Easy to learn syllogism

Easy to Learn Syllogism
Armahedi Mahzar (c) 2015

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

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  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
  
  

  

  
  

  
  A Reflection
  
  1. 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 24 syllogism in the Leibniz table using three consequences in the form algebra of George Spencer-Brown.
  
  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.