Ruggero Pagnan, in two articles ["A Diagrammatic Calculus of Syllogisms", Journal of Logic, Language, and Information, Vol. 21, No. 3 (Summer 2012), pp. 347-364; "Syllogisms in Rudimentary Linear Logic, Diagrammatically", Journal of Logic, Language, and Information, Vol. 22, No. 1 (Winter 2013), pp. 71-113], develops a very nice diagrammatic system for syllogisms. It's particularly handy in that, unlike most of the other good diagrammatic systems, it can easily be typed.
First, for the basic categorical propositions:
All X is Y
X → Y
No X is Y
X → • ← Y
Some X is Y
X ← • → Y
Some X is not Y
X ← • → • ← Y
All of these are commutative; you can do them backwards (this helps for putting them together). So, for instance, you can always change X → Y to Y ← X.
We add two principles that let you add new premises in any argument:
Identity
X → X
Subalternation
X ← • → X
We need to be able to link diagrams by terms. For instance, starting with the premises,
X → Y
Y → Z
You can get
X → Y → Z.
In essence, we just overlap the Y's. And last, we need a rule that lets us delete mediating terms, so that by deleting Y and collapsing the arrows we can change this to
X → Z.
The major restriction is that we cannot delete bullets.
Given this, we can establish the Barbara syllogism:
1: M → P
2: S → M
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S → M → P (by concatenation)
S → P (by deletion)
And so on with all the others.
