The simplest logic

Here is a very simple logic. It has two ingredients. Propositional variables and the Sheffer stroke (which, following Peirce, we could call the amphecore). The logic uses Polish notation to express formulae, which is just to say that the syntax rules require the sentences to be ordered (although there isn't a unique ordering because | commutes).

The syntax looks like this. Any propositional variable is well-formed and if p and q are well-formed then |pq is well formed. We can quickly confirm that all well-formed formulae have n propositional variables and n-1 strokes.

The valuation function is likewise simple. V(|pq)=0 iff V(p)=1 and V(q)=1.

Finally, what are some interesting deductive systems for this logic? A simple modus ponens-style inference rule is the following:

|pq,p ā‡’ |qq

Nicod gives the following as his single NAND rule of inference.[^1]

||rqp,p ā‡’ r

We could use either of these to construct a Hilbert-style axiom system.

Equivalences

Pretty quickly we might get tired of this logic if we're used to thinking in the less sparse language of classical logic. What parts of this logic are equivalent to classical logic?

  • ~p = |pp
  • pāˆ§q = ||pq|pq
  • pāˆØq = ||pp|qq
  • pā†’q = ||qqp

Some Tautologies

  • ||ppp
  • |||pr|pq|pr
  • ||||ppq||ppqp

I suspect that there are some interesting relationships between the length of an axiom in this system and the number and arrangement of unique propositional variables. I mess around with this when I'm doodling, but I don't have much else to say about it.

Some speculation:

  • All tautologies have at least one sentence letter that occurs an even number of times. Maybe an odd number of such sentence letters? Maybe exactly one?
  • No tautology exists for ranks |,|||,|||| but one occurs for every other rank.
  • There is an arrangement where the number of | is weakly decreasing and the number of p is weakly increasing.
  • Probably other stuff too!

[^1]: J.G. Nicod, A reduction in the number of primitive propositions of logic, Proc. Camb. Phil. Soc. 19 (1917), 32-41.

researchAdam Edwardslogic
May Sinclair

Mary Amelia St. Clair (1863-1946), possibly better known by her pseudonym May Sinclair, was a philosopher. She was also a novelist, poet, and suffragist at the turn of the twentieth century. She is responsible for coining the term "stream of consciousness" as it refers to a style of novel in her review of Dorothy Richardson's Pilgrimage.

I know about her work because I picked up a copy of her book A Defense of Idealism (1917) in a used book store in Atlanta a few years ago. She wrote a follow up to that book five years later called The New Idealism (1922).

She begins her Defense as follows:

There is a certain embarrassment in coming forward with an Apology for Idealistic Monism at the present moment. You cannot be quite sure whether you are putting in an appearance too late or much too early.

It does look like personal misfortune or perversity that, when there are lots of other philosophies to choose from, you should happen to hit on the one that has just had a tremendous innings, and is now in process of being bowled out. As long ago as the early 'nineties Idealism was supposed to be dead and haunting Oxford.

So she gets it. The new Hegelians in Britain had just had their whole deal obliterated by the likes of Bertrand Russell and G. E. Moore, so it's odd to see someone come out in defense of the view. This is one thing that makes Sinclair such an interesting figure in the history of philosophy.

I'm troubled by the fact that Sinclair's philosophical work appears to have generated essentially no secondary literature. Like, none. A search on PhilPapers brings up a her books and a few articles, and some contemporary reviews of her books that strike me as... uncharitable. In any case, I think these books deserve attention. File this one away in After-The-Dissertation-Is-Done.

I want to close with this quote from one of Sinclair's feminist pamphlets:

We are dealing less with a psychological portent than with a new sociological factor, the SOLIDARITY OF WOMAN. And there is only one other factor that can be compared with it for importance, and that is the SOLIDARITY OF THE WORKING-MAN.

And these two solidarities are one.

Women's rights = worker's rights.

[EDIT]: Here is a good source on Sinclair's philosophical development by Dr. Charlotte Jones, who works on Sinclair's fiction. Philosophers should be engaging with this work!