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The Mathematics of Charles Sanders Peirce (2001) [pdf] (uic.edu)
55 points by language on Sept 1, 2016 | hide | past | favorite | 11 comments

In a comments subthread that was flag-killed, gballan made the following interesting link:

"Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look.

[0] https://en.m.wikipedia.org/wiki/Conceptual_graph

Peirce also remarked in a letter how electrical relay circuits correspond one-for-one with boolean logic formulas. He seems to have been the first to notice this, almost 50 years before anyone else.

His writings are very tedious to me, though, and I hope this article I just glanced through makes him more palatable.

similar: Lou Kauffman's "Box Arithmetic"


I couldn't get through all of the paper. It seemed to be simply a new, not especially useful, notation for certain boolean operators.

The concept of universal boolean operators, in the sense of functional completeness (see [1]) isn't that deep. A binary boolean operator (function) has four possible combinations of inputs: TT, TF, FT, and FF. For example, the and operation is defined by (T and T) == T, (T and F) == F, (F and T) == F, and (F and F) = F. The familiar truth table [2] is a clearer way to write this out:

    A | B | A and B
    T   T      T
    T   F      F
    F   T      F
    F   F      F
Other boolean operators can be defined by truth tables too. How many unique boolean operators are there? Since there are four possible binary outputs there are 16 possible boolean operators. They are mostly very familiar, but mathematicians use various notations for them.

Again, truth tables make them obvious:

    A B | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
    T T   T  T  T  T  T  T  T  T  F  F  F  F  F  F  F  F
    T F   T  T  T  T  F  F  F  F  T  T  T  T  F  F  F  F
    F T   T  T  F  F  T  T  F  F  T  T  F  F  T  T  F  F
    F F   T  F  T  F  T  F  T  F  T  F  T  F  T  F  T  F
the 16 possible operators are thus:

1: true, sometimes indicated by the "top" character in math, it looks like a capital T

2: or

3: if, (A if B) means B implies A, written <==

4: A, just the value of the first argument

5: onlyif, (A onlyif B) is sometimes written ==>

6: B, the value of the second argument

7: iff, same as ==

8: and

9: nand, not and

10: xor

11: !B, the negation of second argument

12: notonlyif, i.e. not(A ==> B)

13: !A, the negation of the first argument

14: notif, i.e. not(A <== B)

15: nor, not or

16: false, sometimes written in math with the bottom symbol which looks like an upside down capital T

We don't normally use all of these operators in predicate logic or programming or digital circuits because a few familiar ones will do. In fact, if we allow ourselves the negation operator (i.e. not or ! in some programming languages) then all binary boolean operations can be obtain from only and and not or from or and not. For example the operator numbered 5 above is the implication operator (A ==> B), but this is simply (not A or B).

The sign of illation in the original article is simply operation 5 above, A ==> B.

Interestingly, two of the binary boolean operators are universal or functionally complete by themselves, nand and nor. Any stateless boolean circuit can be constructed out of only nand gates or only nor gates. A programming language could get by with only one boolean operator, the binary nand (or the binary nor). For example ((not A) and B) could be rewritten using only nand:

    ((not A) and B) == (A nand A) and B
                    == not ( (A nand A) nand B )
                    == ((A nand A) nand B) nand ((A nand A) nand B)
[1] https://en.wikipedia.org/wiki/Functional_completeness

[2] https://en.wikipedia.org/wiki/Truth_table

This isn't a "new notation" -- Pierce lived in the 19th century and his notation predates the current. Also, the fact that "nor" is functionally complete was actually discovered by Pierce in 1880. Pierce is more famous for his non-mathematical philosophy of "pragmatism", but he actually contributed quite a bit to logic.

Pardon the nerdigression, this feels much like Emacs VS Windows ergonomics.

Wow, I had no idea. Thanks.

Not sure why this is trending on HN (and without any comments or discussion). TBH, its reads like a Sokal Hoax and seems like gibberish to me.


This is not like the Sokal hoax at all. Sure the writer has a few English departmentish affectations, but the article is basically just explaining Peirce's logic. And Peirce is one of roughly two people in history to figure out quantification.

Peirce's diagramatic logic is the forerunner to Sowa's conceptual graphs [0]. Worth a look.

[0] https://en.m.wikipedia.org/wiki/Conceptual_graph

Not sure either. Semiotics is a philosophical branch of linguistics.

De Saussure and Peirce I put conceptually in the same tradition of Cyc and other symbolic/ontological/semantic efforts in knowledge engineering.

That being said, symbol grounding does not get enough attention in current connectionist systems. How to properly present grammar, time, pointers, signs?

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