This page illustrates syllogisms in three different ways:

  • With Venn diagrams, that show in which intersections of the three sets objects do not (black), can (white) or do (red) exist.
  • With Euler diagrams, which are like Venn diagrams with empty regions removed. (Only small diagrams on top of the table.)
  • The gist of this page is the reduction of this first-order logic topic to zeroth-order logic using binary square matrices that are essentially 8-ary logical connectives. There are 8 intersections of the three sets, and each intersection can either contain elements or not. So there are 28 = 256 situations that can be the case. Each statement (premise or conclusion) can be denoted by the set of situations in which it is true.

All Syllogisms (table of contents)

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1  
Barbara
 
Barbari
 
Darii
 
Ferio
 
Celaront
 
Celarent
2  
Festino
 
Cesaro
 
Cesare
 
Camestres
 
Camestros
 
Baroco
3  
Darapti
 
Datisi
 
Disamis
 
Felapton
 
Ferison
 
Bocardo
4  
Bamalip
 
Dimatis
 
Fesapo
 
Fresison
 
Calemes
 
Calemos

Graphical elements

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Venn diagram and corresponding cubic Hasse diagram, used in the diagram on the right
 
The 256 situations that can be the case (minterms)
Light vertices indicate that an area is empty, dark vertices indicate that there is at least one element.

Examples

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Barbara (AAA-1)

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Venn diagrams
 
Minterms


Celarent (EAE-1)

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Venn diagrams
 
Minterms

Similar: Cesare (EAE-2)

Darii (AII-1)

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Venn diagrams
 
Minterms

Similar: Datisi (AII-3)

Ferio (EIO-1)

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Venn diagrams
 
Minterms

Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)

Baroco (AOO-2)

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Venn diagrams
 
Minterms

Bocardo (OAO-3)

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Venn diagrams
 
Minterms

Barbari (AAI-1)

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Venn diagrams

Celaront (EAO-1)

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Venn diagrams

Similar: Cesaro (EAO-2)

Camestros (AEO-2)

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Venn diagrams

Similar: Calemos (AEO-4)

Felapton (EAO-3)

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Venn diagrams
 
Minterms

Similar: Fesapo (EAO-4)

Darapti (AAI-3)

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Venn diagrams
 
Minterms