Introducing the Square of Opposition - Ch. 6.1 in "The Reasonable Person" by Mark Grannis

Equivalent Versus Opposite Propositions 

Equivalent propositions: "propositions that differ in form but not in matter, and which must always carry the same truth value." 

Opposite propositions: "any two propositions that (respectively) affirm and deny the same predicate of the same subject." 

The Square of Opposition
These are formal conclusions, not material ones, that's the power of it. The rules of the formal logic transfer themselves to the material logic world. This structure of propositions is rooted in the Principle of Non Contradiction. "A moment's reflection will convince you that the opposition between these two propositions has nothing to do with the matter and everything to do with the form. If, instead of 'All lawyers are crooks,' we had started with 'All bandersnatches are frumious," then even without knowing what a bandersnatch is or what 'frumious' means, we would still know that the opposite of 'All bandersnatches is frumious' is 'Some bandersnatches are not frumious.' A and O are always contradictories." 

Contradictory Propositions 

A <---> O 

E <---> I 

All S is P <---------> Some S is not P

No S is P <---------> Some S is P 

All (dogs) are (brown) <---------> Some (dogs) are not (brown) 

No (dogs) are (brown) <---------> Some (dogs) are (brown) 

SAME subject and predicate

DIFFERENT quantity 

DIFFERENT quality 

Cannot both be true at the same time. Cannot both be false at the same time. 

It cannot be the case that all dogs are brown and that some wouldn't be brown then. Nor can it can be the case that all dogs aren't brown and yet some are. Therefore, if one is true the other is false, and if one is false the other is true. 

Each propositions is the inverse of the other in terms of BOTH QUANTITY and QUALITY 

All affirmative <--------->  Some negative 

All negative <--------->  Some affirmative

Contrary Propositions 

A <---> E

All S is P <---------> No S is P

All (dogs) are (brown) <--------->  No (dogs) are (brown)

SAME subject and predicate

BOTH are universal in quantity (ALL and ALL)

DIFFER in qualities (ALL and NO) 

Cannot both be true at the same time. CAN both be false at the same time. 

We cannot say that all dogs and no dogs are something ... but we can say that dogs are neither all brown nor all not brown (i.e. there are some brown dogs out there) 

The propositions are the inverse of one another in terms of QUALITY. 

All affirmative <--------->  All negative 

Subcontrary Propositions 

I <---> O

Some S is P <--------->  Some S is not P

Some (dogs) are (brown) <--------->  Some (dogs) are not (brown) 

SAME subject and predicate 

BOTH are particular in quantity (Some and Some) 

DIFFER in qualities (is and is not) 

CAN both be true at the same time. CANNOT both be false at the same time. 

We can say that some dogs are brown and some are not brown. But we cannot say hold that both are false because one would be saying that no dogs are brown (there's not even some) and yet some dogs are brown. 

The propositions are the inverse of one another in terms of QUALITY

Some affirmative <--------->  Some negative