Definitions

Syllogistic relations and connectives

Syllogistic relations and connectives

Relation	Connective	English Sentence
A	A < B	All B are A
E	A | B	No B are A
I	A <> B	Some B are A
O	A |<> B	Some B are not A

The syllogistic connectives are chosen so that:

• Connective < within formula A < B resembles the inclusion arrow, meaning that B is included in A. In case of sets A and B it indicates that B is a subset of A.
• A vertical bar | indicates a negative relation.
• A term facing an open side of a syllogistic connective is distributed, whereas a term facing a pointed side is undistributed. That means, the right side of <, both sides of |, and the left side of |<> are distributed.
• Have the predicate on the left and the subject on the right. This reflects the choice made by Aristotle, modern use that applies the predicate to the subject as in A(B), and has a technical advantage in the use of the chain rule.

Converse syllogistic relations and connectives

Converse relations and connectives

Code	Connective	English Sentence
A*	B > A	All B are A
E*	B | A	No B are A
I*	B <> A	Some B are A
O*	B <>| A	Some B are not A

The choice of the same connective for relations E and E* is possible because E is symmetric. Likewise for I and I*. That is, they obey the following identities:

1. E* = E: B | A = A | B
2. I* = I: B <> A = A <> B

Interpretation of universal syllogistic relations

There are two interpretations of the universal sentence "All B are A": The one by Aristotle implies B to be non-empty, whereas the modern one allows B to be empty. Likewise for the universal sentence "No B are A": The Aristotle interpretation implies either of A and B to be non-empty, the modern one allows either to be empty. We resolve this ambiguity as shown in the table below.

Universal relations and connectives

Relation	Connective	English Sentence
A	A < B	All B are A (B may be empty)
A.	A <. B	All B are A, B is non-empty
A*	B > A	All B are A (B may be empty)
.A*	B .> A	All B are A, B is non-empty
E	A | B	No B are A (A, B may be empty)
E.	A |. B	No B are A, B is non-empty
.E	A .| B	No B are A, A is non-empty
.E.	A .|. B	No B are A, A and B are non-empty

These variants allow a clear formulation of the conversions from universal to particular sentences, which are valid for the "dotted" but not valid for the "undotted" A, A* and E relations.

References

[1]: Paul Halmos and Steven Givant. Logic as Algebra. The Dolciani Mathematical Expositions Number 21, the Mathematical Association of America, 1998.