The Symbolism of PM: Dots

One of the initial challenges in tackling PM is its symbolism. I won't review all of it here, because what W&R say about these operators in the text is fairly straightforward (PM 6-8). The usey of the 'horseshoe' for the material conditional (⊃), the 'triple bar' for material equivalence (≡), and 'wedge' for disjunction (v) --which is all fairly standard. Likewise, the quantificational operators are the same as most standard logic textbooks. If you need more background on these (or any other) operators in PM, Bernie Linsky has a super-helpful entry in the Stanford Encyclopedia of Philosophy here.

W&R use the dot (.) for a dual purpose--both as the symbol for conjunction (the equivalent of the ampersand in contemporary symbolism), and also to denote scope (the equivalent of parentheses in contemporary symbolism). "Dots on the line of the symbols have two uses, one to bracket off propositions, the other to indicate the logical product of two propositions" (PM 9).

On this they say:

Dots immediately preceded or followed by “v” or “⊃” or “≡” or “⊢”, or by “(x)”, “(x,y)”, “(x,y,z)” … or “(∃x)”, “(∃x,y)”, “(∃x,y,z)” … or “[(x)(φx)]” or “[R‘y]” or analogous expressions, serve to bracket off a proposition; dots occurring otherwise serve to mark a logical product. The general principle is that a larger number of dots indicates an outside bracket, a smaller number indicates an inside bracket. The exact rule as to the scope of the bracket indicated by dots is arrived at by dividing the occurrences of dots into three groups which we will name I, II, and III. Group I consists of dots adjoining a sign of implication (⊃) or equivalence (≡) or of disjunction (v) or of equality by definition (=Df). Group II consists of dots following brackets indicative of an apparent variable, such as (x) or (x,y) or (∃x) or (∃x,y) or [(x)(φx)] or analogous expressions. Group III consists of dots which stand between propositions in order to indicate a logical product. Group I is of greater force than Group II, and Group II than Group III. The scope of the bracket indicated by any collection of dots extends backwards or forwards beyond any smaller number of dots, or any equal number from a group of less force, until we reach either the end of the asserted proposition or a greater number of dots or an equal number belonging to a group of equal or superior force. Dots indicating a logical product have a scope which works both backwards and forwards; other dots only work away from the adjacent sign of disjunction, implication, or equivalence, or forward from the adjacent symbol of one of the other kinds enumerated in Group II (PM 9-10).

There are several observations one must make here. First, the highest group (the one with the most "force") consists of the disjunction, the material implication and material equivalence, plus the definition relation and the assertion sign. The conjunction seems to lack the standing that the other three binary operators have. (One should interpret this as a convention, not as a rule with is provided by deeper philosophical reasons--since the operators are inter-definable in PM). So when one sees dots flanking an operator, those are the ones of highest priority.

The next level are those of "apparent variables" (or, as one might call them: "bound" variables). Only then do we reach the conjunction operator itself, at the lowest level of force.

Secondly, there are four possible configuration of dots themselves:., :, : ., : : Each configuration is a higher level, like round brackets (parentheses), square brackets and curly brackets.

The problem is, that when a conjunction is used, the dots indicating parentheses do not jump to a higher number of dots. Consider:

*3·12 ⊢ : ~p . v . ~q . v . p . q

Recall the rule: "The scope of the bracket indicated by any collection of dots extends backwards or forwards beyond any smaller number of dots, or any equal number from a group of less force, until we reach either the end of the asserted proposition or a greater number of dots or an equal number belonging to a group of equal or superior force." The double dots after the assertion sign indicate the scope of the entire wff. the second set of dots around the first disjunction are read afterward, and since that's a Group I type, it takes prominence. They extend to the next (third) single dot, of equal group, around the second disjunction. It's not easy to see at first--but the fifth dot is a conjunction, not a scope mark. So it should be read as:

⊢ [(~p v ~q)] v (p & q)]

 

Reading Principa

For logic dorks like myself, I don't have to remind you that the first volume of Russell and Whitehead's masterwork Principia Mathematica is having its centennial anniversary this year. I thought I'd commemorate the occasion by trying to read through the first volume of PM.

I'm going it alone on this. But I hope there would be people interested in reading along. Several months ago, when I started considering reading PM this year, I began by emailing my colleagues at the regional state institution which I teach. None were interested--not that I can wholly blame them. I thought about possibly emailing faculty at other institutions in the region, but after perusing the websites and finding nobody really interested in early analytic philosophy or the history/philosophy of logic, I thought that the reception to such an idea would be just as tepid. And besides, they have their own projects going on, and I'm not sure if they'd be bothered by my pet project. But I was sure whether there'd be be interest in such an endeavor else--nationally or even internationally. So I began this blog.

This will be (I hope) a periodic update of my progress, and my difficulties, reading the first volume of Principia Mathematica. And I hope others might be interested in picking up and reading along with me.

Enjoy!