The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: let us calculate!, without further ado, to see who is right.

— Leibniz, The Art of Discovery (1685)

Leibniz was a lawyer, diplomat and philosopher. He spent his life watching arguments that never ended, between religions, between countries, between schools of philosophy. A mathematician, on the other hand, does not argue about whether $2+2$ is 4: he does the sum. What if arguments could be calculated like numbers?

An alphabet of thought

The plan had two parts.

The first was a universal language, which Leibniz called characteristica universalis. In it, each simple idea would have a symbol, and complex ideas would be combinations of these symbols, just as words are combinations of letters. Leibniz began thinking about it at 20, in De arte combinatoria of 1666.

The second was a calculus of reasoning, the calculus ratiocinator: rules for manipulating the symbols and reaching correct conclusions mechanically, as one does an addition. In that world, an argument would be settled with pencil and paper.

Concepts as prime numbers

Leibniz actually tried a version with numbers. The idea: give each simple concept a prime number, and each compound concept the product of the primes of its parts.

For example, in the classical definition, “man is a rational animal”. If animal $=2$ and rational $=3$, then man $=2\times3=6$.

Now the sentence “every man is an animal” becomes a question of arithmetic: does 2 divide 6? Yes, so the sentence is true. And “every animal is a man”? Does 6 divide 2? No, so it is false. A piece of reasoning about concepts has become a division.

The system had serious problems. It did not handle negations or sentences with “some” well. Leibniz knew this and kept trying other forms. But the central idea, that logic can be algebra, had been launched.

An example in today’s notation

Today we write reasoning with logical symbols. Consider three statements about a figure:

  • $P$: the figure is a square;
  • $Q$: the figure is a rectangle;
  • $R$: the diagonals of the figure have the same length.

If $P\Rightarrow Q$ (every square is a rectangle) and $Q\Rightarrow R$ (every rectangle has equal diagonals), then $P\Rightarrow R$. The conclusion follows from the form of the argument, without having to think about squares. That is exactly what Leibniz wanted: to check a line of reasoning by looking only at the symbols.

But symbols work no miracles. If a premise is wrong, the conclusion can be wrong, however correct the calculation. And reversing an implication is a common mistake: from “every square is a rectangle” it does not follow that every rectangle is a square.

Combining is not interpreting

Three statements, each true or false, give $2^3=8$ possible combinations: 000, 001, 010, 011, 100, 101, 110, 111. Notice that these are the numbers from 0 to 7 in binary. The link between logic and numbers made of 0 and 1, which Leibniz sensed, would become central two centuries later.

The dream, partly fulfilled

Leibniz’s writings on logic remained almost entirely unpublished until the 19th century. Even so, the idea returned:

  • In 1854 George Boole published The Laws of Thought, turning logic into an algebra with 0 and 1.
  • In 1879 Gottlob Frege created a complete symbolic language for logic, very close to what Leibniz had imagined.
  • In 1936 Alan Turing described an abstract machine able to carry out any calculation described by rules.

In 1931, however, Kurt Gödel proved that no system of rules can decide every truth of mathematics. Leibniz’s complete dream is impossible. But the possible part of it is in every computer: a mechanism that manipulates symbols by following rules.

Exercise: from P ⇒ Q, can we conclude Q ⇒ P?

No. Every square is a rectangle, but not every rectangle is a square. The converse of an implication needs its own proof.

Exercise: in the prime system, “animal” = 2, “rational” = 3 and “mortal” = 5. What number represents “mortal man”? Is it “animal”?

“Mortal man” $=2\cdot3\cdot5=30$. Since 2 divides 30, the sentence “every mortal man is an animal” is true in the system.

Sources: Leibniz, logic and philosophy, MacTutor.

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