# In what sense was it thought (at one point) that every lattice was distributive?

Blyth writes in his 2006 book Lattices (p. 65):

It is a curious historical fact that it was originally thought that every lattice was distributive! That this is not so is shown by the following example [M3] of a modular lattice that is not distributive.

(I'll spare you the Hasse diagram of M3 here since it's well-know/obvious and found on Wikipedia too.)

What I want to ask here is: what does Blyth mean by the first sentence highlighted? Did people at one point define a lattice in a way that made it distributive, or was M3 (or N5) really that non-obvious as a counter-example to some early writers on the topic even under a more general defintion?

• Bilova mentions in Lattice Theory – Its Birth and Life that Peirce thought distributivity follows from Boole's other axioms and Schröder proved otherwise. – Conifold Mar 23 at 8:34
• @Conifold: interesting. I guess the Hasse diagrams we so commonly use today were not known (or clear) to Peirce etc. But I could make that another question... – Fizz Mar 23 at 8:40

The original approach to lattices, following Boole, was algebraic rather than graphic or order-theoretic, and simple examples that quickly come to mind today were not really in the offering. The pioneers, Boole, Peirce and Schröder, operated with a fluid set of algebraic rules and logic related principles, and examples of logical systems they operated with were hard to visualize and distinguish from each other. Blyth likely refers to the episode with Peirce's casual remark in On the algebra of logic (1880) that was disputed by Schröder. It was instrumental in sorting out which sets of rules lead to which structures, and coining the lattice concept in particular. The relations between the syllogistic, propositional logic and "the logic of relatives" (predicate calculus that Peirce discovered independently of Frege) were clarified around the same time.

Ma and Pietarinen describe the episode as follows in Peirce’s Sequent Proofs of Distributivity:

"In the 1880 paper, Peirce stated the following distributive laws... He casually mentions that “they are easily proved . . ., but the proof is too tedious to give” [11, p. 33]. This passage provoked a challenge by Schröder, who took the laws of distributivity to be independent from the lattice axioms. A rejoinder and a lively discussion ensued and Peirce’s lost and subsequently recovered version of the proof was finally added to Huntington’s 1904 paper... Huntington published in 1904 the proof Peirce had sent to him, including Peirce’s footnote about it. In the published proof, the axiom — Huntington’s “postulate” — number 9 is crucial. Without it, the axioms 1–8 only define uniquely complemented lattices which need not be distributive.

Here is Peirce's own description from the follow-up paper On the algebra of logic (1885), where he admits slipping in a "non-syllogistic" principle (dilemmatic argument) into his proof of the "wrong" direction of distributivity ($$-\!\!\!<$$ is Peirce's symbol for the material implication):

"The dilemma was only introduced into logic from rhetoric by the humanists of the renaissance; and at that time logic was studies with so little accuracy that the peculiar nature of this mode of reasoning escaped notice. I was thus led to suppose that the whole non-relative logic was derivable from the principles of the ancient syllogistic, and this error is involved in Chapter II of my paper in the third volume of this Journal [the 1880 paper]. My friend, Professor Schröder, detected the mistake and showed that the distributive formulae $$(x + y)z\ -\!\!\!<\ xz + yz$$ $$(x + z)(y + z)\ -\!\!\!<\ xy + z$$ could not be deduced from syllogistic principles. I had myself independently discovered and virtually stated the same thing. (Studies in Logic, p. 189.)"

Schröder developed what we would now call theory of lattices in Vorlesungen Über die Algebra der Logik (1891), the first systematic exposition of modern logic that laid the foundation of Russell's Principia. There he gave the first example of a non-distributive lattice, but it was not an elementary one one would think of today. Here is from Peckhaus's Mathematical Origins of 19th Century Algebra of Logic:

"In the sections dealing with statements without negation he proves one direction of the distributivity law for logical addition and logical multiplication, but shows that the other side can not be proved, he rather shows its independence by formulating a model in which it does not hold, the so-called “logical calculus with groups, e. g. functional equations, algorithms or calculi”. He thereby found the first example of a non-distributive lattice."