Existential vs Universal Quantifiers

MICH1234

I should clarify I am confused about the negations! How does it result that from a statement such as "All cats are black" to Neg Some cats /black OR (this is the one I'm confused about)... cats and /black. I don't understand the "and"?

keets993

Hey there!

So universal quantifiers generally consist of conditional relatioships. That means if I tell you "all apples are red" then our relationship looks like A -> R. If i tell you 'jessica is an apple' it must be 100% true that jessica is red. Because that's what our conditional relationship tells us of. You can think of a negation either as: (1) something that negates the relationship, or; (2) a counter-example. These are both the same thing, but one method may be easier to understand than the other.

So what does it mean to say that, not [all apples are red]? Well, it's not the case that all apples are red. It means if I tell you "jessica is an apple" you cannot necessarily conclude that "jessica is red." A thing can be an apple and be not red. The reason it's "some" or "and" is because, if we put A -> /R then that implies that all apples are not red. Which is not the proper negation. When we negate a conditional relationship, we are basically putting a slash through the arrow. The conditional arrow is just transformed into either the "some" or the "and"

If someone tells us "all apples are red" we can say, hey that's wrong! I know of at least one apple that is not red, which would be our counter-example.

There's two ways to properly negate and they both imply the same thing.

Apples < some> /red

Some apples are not red. At least one apple is not red.

Apple and /red

The "and" here is a descriptor. You can have an apple AND have it be not red.

We are negating the necessary indicator and not the sufficient one because our original relationship is about the color red (necessary) in relation to the sufficient (apples). So if we said a thing can be red and not be an apple that would be incorrect because we're not negating the right relationship. A -> R could already mean that some red things are not apples.

Hope that helps, let me know if you have any other questions.

MICH1234

Wow thank you for the very thorough reply!

mynameisjeff

@keets993 you the real MVP

macknzx3

very helpful!!