The Great Conversation:
Logic

By Matt McKeown

Why don't they teach logic in these schools?

Logic is the name of a few closely related things. It is often used to refer exclusively to deductive logic; inductive logic, if it is graced with the name of logic at all, generally has to be specified as such. It is also a shorthand for the dialectic phase of the Trivium, the classical curriculum inherited from antiquity by the Medievals and handed down to modernity (with varying levels of receptiveness on our part). Whether by analogy or because it originates in the Greek λόγος—which has a range of meanings, from “speech” or “word” to “reason,” “calculation,” and “study”—it can sometimes mean any systematic arrangement of things or ideas. Puzzles and word games have a very long history as an offshoot of logic, often creating humorous wordplay by the sly use of technicalities, or by flat-footedly literal interpretations of idioms.

Our earliest texts in the discipline of deductive logic are arguably the Platonic dialogues. Socrates does not necessarily approach his cross-examinations (whether of Sophists or his own disciples) with a system in mind; but the way he pursues any given line of argument, insisting on precise definitions and distinctions and testing things with counterexamples, is a classic demonstration of logical analysis. Plato’s dialogues naturally continue the tradition, as did the Academy itself.

Here, of course, we must address Aristotle, whom Dante would later call “the master of those who know.” His textbook on logic, the Organon (roughly, “The Toolbox”), consisted actually in six distinct works: Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, and On Sophistical Refutations. These detail a thorough system of definition, predication, and deduction that all subsequent logical systems have drawn from. He detailed the logical relationships of both categorical and consequential statements, articulating what conclusions did and did not follow from various kinds of premises, and setting out the limited number of syllogisms that are valid (i.e., that necessarily result in true conclusions if the premises are true). Sophistical Refutations was an important element of the Organon too, exposing reasoning that might appear valid at first glance but was, on analysis, specious. Only after a thorough grounding in logic were his students ready to advance to rhetoric, or the art of persuasion, which presupposed the discovery of truth that logic made possible.

"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."

Lewis Carroll

Beginning in the eighth century, the golden age of Islamic learning came about in large part due to its interest in Aristotle, particularly in Spain. This in turn, via sometimes-peaceful contact in both Spain and the Levant, led to the Scholastic renaissance in Western Europe. The last four books of the Organon were translated into Latin in the twelfth century—only the first two had been available in that language before then (thanks to the efforts of Boethius).

Aristotle’s system, however, has been both advanced upon and questioned. The late Renaissance and Enlightenment significantly reframed logic, partly due to major shifts in epistemology, or the philosophy of thought. The idea that logic discovers truth drew special criticism—not so much on the grounds that logic is impotent, but on the grounds that all logic is, on some level, an “if-then” statement. If X is true, then Y follows from it; but what reason do we have to think X is true? That question is normally the domain of investigation, rather than of deductive logic. Whether the primary axioms of logic, such as the law of non-contradiction, are known through logic or are properly part of a separate discipline or mental faculty, is another matter that later philosophers disputed.

An interesting change in the history of logic has been in its relation to mathematics. The Medievals classified logic as one of the “humanities,” alongside grammar and rhetoric, while arithmetic and geometry were “sciences,” like astronomy and music. However, the fact that both rely on intuitive recognition of inherent relationships—we just see that two and two make four in the same way that we just see that direct contradictions cannot both be true—prompted some philosophers and mathematicians to rethink them. Logic tends to serve metaphysics, as mathematics tend to serve physics; once upon a time, both of these disciplines were districts of the same single province of wisdom. Could studying the common root of the two disciplines, mathematics and logic, lead to a reunification of what the ancients meant by the term philosophy?

Suggested reading:
Aristotle, Organon
Francis Bacon, The New Organon
George Boole, The Laws of Thought
Lewis Carroll, Alice’s Adventures in Wonderland
Bertrand Russell and Alfred North Whitehead, Principia Mathematica
Umberto Eco, The Name of the Rose

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