Negations of Categorical Propositions

Before we look at how to form the negation of our four types of categorical propositions, let's review the translation of them into QL:

All S are P

(x) (Sx ⊃ Px)

No S are P

(x) (Sx ⊃ ~Px)

Some S are P

(∃x) (Sx & Px)

Some S are not P

(∃x) (Sx & ~Px)

We know how to negate a proposition: We simply throw a tilde in front of it. So we can form the negations as follows:

~(x) (Sx ⊃ Px)	negation of A categorical proposition
~(x) (Sx ⊃ ~Px)	negation of E categorical proposition
~(∃x) (Sx & Px)	negation of I categorical proposition
~(∃x) (Sx & ~Px)	negation of O categorical proposition

Indeed, these are the appropriate negations. However, we can also form the negations by using the information from the Traditional Square of Opposition. Remember that we find "opposites", i.e. negations at opposite corners of the square. So we have the following negations:

The negation of	(x) (Sx ⊃ Px)	is	(∃x) (Sx & ~Px)
The negation of	(x) (Sx ⊃ ~Px)	is	(∃x) (Sx & Px)
The negation of	(∃x) (Sx & Px)	is	(x) (Sx ⊃ ~Px)
The negation of	(∃x) (Sx & ~Px)	is	(x) (Sx ⊃ Px)

The following exercise asks you to use the information you're just acquired to draw some conclusions about logical equivalence involving categorical propositions.