Dialectic: reasoning soundly
The second art: when does something really follow, and when does it only sound as if it does? The difference between true and valid, the commonest fallacies to test oneself on, and what 'As above, so below' can and cannot do.
Dialectic comes from the Greek word for “conversing”. In conversation Socrates tested whether the claims of the people he spoke with could withstand questioning. Aristotle set beside it analytics, the doctrine of sound inference, which became logic. In the Middle Ages this second art of the trivium was called Dialectica, and it was the sharpest of them all.
True and valid
The most important distinction in logic is that between true and valid. Single statements are true or false. An inference, the way from premises to conclusion, is valid or invalid. An inference is valid if the conclusion must be true as soon as the premises are true.
Two things follow, and both matter in everyday life. A valid inference from a false premise can lead to a false conclusion: “All birds can fly, penguins are birds, so penguins can fly.” And an invalid inference can happen to reach a true conclusion. That someone is right does not yet mean he has thought rightly.
Mathematics is the school of valid inference. Every proof in the Foundations is a chain of valid inferences from a few premises. The following tests, by contrast, come from everyday life. Decide in each case whether the conclusion follows from the premises:
If it rains, the street gets wet. It is raining. So the street gets wet.
Modus ponens. From "if A, then B" and "A", "B" always follows. This is the basic form of every proof.
If it rains, the street gets wet. The street is wet. So it has rained.
Affirming the consequent. The street may be wet for another reason, say from a street cleaner. From "if A, then B" and "B" nothing follows about A.
Every prime greater than 3 lies right next to a multiple of 6. 25 lies next to 24. So 25 is a prime.
Affirming the consequent. The same mistake as with the rain, this time with numbers. The condition is necessary but not sufficient: 25 = 5 × 5. See the page on the order of the primes.
Every whole number is even or odd. 7 is not even. So 7 is odd.
Disjunctive syllogism. If there are only two possibilities and one is ruled out, the other remains. Careful in everyday life: there are often more than two.
Since I started wearing this stone I have felt better. So the stone helps.
After this, therefore because of this. That something happens after something else does not mean it happens because of it. Many things get better on their own.
A famous master of Hermeticism taught it. So it is true.
Appeal to authority. A great name can be a reason to look more closely, but it does not replace a reason. Masters err too.
No one has ever proved that Atlantis did not exist. So Atlantis existed.
Appeal to ignorance. The absence of a disproof is not a proof. The burden of proof lies with whoever makes the claim.
All men are mortal. Socrates is a man. So Socrates is mortal.
Syllogism. The classic textbook example of logic. Valid means: if the premises are true, the conclusion must be true.
Eight inferences to test yourself on. The question is not whether the conclusion is true but whether it follows from the premises.
The commonest fallacies
Whoever deals with secret teachings meets certain fallacies especially often. It is worth knowing them by name:
- Affirming the consequent. “If an ancient wisdom had existed, we would find its signs everywhere. We find the signs everywhere. So it existed.” The signs can have quite different reasons, for instance that circle, triangle and square are simply the simplest figures there are.
- After this, therefore because of this. What happens after a ritual, a stone or an exercise does not happen because of it.
- Appeal to authority. A great name is a reason to look more closely, but no proof. The Rosicrucians, Paracelsus and Newton erred too.
- Appeal to ignorance. That something has not been refuted does not make it true. The burden of proof lies with whoever makes the claim.
- The circle. “The scripture is true because it is divine, and it is divine because it says so itself.”
- The false either-or. “Either science explains everything, or it is magic.” There are usually more than two possibilities.
What a correspondence can do
A guiding principle of Hermeticism reads, in the familiar short form after the Emerald Tablet, “As above, so below”. It says that correspondences exist between the great and the small, between heaven and man. Logically this is an inference from similarity, an analogy. Analogies are not proofs. But they are a splendid tool for finding conjectures.
The best example is Johannes Kepler. All his life he looked for the harmony of the world, for correspondences between the planetary orbits, the Platonic solids and the musical intervals. On this search he found his third law, which he published in 1619 and which holds to this day. But Kepler tested every conjecture against the measurements. Years earlier, when his calculation for the orbit of Mars deviated by eight minutes of arc from Tycho Brahe’s observations, he gave up the sacred circular orbits and found the ellipse. These eight minutes, he wrote, had pointed the way to the reform of the whole of astronomy.
That is the right attitude to analogy: find by similarity, test by proof or observation. Whoever takes only the first step stays with beautiful pictures. Whoever takes only the second finds nothing new.
Exercise
Write down a conviction that matters to you, and beneath it one sentence: “I would change my mind if …” If nothing occurs to you, that is no sign of strength but a warning sign, for then no experience in the world can reach you any more.
And a second exercise, which is harder: take a view you reject and state it so strongly and so fairly that someone who holds it would say: “Yes, that is exactly what I mean.” Only then contradict it.
Sources and further reading
Sources
- Aristotle, Prior Analytics and Sophistical Refutations
- Johannes Kepler, Astronomia nova, Heidelberg 1609
- Johannes Kepler, Harmonices mundi libri V, Linz 1619
Further reading
- Arthur Schopenhauer, Eristische Dialektik oder Die Kunst, Recht zu behalten, around 1830