## Peirce’s 1903 Lowell Lectures • Comment 3

Peirce’s use of the “scroll” as a graphical syntax for implication continued to raise many questions at this point in the Peirce List reading.  I think a lot of what bothers people has more to do with general misunderstandings about material implication than anything peculiar to Peirce’s graphs.  I suggested a way of reading the “scroll” $\texttt{(} a \texttt{(} b \texttt{))}$ that makes it crystal clear to me, namely, “not a without b”, and then I added the following comment.

Peirce’s approach in these lectures appeals to the line of thinking that takes implications and the corresponding subject-predicate form as basic, but that is not the only possible basis for a system of logical syntax and not the only basis that Peirce himself took up in his many syntactic experiments.  In relating logical signs to logical objects it normally proves best to remain flexible and to consider the object of logic that is common to all its avatars.

## Peirce’s 1903 Lowell Lectures • Comment 2

### How Logic Got Its Blots

Taking positive implication as a basic construct, as Peirce does in the lectures at hand, one has to find a way to rationalize the introduction of negative concepts, in the first instance, logical negation and a logical constant for falsity.  Questions about this naturally arose in the Peirce List reading, prompting me to make the following comment on Peirce’s just-so-story, especially as it bears on the link between primary arithmetic and primary algebra.

Peirce’s introduction of the “blot” at this point as a logical constant for absurdity or falsity is one of the places where he touches on the arithmetic of logic underlying the algebra of logic, a development that began with his taking up the empty sheet of assertion, a tabula rasa or uncarved block, as a logical constant for truth.

The radical insight involved in this move would later be emphasized by George Spencer Brown when he revived Peirce’s graphical approach to logic in the late 1960s.

More to follow, as I find the opportunity …

## Pragmatic Traction • 7

It’s good to remember that observation, perception itself, has an abductive character in Peirce’s analysis and induction for him is more a final testing than initial conception stage.  Yes, it’s wheels upon wheels but some steps are logically more primitive in the recursion.

## Peirce’s 1903 Lowell Lectures • Comment 1

A question arose concerning one of Peirce’s ways of explaining logical negation.

I commented as follows.

One way of saying “not x” or “x is false” is to say “x implies α” where “α” is taken to mean “any proposition whatever”.  This is the hoary old rule of ex falso quodlibet, more lately going under the name explosion principle.  It is related to the definition of an inconsistent logical system as one in which every formula is a theorem, and thus in which no line of distinction can be drawn between true and false.

One place where Peirce makes use of this style of negation is in his comments on a logical formula we now call Peirce’s Law.

## Peirce’s 1903 Lowell Lectures • Preliminaries

In September the Peirce List began a reading of Peirce’s 1903 Lowell Lectures (“Some Topics of Logic Bearing on Questions Now Vexed”).  I’ve had opportunities for only a few desultory comments from time to time but as it turned out most of those thoughts had to do with the algebraic, graph-theoretic, and logical ideas exhibited by Peirce’s systems of logical graphs and Spencer Brown’s Laws of Form.

At any rate, I thought there might be something in those remarks worth recycling to the Laws of Form discussion group and other interested parties.

## Ask Meno Questions • Discussion 4

The questions raised under the heading of “Foundations of Mathematics” are generally considered to fall under the “Philosophy of Mathematics”, in particular, critical reflection on the possibility of mathematical knowledge and how we come to acquire it, or imagine we do.  Now that’s a question of epistemology, or how we may be able to learn anything at all, and Plato’s dialogue Meno is one of the earliest and finest examinations of that question.