and" by using algebraic notation. The letters p, q, r are generally used to denote arbitrary propositions, and a symbol such as & is used to abbreviate the word "and". Thus, [p&q] denotes the proposition [p and q].

It is of interest to note that at no time did the Stoics themselves hit upon the idea of using algebraic notation, with letters denoting arbitrary propositions and symbols denoting connectives. They wrote everything out in ordinary language. This often resulted in their having to write down long and complicated sentences that are difficult to follow, and that almost certainly hampered their possible progress in logic. The modern, algebraic way of expressing the notions of propositional logic is much better. The algebraic expressions are far shorter and much easier to read. (It is one of life's many ironies that a linguistic device introduced to make things clearer and simpler, namely algebraic notation, should have precisely the opposite effect on so many people.)

Other means of combining propositions analyzed by the Stoics were "or" (abbreviated by the symbol v in modern logic), "not" (often denoted by the symbol ~), and "implies" (denoted by ->).

As alluded to above, the key idea behind the Stoics' approach to logic was that you do not know what the constituent propositions are about, or even whether each constituent proposition is true or false. All that you know is that any proposition must be either true or false. When the Stoics came to analyze the combining of two propositions by one of the connectives, they did so by looking at the pattern of truth and falsity. For example, in the case of "and", the pattern is straightforward: if both p and q are true, then the proposition [p&q] will be true; if one or both of p and q are false, then [p&q] will be false.

Modern logicians generally display such a 'pattern of truth' in a tabular form, using a truth table, a nineteenth century device not available to the Stoics. The Stoics had to express the truth pattern for [p&q] in the following cumbersome fashion:

"If the first and if the second, then the first and the second. If not the first, then not the first and the second. If not the second, then not the first and the second."

If you replace "the first" by the letter p, "the second" by the letter q, and abbreviate "and" by &, this rather perplexing looking sentence becomes:

"If p and if q, then [p&q]. If not p, then not [p&q]. If not q, then not [p&q]."

From this version, it is but a small step to the truth pattern for "and" as originally expressed above (or to the truth table for "and").

The Stoics realized that there is ambiguity in the meaning of "or": does it mean the inclusive-or or the exclusive version? They tended to prefer the exclusive variant; present-day logicians plump for the inclusive version.

The Stoics had great trouble trying to understand the nature of "implies". As any present-day student of logic will know, when you try to define the meaning of the symbol -> in terms of truth values alone, the result is a logical connective that only partially captures the notion of implication.

The Stoics formulated five rules of inference. Expressed in modern-day algebraic notation (but with the symbol v denoting exlusive-or), they are:

From [p -> q] and p deduce q.

From [p -> q] and ~q deduce ~p.

From ~[p&q] and p deduce ~q.

From [pvq] and p deduce ~q.

From [pvq] and ~q deduce p.

The first of these rules is the modern-day logical inference rule of 'modus ponens'. Here is how the Stoics themselves expressed this rule:

"If the first then the second, and if the first, then the second."

Starting with their five inference rules, the Stoics were able to deduce a number of other patterns of reasoning. For example, they showed that the following deduction is valid:

From [p -> (p -> q)] and p deduce q.

Using the Stoics' own terminology, this was expressed like this:

"If the first then if the first then the second, and if the first, then the second."

Given algebraic notation and the modern technique of truth tables, much of Stoic logic reduces to some simple algebraic manipulations together with the filling-in of truth-values in a table. However, it took over two thousand years for Mankind to reach that stage. Not having access to such modern tools, the Stoics had a much harder time establishing their results. But establish them they did.

By singling out propositions as the building blocks for reasoning and identifying some of the abstract patterns involved in reasoning with propositions, including modus ponens, the Stoics' contribution to logic was a major intellectual achievement. Together with Aristotelean logic, it paved the way for all subsequent work in logic, right up to the present day, and led to much of twentieth century logic and computer science.

Clearly, it's high time the other Zeno was given proper credit for his role in the development of modern logic.

And in case you are wondering, yes, these are the Stoics from whom we get our present-day word "stoical". The Stoics were such enthusiastic believers in the power of formal logic that if formal reasoning led them to adopt a particular course of action, they would pursue that course even if it involved hardship, pain, or suffering.

-Keith Devlin

The above article is adapted from the book "Goodbye Descartes: The End of Logic and the Search for a New Cosmology of Mind", by Keith Devlin, to be published by John Wiley and Sons in January 1997.

Devlin's Angle is updated at the start of each month.

Keith Devlin ([email protected]) is the editor of FOCUS, the news magazine of the MAA. He is the Dean of Science at Saint Mary's College of California, and the author of Mathematics: The Science of Patterns, published by W. H. Freeman in 1994.