
The Mathematics of Charles Sanders Peirce (2001) [pdf] - language
http://homepages.math.uic.edu/~kauffman/CHK.pdf
======
pavlov
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](https://en.m.wikipedia.org/wiki/Conceptual_graph)
_

------
abecedarius
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.

------
mrcactu5
similar: Lou Kauffman's "Box Arithmetic"

[http://homepages.math.uic.edu/~kauffman/Arithmetic.htm](http://homepages.math.uic.edu/~kauffman/Arithmetic.htm)

------
todd8
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](https://en.wikipedia.org/wiki/Functional_completeness)

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

~~~
jhbadger
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.

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

------
dmfdmf
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.

[https://en.wikipedia.org/wiki/Sokal_affair](https://en.wikipedia.org/wiki/Sokal_affair)

~~~
ykler
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.

