Tag: soundness

What the Fuck Does “Proof” Even Mean? (And Why It Doesn’t Guarantee Truth)

This is the kind of weird shit logicians study in something called “metalogic.”

Briefly: Much confusion arises from the term “prove.” If I can prove something, doesn’t that just mean that that thing is true?

Not necessarily.

In logic (and analytic philosophy), “prove” has a different meaning from everyday usage. All it means here is: I can arrive at some conclusion if I follow the rules in a system.

Imagine that you’re playing a game called “Math for Morons.” The game is the aforementioned system, and the rules of that system include the following:

  • 1+1 = 2
  • 1 + 1 = 3

According to the rules of the “Math for Morons” system, you can then prove 2 = 3. But that doesn’t mean that it’s true that 2 = 3. The point being? Proof (according to the definition of “proof” in logic class) doesn’t guarantee truth.

And if you can prove something in the logical sense but still not arrive at the truth? That means the system is broken. Logicians call a broken system like this inconsistent.

Logicians even have a symbol for this kind of provability: ⊢ (the single turnstile).

But what if it’s the other way around? That is, something is true, but you can’t prove it.

This is what logicians call incompleteness.

Incompleteness:

  • does NOT mean “true but very hard to prove” (like the fact that my dad is an asshole—hard to prove, but not what logicians mean here)
  • means “true but you can’t prove it using the rules you have in your system/game” In other words: the truth is out there, but your system is too weak to reach it.

So imagine that in your “Math for Morons” system, there are no rules at all about even and odd numbers.

But it’s TRUE that

“2 is an even number.”

Then even though it’s true that 2 is even, you can’t prove it within the system, because the system simply doesn’t have the tools to express or derive that fact.

Logicians also have a different symbol for completeness: ⊨ (the double turnstile).

Roughly speaking, this doesn’t mean “you can prove it.”

It means:

“This has to be true, no matter how you interpret things.”

So metalogic asks a deeper question:

Do the things you can prove (⊢) line up with the things that are actually guaranteed to be true (⊨)?

Or are you just pushing symbols around and hoping for the best?

By the way, in standard (“classical”) logic, there’s a nice result:

If you can prove something using the rules (⊢), then it really is true in the relevant sense (⊨).

Logicians call this soundness.

But that’s another can of worms for another day.