Words Rewriting Method in Syllogism

Words Rewriting Method in Syllogism

Armahedi Mahzar (c) 2015

In my last blog it was shown that we can derive the conclusion all valid syllogism as it is tabulated by Leibniz, using rewriting rules for the combination of the premises which are codified with alphabets as it was done by medieval logicians. However, the alphabetic codifications has the same pattern as the English sentences of categorical proposition of Aristotle  .

Asp is the code for “all s is p”
Esp is the code for “no s is p” or “all x is not p”

Isp is the code for “some s is p”

Osp is the code for “some s is not p”

So, it seems that the rewriting rules can also be translated to the verbal formulation.

R1. Commutation: Exchange sequence of  premises
R2  Transposition: “all x is y” = “all not y is not x”        
R3. Conversion:
       “all x is not y = “all y is not x” (Exy = Eyx) and
       “some x is y” = “some y is x”    (Ixy = Iyx)
R5. Deletion of “x all x is” or “not x all not x is”

For example the premises combination of Barbara is “all s is m and all m is p”. If we delete the words “m and all m is” from it we get “all s is p” which is nothing but the conclusion.

For other valid syllogisms we can also use the four rules selectively so the mid terms are in the middle of the premises conjunction and then we apply the fourth rule to get the conclusion of the syllogisms. The following table will help us to do such words rewriting: 

For example for the Darapti syllogism, it is done like this
“all m is p and all m is s and some m”
is rewritten following commutation rule R2 as
“some m is m and all m is s and all m is p”
that can be rewritten following deletion Rule R4 as

“some m is s and all m is p”
that can be rewritten following conversion rule R3 as

“some s is m and all m is p”
that can be rewritten following deletion rule R4 to get
“some s is p”
which is nothing but the conclusion of Darapti.

Another example is getting the conclusion of Felapton
“all m is not p and all m is s and some m is m”
is rewritten following commutation rule R1 as
“some m is m and all m is not p and all m is s”
which can be deleted (R4) to get
“some m is not p and all m is s”
which can be converted (R3) to get
“some not p is m and m is s”

which can be deleted (R4) to get

“some not p is s”

which can be converted (R3) to get

“some s is not p”
which is nothing but the conclusion of Felapton.

Afternote

It is a surprise that finally I discover a verbose method after a long exploration of algebraic (Brownian, Brickenian and Kauffmanian),  arithmologic (Boolean, Peircean and Sommersian) and combinatorics methods with formulas (Ploucquetian) and combinatorics games of things that I discovered, when in fact I was starting the journey ignited by my incomprehension of verbose formulation of logic by Aristotle. It turns that my journey is actually a circle.

However, my verbose method differs from Aristotle method of reducing all syllogisms to the perfect syllogism as the necessary syllogism which are true by themselves. My method is a reduction of the premises to the conclusion using rewriting rules. I have never seen before that the rewriting rules is also useful for a deduction in syllogistic.

All my methods can be simulated into a game of things, including the alphabetic and words rewriting method by replacing alphabets in a string, and words in a sentence, into different kinds or color of duplicate things such as colored beads in a chain.

Later on, I will present you such chains of colored beads, but firstly I like to examine other rewritten methods on the strings of alphabets and non-alphabetic  as it is implicit in historical systems of logic. That issues I will discuss in my following blogs.