Literal Combinatoric of Plouquet

Literal Combinatoric of Plouquet

 Armahedi Mahzar (c) 2015

Gottfried Ploucquet
(1716-1790)

In previous blog, I pointed out that there are three kinds of arithmologic (Boole, Peirce and Sommers) but all three are structurally similar to each other. Essentially any arthmological statement is expressed as string of letters and symbols of mathematical operations. \

In this blog I will present a simpler literal combinatoric method which is inspired by Ploucquet nethod in the 18th century, before Boole introduced the algebraic symbolism for logic in the 19th century. The method is presented as a combinatoric literalization of Sommersian arithmologic.

Arithmologic

The following are the arithmological symbols

+===============================================+
|           Concept  TRUE FALSE  NOT  OR  AND   |
+-----------------------------------------------+
| Boole     letter     1    0    1-    +  -1+   |
| Peirce    letter     0    1    1/    +        |
| Sommers   letter               -     +   +    |
+-----------------------------------------------+  

Similarity of of the formulas can be shown by the following table of syllogism

+===========================================+
| Syllogism IF p AND q THEN r               |
|                 = NOT (p AND q) OR r      |
+-------------------------------------------+
| Boole             1-(p-1+ q)+r = 1-p+1-q+r|
| Peirce            1/(pq/r)     = 1/p 1/q r|
| Sommers           -((p+q)-r)   =  -p  -q+r|
+-------------------------------------------+

Proving its validity has also a similar procedure, namely:

the annihilation of pairs of oppositely signed variables.

Seeing the structural and procedural similarities, we can expect that there is a simpler symbolic formulation. The following is one of its simplification. For simplicity, I will use the Sommersian arithmologic as a reference, because it is the simplest.

Combinatoric Simplification of Arithmologic

To simplify the Sommersian arithmologic, we can use the following literalization conventions:
- Write the + sign with no spaces or symbols at all
- Write -x as the uppercase X.

Categorical statement Aristotle

By shortening convention, we can write Aristotle fundamental categorical statement as follows:

(1) Universal affirmative
Aab = 'all A is B'
can be written as Ab

(2) Universal negative
Eab = 'no a is b'
can be written as AB

(3) Particular Affirmative
Iab = 'some a are b'
can be written as ab

(4) Particular Negative denial
OAB = 'some a are not b'
can be written as aB.

Validity proof of syllogism

In this notation the verification algorithm of the validity of 24 syllogism Leibnitz in the following table,
becomes very simple:

Step 1: write a joint symbol for premises
Step 2: delete the upper/lower case pair of letters.

The algorithm is becoming more concise and easier than the arithmologic.
Even elementary school children can do :)

We will make the proofs of  all valid syllogism in column 1 in the Leibniz table now.

Barbara proof  is

Abc AND Aab = Ab Bc = Ac = Aac

Celarent proof  is as follows

Ebc AND Aab = Ab BC = AC = Eac j

and Darii proof is like this

Abc AND Iab = ab Bc = ac = Iac

Similarly, the proof of Ferio is
EBC AND Iab = ab BC = aC = ​​Oac
 
To prove the existential syllogism proof is also simple.
Barbari proof is as follows

Iaa AND Aab AND Abc = aa Ab Bc = ac = Iac

and the Celaront proof islike this
Iaa AND Aab AND EBbc = aa Ab BC = aC = ​​Oac.
Syllogisms in the other columns also can be proven in the same way. Just use the Leibniz-Ploucquet table below

Well, this is very easy expression. The formula, with no signs of any math, is a mere string of small and capital letters. While the algorithm is not  arithmetical but purely combinatorical.
Thus the method becomes very easy as well: merge premises and delete letter pairs.

End notes

Because this method is similar to the letter method of Ploucquet, I shall call it as Ploucquet logical method of literal combinatoric. The notation for the negative is similar to the notation of Ploucquet for universality. 

However, Ploucquet also have a more visual method, namely the method of lineal combinatorics that, I think,  is easier than the method of literal combinatoric. Hopefully, I will  show a simplification of the literal Ploucquet method with the line pictorial in my next blog. :)