Tradition Square of Opposition

Parsons, Terence, "The Traditional Square of Opposition", The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.), URL = .

Contrary- All S are P, No S is P All s is P is contrary to the claim NO S is P. 

----------------------- 

A contrary can be true as well as false. Contraries can both be false. Contraries can't both be true. 

The A and E forms entail each other's negations 

Subcontrary Some S are P, Some S are not P

 -------------------------------------------- 

Sub contraries can't both be false. Sub contraries can both be true. The negation of the I form entails the (unnegated) E form, and vice versa. 

Contradiction- All S are P, Some S are not P, Some S are P, No S are P

 ----------------------------------- 

For contradictions -Two propositions are contradictory if they cannot both be true and they cannot both be false. Contradictory means there is exactly one truth value and if one proposition is true the other MUST be false. If one is false the other MUST be true. 

The propositions can't both be true and the propositions can't both be false. 

The A and O forms entail each other's negations, as do the E and I forms. 

The negation of the A form entails the (unnegated) O form, and vice versa; likewise for the E and I forms.
 Super alteration[- Every S is P, implies Some S are P No S is P, implies Some S are not P 

-------------------------------------------- 

The two propositions can be true.

 Sub alteration- All S are P, Some S are P No S are P, Some S are not P 

---------------------------------- 

A proposition is a subaltern of another if it must be true The A form entails the I form, and the E form entails the O form. 

"The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'some S are P'" . 

As in the first(Proposition 1) or the "I" "To be clear the I proposition is SOME S is P. This is what is meant by a I proposition. Well you can certainly infer if an I proposition is true that an E proposition is false because they are contradictory. Unfortunately there is NOTHING else to infer with certainty. That is there will be times where the proposition will be true and different times it will be false. This is called contingent truths. That is the proposition is not true 100% of the time. It has false cases. Deductive logic tries to stay away from contingent truths." 

"The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'some S are P'" 

Summary

Universal statements are contraries: 'every man is just' and 'no man is just' cannot be true together, although one may be true and the other false, and also both may be false (if at least one man is just, and at least one man is not just).

Particular statements are subcontraries. 'Some man is just' and 'some man is not just' cannot be false together

The particular statement of one quality is the subaltern of the universal statement of that same quality, which is the superaltern of the particular statement, because in Aristotelian semantics 'every A is B' implies 'some A is B' and 'no A is B' implies 'some A is not B'. Note that modern formal interpretations of English sentences interpret 'every A is B' as 'for any x, x is A implies x is B', which does not imply 'some x is A'. This is a matter of semantic interpretation, however, and does not mean, as is sometimes claimed, that Aristotelian logic is 'wrong'.

The universal affirmative and the particular negative are contradictories. If some A is not B, not every A is B. Conversely, though this is not the case in modern semantics, it was thought that if every A is not B, some A is not B. This interpretation has caused difficulties (see below). While Aristotle's Greek does not represent the particular negative as 'some A is not B', but as 'not every A is B', someone in his commentary on the Peri hermaneias, renders the particular negative as 'quoddam A non est B', literally 'a certain A is not a B', and in all medieval writing on logic it is customary to represent the particular proposition in this way.

These relationships became the basis of a diagram originating with Boethius and used by medieval logicians to classify the logical relationships. The propositions are placed in the four corners of a square, and the relations represented as lines drawn between them, whence the name 'The Square of Opposition'.