with Categorical Propositions
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Negations & Complements with Categorical Propositions
Four Basic Categorical Propositions. . .
A – Affirmative, Universal
I – Affirmative, Particular
E – Negative, Universal
O –Negative, Particular
Basic Examples. . .
A – “All men are mortal.”
I – “Some men are mortal.”
E – “No men are mortal.”
O – “Some men are not mortal.”
There are two negative propositions, namely, E and O propositions.
Essentially the copula (i.e., the “to be” verb) is being negated.
The predicate is not affirmed of the subject.
For example, “No atheist is a believer in God” is an E proposition (as it is a negative, universal proposition). It denies the predicate “a believer in God” as an attribute of the subject “atheist.”
Or consider: “Some logicians are not mathematicians.” It is an O proposition (as it is a negative, particular proposition). It denies the predicate “mathematician” as an attribute of “some logicians.”
Some O propositions are thorny, by the way.
Consider this proposition: “Not all planets of this solar system have been discovered.”
That’s the same thing as: “Some planets of this solar system have not been discovered.”
(Okay, for those of you who are fussy, neither are in the “standard form” of a categorical proposition. Allow me to re-write: “Some planets of this solar system are not those things that have been discovered.”)
“Not all S is P” means the same thing as “Some S is not P.” The “not” contradicts the “all.” This is what makes the proposition a particular proposition, not a universal proposition. On the Square of Opposition, which is covered on another page of this website, when we contradict an A proposition (“All S is P”) we then will get an O proposition (“Some S is not P”). This explains why “Not all S is P” means “Some S is not P.”
“All S is not P” likewise means “Some S is not P.” Other possible equivalent expressions include “Not every S is P” and “Every S is not P.”
Terms might also be negated.
When a term is negated, it’s said to be “complemented.”
For example, “All men are non-angels.” The prefix “non-” affects the predicate in this example, in that there is a denial of “angels.” This example is an A proposition (as it is an affirmative, universal proposition). Negating a term doesn’t negate the proposition.
We can symbolize a complemented term with an apostrophe.
For example, angels can be symbolized as x.
Let’s negate or complement that term.
Non-angels can be symbolized as x’. (Pronounce that as “x prime.”)
More Examples. . .
Please notice the apostrophe for complemented terms. We can have complemented subjects, complemented predicates, or both complemented subjects and complemented predicates.
Incidentally, look closely at those examples. Some of them are equivalent to each other! I’ll cover this on another page. “No men are angels” really means the same thing as “All men are non-angels.” This is an example of what’s called obversion. It’s an immediate inference or so-called eduction.