Matrices of valid syllogisms

The form of categorical syllogisms can be represented as a reduction on the space of the syllogistic relations (A*, A, E, I, O, O*), as indicated below for the syllogisms of figure 1.

  1. A < B, B < C => A < C is represented as: A A => A
  2. A < B, B <> C => A <> C is represented as: A I => I
  3. A | B, B < C => A | C is represented as: E A => E
  4. A | B, B <> C => A |> C is represented as: E I => O
  5. A <> B, B | C => A <| C is represented as: I E => O*
  6. A <. B, B .| C => A <| C is represented as: A. .E => O* (middle term non-empty)

The reduction is non-commutative.

A conjugation operation on reductions can be defined as follows:

  1. (X Y)* = Y* X*
  2. X** = X
  3. (X => Y)* = (X* => Y*)

Application of the *-operation transforms valid syllogisms into valid ones. For example (E I => O)* yields I E => O*.

Statement of matrices

Arranging reductions X Y => Z into matrices with X on the left (labeling the row), Y on the top (labeling the column), and the result Z in the cell of the X-row and the Y-column yields the matrices shown below.

Matrix for figure 1
A E I O
A A O* I
E E O
I O*
O
Matrix for figure 2
A E I O
A* E O
E E O
I O*
O* O*
Matrix for figure 3
A* E I O*
A I O* I O*
E O O
I I O*
O O
Matrix for figure 4
A* E I O*
A* A* E
E O O
I I O*
O*

We display a separate matrix for each syllogistic figure.

Reductions that require the middle term to be non-empty are included in the above matrices, indicated by the resulting relation symbol in italics. In cases where a non-emptiness assumption cannot be made, such cells would be empty.

Definition of matrix operations

The conjugate matrix is obtained by applying the *-operation individually to each matrix element (including the table headers).

The transpose matrix is obtained by exchanging rows and columns (including the table headers).

The adjoint matrix is obtained by taking the conjugate transpose.

A matrix is self-adjoint (Hermitian) when it equals its adjoint.

Properties of syllogism matrices

  • The matrices for figure 1 and figure 4 are adjoint to each other.
  • The matrix for figure 2 is self-adjoint. Same for figure 3.