A Second Theory of Term Logic

I added Term Logic to somerby.net/mack/logic for fun. While doing the necessary research, I discovered the logical Square of Opposition, which is kind of cool. Terence Parsons wrote an illuminating article on the Square. In it, he argues convincingly for an interpretation of 2-term propositions that affirms the Square of Opposition, and also convincingly that this interpretation is Aristotle's intended interpretation. I like the article so much that I've chosen to use this interpretation in my application, defining the four forms of propositions (SaP, SeP, SiP, SoP) just as he does. Even so, I doubt that this is the only coherent theory of Term Logic held by premodern logicians. Here I shall explain why. You can click on any of the symbolic statements in this post to test them in somerby.net/mack/logic.

When explaining why the interpretation of the O-form as "Some S is not P" did not cause problems for premodern logicians, Parsons dismisses the possibility that they assumed that the S-term was not empty, stating "Explicitly rejecting empty terms was never a mainstream option, even in the nineteenth century". But I'm not so sure. First of all, just because they did not explicitly reject empty terms does not mean they implicitly rejected empty terms. Second, they did not have to reject empty terms altogether to make this interpretation of O-form compatible with the traditional Square of Opposition. They only needed to assume a priori (and perhaps unconsciously) that the S-term was not empty whenever they were making an argument. This isn't a very rigorous thing to do, but it's a natural thing to do. Usually, if we are making assertions about some kind of thing, it's because some such thing exists and we want to say something meaningful about it. Reasoning about unicorns may have its uses, but they are not obvious.

Suppose that some pre-modern philosophers, like Boethius and Peter of Spain, did not interpret the propositional forms as Aristotle intended. Instead, they assumed a priori that the S-term was nonempty, and let the O-form have existential import, just as Boethius seemed to be doing when he translated "Some S is P". Then, instead of the Square of Opposition being this:

// Aristotle's Square of Opposition

A <=> ((x,Sx->Px) & (3x,Sx))
E <=> (x,Sx->~Px)
I <=> 3x,Sx&Px
O <=> ((3x,Sx&~Px)|(~3x,Sx))

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I| O

// Subalterns
A -> I
E -> O

they believed the Square of Opposition was this:

// A Hypothetical Alternative to
// Aristotle's Square of Opposition

3x,Sx // Assume a priori that S is not empty.

A <=> (x,Sx->Px)  // (Existential import here would be redundant.)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px) // Assume O has existential import.

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

The relationships of the Square hold in this interpretation as well as in Aristotle's.

And then there is the matter of the Principle of Obversion and the Principle of Contraposition. Parsons says that some medieval logicians advocated these principles, though they are both fallacious under Aristotle's interpretation of the four forms. The following is not necessarily true:

// The Principle of Conversion by Contraposition,
// with Aristotle's interpretation
// of the A-form and the O-form
((x,Sx->Px) & (3x,Sx)) <=> ((x,~Px->~Sx) & (3x,~Px))
((3x,Sx&~Px)|(~3x,Sx)) <=> ((3x,~Px&~~Sx)|(~3x,~Px))

This is not necessarily true, either:

// The Principle of Obversion,
// with Aristotle's interpretation
// of the A-form and the O-form

// Every S is P = No S is non-P (SaP <=> Se~P)
((x,Sx->Px) & (3x,Sx)) <=> (x,Sx->~~Px)

// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> ((x,Sx->~Px) & (3x,Sx))

// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> ((3x,Sx&~~Px)|(~3x,Sx))

//Some S is not P = Some S is non-P (SoP <=> Si~P)
((3x,Sx&~Px)|(~3x,Sx)) <=> (3x,Sx&~Px)

Why did some logicians make these mistakes? And why did other logicians like Peter of Spain endorse them? Maybe to them, they weren't mistakes. Under what we call the modern interpretations of the four forms, these principles are necessarily true.

// The Principle of Conversion by Contraposition,
// with the modern interpretations of the forms:
(x,Sx->Px) <=> (x,~Px->~Sx)
(3x,Sx&~Px) <=> (3x,~Px&~~Sx)

// The Principle of Obversion,
// with the modern interpretations of the forms:

// Every S is P = No S is non-P (SaP <=> Se~P)
(x,Sx->Px) <=> (x,Sx->~~Px)

// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> (x,Sx->~Px)

// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> (3x,Sx&~~Px)

//Some S is not P = Some S is non-P (SoP <=> Si~P)
(3x,Sx&~Px) <=> (3x,Sx&~Px)

Being necessarily true, they will still, of course, be true under an a priori assumption that the S-term is nonempty. So maybe there was a theory of term logic floating around Medieval Europe that looked like this:

3x,Sx

A <=> (x,Sx->Px)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px)

->

// Contraries
~(A&E)

// Contradictories
A ^ O
I ^ E

// Subcontraries
I | O

// Subalterns
A -> I
E -> O

If so, then they really did have a coherent theory of Term Logic which affirmed the Principle of Conversion by Contraposition and the Principle of Obversion. I can't be sure, since I haven't looked for evidence to the contrary, e.g. Peter of Spain discussing empty terms in Summulae Logicales Magistri Petri Hispani, but as far as I know, it makes sense. I guess I'll have to read some Medieval logic to find out. It's too bad I don't know Latin.