Sheffer stroke the most important advance in logic?

Question

I think I once read, or heard, that Bertrand Russell once said that the discovery that all logical operators are expressible in terms of the Sheffer stroke was the most significant advance in logic since the publication of Principia Mathematica, and that had he and Whitehead known about it beforehand, they would have proceeded completely differently.

This is a really strange claim, so I imagine that I (or the person who told me) misunderstood, or misheard, or made it up completely.

If Russell really did say something like this, what exactly did he say, and where and when?

Answer 1

Russell apparently made this comment in the 1925 edition of PM, where he introduced the (NAND) Sheffer stroke. See page 1 of The Evolution of Principia Mathematica. It says that Russell called it "the most definite improvement resulting from mathematical logic during the past fourteen years." But as the author notes, this is an odd comment, and the change is a technical triviality. I don't understand why Russell was so impressed with this either. I have never found it useful for proving anything, and I have never seen others use it either. It gets a passing mention at most.

Answer 2

The focus that Russell had was very different than the focus logicians have today. Russell's viewpoint of "logic" was tightly connected to Principia Mathematica, and his interpretations of other results depended on their relation with PM, because that system was the basis for his research program in logic and philosophy.

Russell's work was the peak of the logicist program: to argue that all of mathematics can be expressed in pure "logic" and to study the properties of that "logic". The existence of a single operator to which all other "logical" operators can be reduced would be wonderful from that point of view - just as a single physical law that explains all of physics would be wonderful for the foundations of physics. The use of a single primitive would have reduced the number of "logical notions" that were needed in the foundations of PM, which was an important philosophical goal.

So, in the context of PM, Russell's remarks were perfectly reasonable. The key is that his viewpoint of "logic" was directly focused on PM.

Answer 3

I have been reading the recent book The Evolution of Principia Mathematica, by Bernard Linsky.

The quote in the post is in significant ways not quite right. It comes in the Introduction to the 1925 Edition of Principia, first page. Here is what was actually written, in full.

"The most definite improvement resulting from work in mathematical logic is the substitution, in Part I, Section A, of the one indefinable "$p$ and $q$ are incompatible" (or, alternatively, "$p$ and $q$ are both false") for the two indefinables "not-$p$" and "$p$ or $q$." This is due to Dr. H.M. Sheffer. consequently, M. Jean Nicod showed that one primitive proposition could replace the five primitive propositions $\ast$1.2.3.4.5."

So Russell is saying that the greatest improvement in the second edition of Principia that is due to recent advances in mathematical logic, presumably by others, came from using the Sheffer stroke and an axiomatization based on the work of Nicod. There are also significant changes in the description of type theory, but presumably that is not mentioned here because it is based on ideas of Russell. (By this time, Whitehead had bailed.)

Russell is not saying that Sheffer's work is the most important development in mathematical logic of the previous fourteen years.