Easy to Learn Syllogism

Easy to Learn Syllogismu

Armahedi Mahzar (c) 2015

Last night I woke up and saw formulas of both premises of a syllogism were flying in front of my eyes, merging and retracting becoming the conclusion with a very simple way: eliminate xAx.

Syllogistic inference

Example: Barbara IF Aab AND Abc THEN Aac. Combining Aab and Abc we get AabAbc. If we remove bAb, then Aac is staying and that is the conclusion.For the syllogism that involves negative statements E also done the same thing, if Exy is rewritten as AxNy, where Nx means NOT x. Meanwhile IxNy in the conclusion is rewritten as Oxy.Examples for figure-1 syllogism:

Barbara: Abc Aab = Aab Abc = Aac
Celarent: Ebc Aab = AbNc Aab = Aab AbNc = AaNc = Eac
Darii:        Abc Iab = Iab Abc = Iac
Ferio:       Ebc Iab = Iab Ebc = Iab AbNc = IaNc = Oac
Barbari:   Iaa Aab Abc = Iac
Celaront: Iaa Ebc Aab = Iaa AbNc Aab = Iaa Aab AbNc = IaNc = Oac
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Rules of Rewriting
It turns out that syllogism can be solved with simple rules
R1. Exchange sequence of  premises {AND commutativity}
R2. Identities Exy = AxNy; Oxy = IxNy           {definition}
R3 Axy=ANyANx                                         {transposition}
R4. Identity Exy = Eyx and IXY = Iyx {simple conversion}
R5. Deletion of x Ax

The proofs of the validity of all syllogistic mood are shown in the following table

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A Reflection

Then I felt so stupid. You see, I have long been aware of the formulas A, E, I and O of the medieval logicians. Those formulas  are useful to me, because it made me able to prove the validity of the 24th syllogism in the Leibniz table using three consequences in the form algebra of George Spencer-Brown.

For that purpose, I transform the symbolic algebra of Spencer-Brown into the game of drawing and erasing pictures that I call the object algebra of logic  Objective primary algebra

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Then I had found Arithmologic and its simulators in the form of various combinatorial games of logic. After a long time exploring  the world of logic, I finally found a new rewrite method described above.

1. Actually, I firstly found the rewriting method for the symbolic method of  De Morgan Syllabus of a Proposed System of Logic : Augustus De Morgan : 

			Syllabus of a Proposed System of Logic : Augustus De Mor… Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.
			View on archive.org

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(colleagues during George Boole) in the 19th century and in the 20th century Pagnan method ( http://arxiv.org/abs/1302.7111 ) . this rewriting method was embedded in my subconscious, so they work on the medieval alphabetical symbolism.

2. The rewriting method can also be simulated by the linear game of things. In this game, each letter in the string of alphabets is replaced with a chain of colorful objects. Hopefully I will present it in my next blog.