Christine Ladd-Franklin

1847 – 1930 · Connecticut & Johns Hopkins & Columbia · student of C. S. Peirce · logician & vision scientist

Aristotle left a sprawling catalogue of valid and invalid syllogisms; Boole turned logic into algebra. Ladd-Franklin — studying under [[charles-peirce]] — found the single test that unifies all of it. Her insight: forget memorising which moods are valid. A syllogism is valid exactly when its two premises, taken together with the contradictory of its conclusion, form an inconsistent triad — three statements that cannot all be true at once. She called it the antilogism, and it is gorgeously symmetric: from one inconsistent triad you can read off three valid syllogisms. It was, for decades, the cleanest decision procedure for the syllogism in existence.

✓ STRONG

A real, elegant result. The antilogism is a correct and unifying test for syllogistic validity — a genuine advance in algebraic logic, recognised by Peirce and standard in logic teaching for generations.

◐ ONE PIECE OF A FIELD

Syllogistic, not all logic. The antilogism settles the classical syllogism beautifully; it is not, and was not meant to be, the whole of modern predicate logic ([[gottlob-frege]]). A sharp tool with a defined edge.

◔ THE DEGREE WITHHELD

Excluded by rule, not merit. She completed every requirement for a Johns Hopkins PhD in 1882; the university refused the degree to a woman. She got it in 1926 — forty-four years later, at 78. The injustice, named.

I · The antilogism — one test for every syllogism

Pick a syllogism. The machine forms its antilogism — the two premises plus the contradictory of the conclusion — and checks whether those three can all be true together. If they cannot (an inconsistent triad), the original argument is valid: denying its conclusion while granting its premises is impossible. If they can all hold, the argument is invalid. (Checked live by enumerating every possible Venn arrangement of the three terms.)

its antilogism — premises + the contradictory of the conclusion

The whole of Aristotle's syllogistic collapses into one question: is this triad inconsistent? ⚑ And the symmetry is the prize — any inconsistent triad gives three valid syllogisms (drop any one statement, negate it, and conclude it from the other two). One inconsistency, three theorems. This is the syllogism finally made algebraic and mechanical — a decision procedure, the kind of thing a computer runs.

"The cause of [a syllogism's validity] is that the three propositions … form an inconsistent triad." — C. Ladd, "On the Algebra of Logic", 1883

II · One inconsistency, three syllogisms

Here is the antilogism's deep economy. Take the current inconsistent triad. Drop any one of its three statements and assert its opposite as a conclusion from the remaining two — you get a valid syllogism, every time. Three buttons, three theorems from a single fact of inconsistency.

This is why she could throw away the medieval mnemonics (Barbara, Celarent, Darii…): they are just the faces of inconsistent triads. Find the inconsistencies, and you have found every valid syllogism there is. ⚑ A unifying principle in place of a list — the move that turns a catalogue into a science.

III · The logic, and the forty-four years

Gate kept on. The mathematics is clean and real: the antilogism is a correct, elegant decision procedure for the categorical syllogism, praised by Peirce and taught for generations — a genuine contribution to the algebra of logic, even if it addresses the classical syllogism rather than the full predicate logic that [[gottlob-frege]] was building in the same years. What must be told without softening is the cost. Christine Ladd entered Johns Hopkins in 1878 — when it did not admit women — by the intervention of the logician J. J. Sylvester, who recognised her work; she was permitted to attend on the condition that her name not appear in the register. She completed every requirement for the doctorate and wrote a dissertation Peirce published in his landmark 1883 volume — and the university refused to grant her the degree because she was a woman. She was finally awarded it in 1926, forty-four years later, at the age of 78, four years before her death. In between she did major work in the theory of colour vision as well, and spent decades barred from the regular faculty positions her male peers held automatically. Her logic stands entirely on its own; the four-decade delay stands as the plainest possible illustration of why this lineage was, for nineteen spheres, "all men." ⚑ [[ada-lovelace]] ← · next → Rózsa Péter.