Friday, March 15, 2019

Diagramming Syllogisms

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
--------------
S → M → P (by concatenation)
S → P (by deletion)

And so on with all the others.

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