Talk:Syllogism

     (1.)  The boxes are given  characteristic borders. 
     (2.)  Diagram has boxes labeled with example predicates.  
        ……………
   S = /     \
       \_____/

         ……………
   M =  |     |
        |_____|

         …………… 
   P =  {     }
        {_____}

The joint presence of three boxes in a syllogistic diagram below implies three pairs of two boxes, the configuration of each pair illustrating one of the three propositions of the syllogism.

REDUCTION TO SEVEN (7) ESSENTIALLY DISTINCT SYLLOGISMS:

As probably is fairly well-known, we can reduce the count of syllogisms from 24 to 11 by

 (1.) dropping five "weak" syllogisms Barbari (AAI-1), Celaront   
      (EAO-1), Cesaro (EAO-2), Calemos (AEO-4), Camestros (AEO-2);
 (2.) identifying equivalent propositions due to the symmetries
            AeB <=> BeA,    AiB <=> BiA.

Thus for example all four EIO syllogisms, through the 4 figures are really the same or at least equivalent, and all four may be represented by FERIO (EIO-1). This leads to the following table of eleven (11) "reduced" syllogisms or equivalence classes.

However, even 11 syllogisms is too many, there are really only seven (7) essentially truly distinct syllogisms: Barbara, Darii, Felapton, Ferio, Celarent, Bocardo and Baroco. Please see below for discussion ("Homework").

To simplify reading of diagrams, premises are stated next in non-traditional order:

  First premise (=minor premise)
  Second premise (=major premise)

However,for the three-letter codes we retain the traditional form, placing the major premise first, before minor premise:

 1.  Barbara (AAA-1): SaM,MaP:SaP 
 2.  Darii (AII-1): SiM,MaP:SiP         
 3.  Felapton (EAO-3):  MaS,MeP,SoP      
 4.  Ferio (EIO-1): SiM,MeP:SoP         
 5.  Celarent (EAE-1): SaM,MeP:SeP      
 6.  Bocardo (OAO-3):  MaS,MoP:SoP 
 7.  Baroco (AOO-2):SoM,PaM:SoP

It is interesting to exhibit the seven (7) simplified Venn or Euler diagrams for this list with examples showing both, similarities and changes, as we pass from one to the next

DIAGRAMS OF SEVEN REDUCED SYLLOGISMS.

- - - - -    

1. Barbara (AAA-1): SaM,MaP:SaP 

 all S is M; all M is P: all S is P 

 S:Greeks M:Men P:mortal

    All Greeks are men.    (SaM)
    All men are mortal.    (MaP)
  ∴ All Greeks are mortal. (SaP)

   …………………………………………………………………………
  {     mortals               }
  {    …………………………………………       }  
  {    |     men       |      }
  {    |   ………………………   |      }   
  {    |  / Greeks  \  |      } 
  {    |  \_________/  |      }
  {    |_______________|      }
  {___________________________}

- - - - -

2. Darii (AII-1): SiM,MaP:SiP  (=Datisi (AII-3), 2 in class)

 some S is M; all M is P: some S is P

 S:fragrant; M:flower; P:lifeform
 
       Some fragrant things are flowers  (SiM)
       All flowers  are lifeforms         (MaP)
     ∴ Some fragrant things are lifeforms (SiP)

         …………………………………………………………………
        {                         } lifeforms
        {   ………………………………………………    }   
        {   |  flowers        |   }
        {   |     ………………………………|………}………………     
        {   |    /  ($)       |   }  ?    \                              
        {   |    \____________|___}_______/                               
        {   |_________________|   }  fragrant things  
        {_________________________} 

- - - - -

3. Felapton (EAO-3):MaS,MeP:SoP (=Fesapo (EAO-4), 2 in class)    
      

 all M is S; no M is P: some S is not P

 S:lifeform; M:flower; P:heavy weight

     All flowers are lifeforms.         (MaS)
     No flowers are heavy weight.       (MeP)
  ∴ Some lifeforms are not heavy weight. (SoP)

    …………………………………lifeforms  
   /  ……………………      \     
  /  |        |  …………\…………………
 /   |  ($)   | {  ?  \      }
 \   |flowers | {_____/______} 
  \  |________|      / heavy weight things  
   \________________/   

The symbol ($) means that set S of flowers is not empty, there is stuff in it (flowers) . .!!

- - - - - 
4. Ferio (EIO-1):SiM,MeP:SoP (=Festino (EIO-2)=Ferison(EIO-3)=Fresison (EIO-4), 4 in class)

 some S is M; no M is P: some S is not P

 S:mammal; M:carnivore; P:ruminant

     Some carnivores are mammals. (MiS)
     No herbivores are carnivores. (PeM)
  ∴ Some mammals are not herbivores. (SoP)

       ……………………………………
      |             |
     …|…..…….…………………|………………  mammals 
   /  |             |       \     
  /   |             |    …………\……………
 /    |   ($)       |  {      \  ?  }
 \    | carnivores  |  {______/_____} 
  \   |_____________|        /  herbivores
   \________________________/   

- - - -
5. Celarent (EAE-1): SaM,MeP:SeP (=Cesare (EAE-2), 2 in class)

 all S is M; no M is P: no S is P

 S:flowers; M:lifeforms; P:fulgurite

     All flowers are lifeforms.   (SaM)
     No lifeforms are fulgurites. (MeP)
   ∴ No flowers are fulgurites.   (SeP) 
   
    ………………………………………………          
    |                 |       …………………
    |   …………………       |      {        }  fulgurite
    |  /flowers \     |      {________} 
    |  \________/     |     
    |_________________| 
               lifeforms 
 
- - - - - 
6. Bocardo (OAO-3): MaS,MoP:SoP 

 all M is S; some M is not P: some S is not P

 S:plant; M:flower; P:fragrant

    All flowers are lifeforms.     (MaS)
    Some flowers are not fragrant.  (MoP)
  ∴ Some lifeforms are not fragrant. (SoP)

    …………………………………  lifeforms
   /  ……………………………  \     
  /  |        ………|……\………
 /   | ($)    {  |   \  ?}
 \   |flowers {__|___/___} 
  \  |___________|  / fragrant things
   \_______________/   

- - - - -

7. Baroco (AOO-2): SoM,PaM:SoP

 some S is not M; all P is M: some S is not P

 S:fragrant; M:lifeform; P:flower

    Some fragrant things are not lifeforms. (SoM)
    All flowers are lifeforms.             (PaM)
  ∴ Some fragrant things are not flowers.  (SoP)

   …………………………………………………… lifeforms
  |    ………………………………    |  
  |   {   flowers  }   |       
  |   {     …………………}………|……………….      
  |   {    /       }   |        \      
  |   {   /    ?   }   |  ($)    \      
  |   {   \        }   |         /    
  |   {    \_______}___|________/    
  |   {____________}   |  fragrant things
  |____________________|

List of the seven (7) essential ones, with the above examples in short form "syllogism (P S)":

  Barbara (Mortal Greeks) 
  Darii (Lifeform Fragrants) 
  Felapton (Non-Heavy Weight Lifeforms) 
  Ferio (Non Herbivore Mammals)
  Celarent (Fulgurite Non Flowers)
  Bocardo (Non-Fragrant Lifeforms)  
  Baroco (Non-Flower Fragrants)  

Homework For Syllogism Tutorial: Please convince yourself that only these seven (7) syllogisms are truly distinct, as in two pairs there is no actual distinction in the inference:

   Darrii & Disamis are identical inferences.
   Celarent & Camestres are identical inferences.

Furthermore, two additional syllogisms can be eliminated, being weakened forms:

Frst, Bamalip AAI is weaker than Barbara AAA, having a weaker conclusion.

    Bamalip:  If MaS & PaM, then SiP.  
    Barbara:  If SaM & MaP, then SaP.

Perform a substitution of variables in Barbara:

    Barbara:  If PaM & MaS, then PaS.

But PaS for nonempty P implies SiP, hence Bamalip is weaker than Barbara. If P is empty, then Bamalip is invalid, as is well-known.

Second, Darapti is weaker than Darii, having stronger premises:

     Darapti: If MaS & MaP, then SiP.  
     Darii: SiM, MaP: SiP.

For Darapti to be true, M must be nonempty, as is well-known. If so, the premises imply SiM & MaP, hence SiP by Darii.