Inquiry Into Inquiry

How Logic Got Its Blots

Cf: Laws Of Form Discussion • JA

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.

Re: Peirce List Discussion • GF

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 …

Cf: Laws Of Form Discussion • JA

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

Re: Peirce List Discussion • GF

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.

Cf: Laws Of Form Discussion • JA
Re: Peirce List Discussion • GF

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.