The Limits of Logic #3

in #punchlinelast year

Suppose we take a simple logical argument that A implies B. This means that not A implies not B. A simple case could be that if it is raining then that implies that it is cloudy. Therefore, if it is not raining then that implies that it is not cloudy. No confusion there, seems to make perfect sense. Let's now say that if you have a crow, that implies that it is black. All ravens are black.


All nonblack birds are not ravens

What if however we see some other creature, say a bluejay. A bluejay is neither a crow nor is it black. We could then say that this satisfies our statement that all nonblack birds are not ravens. We are seemingly reinforcing our concept of ravens being black by saying that a bluejay is not black. Obviously, bluejays are blue.

However, what if then we see a red cardinal? This is also reinforcing a statement that all nonred birds are not ravens. The red (nonblue) bird is a cardinal (not raven).

This leads to the statement:

All ravens are blue

Obviously there is a problem with this! This shows that there is a fault in our system of logic. Can you see what it is? This is known as the ravens paradox.



It's late and I may be having a big brain fart, but I don't see the "A implies B" meaning "not A implies not B". Even the rain example: it can be cloudy and not raining, right? I would see something like "if something is a X, then it is Y" equivalent to "if something is not Y, then it is not a X". Now you can replace X with raven and Y with black and that's your raven paradox. Am I missing something?
Anyways, thanks for the brain teaser.

From Wikipedia:

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely.It is very closely related to the rule of inference modus tollens. It is the rule that:

P -> Q <=> (~P -> ~Q)

This means that you can replace one statement for the other. What the ravens paradox shows us is there is a flaw with induction, which is what we use everyday to make sense of the world.

Assuming that every time that it rains we see that it must be cloudy, we can see that:

Raining -> cloudy
Not raining -> not cloudy

How could it rain unless it is cloudy?

When you say it can be cloudy but not raining, notice the order is different. It could perhaps be snowing when it is cloudy or no precipitation happens. But that doesn't take away from our first statement that:

Raining -> cloudy

We obviously must be careful with this, because using this logic we showed that all ravens are blue, which is obviously false.

Yes that is what I am saying, the order is different: "A implies B" equivalent to "not B implies not A" but not "not A implies not B". The order is critical. And that would be P -> Q <=> (~Q -> ~P) as can be seen on Wikipedia too.
Going back to the rain example, not raining does not mean not cloudy, but not cloudy means not raining. Just like you're saying: snowing (not raining) means cloudy which means not raining -> not cloudy is not correct. So I would think it is:
raining -> cloudy
not cloudy -> not raining

I like these logic questions!

Yes, you are correct. My order was:

P -> Q

Which should lead us to

~Q -> ~P

Not the way I wrote it as:

~P -> ~Q

For the raining/cloudy example, try thinking about another one:

All triangles are 3-sided
Not a triangle -> not 3-sided

Again, I think the order is critical and the correct statement would be:
All triangles are 3-sided
not 3-sided -> not a triangle

For instance, a shape with 3 curved sides is not a triangle, but it is 3-sided