Logic - other worlds and propositions?

Question

I am reading a book on logic called "Possible Worlds"; Question: I assume that 2+2=4 is necessarily true, what if someone said that they can conceive of a world where 2+2=5; would this be enough to consider 2+2=4 a contingent truth? Same with square circles; how do you defend the position that circles can't be square when someone says that there could be a world (Limbo or Heaven...) where circles are square. How do you disprove such a statement? What or how do we demarcate between "All possible worlds" and the impossible?

Answer 1

All right. Let's take this slowly.

First of all, other than an 'a priori' truth, pretty much everything is contingent. And 'a priori' truths are only so because such a truth is true by definition... if you change the definition then you just end up with a different 'a priori' truth.

So when you're talking about math, normally we assume a certain number of things. You can even look these up, if you like (one example is link 1)... these are some of the things the mathematicians just assume or define to be true and cannot prove. If you choose not to assume those things, you may be talking about a systematic field of some type, but you're not really talking about math any more.

In math, '2' and '4' refer to certain quantities of stuff, while '+' and '=' refer to certain operations performed on that stuff. All 'a priori'... the conclusion necessarily follows with those definitions, if you change definitions then you're in a different system.

A good example of this is the base system. When we use numbers, we are normally using base ten. It's assumed. If you want, you can use base three; then 2+2=11. In this sense, we haven't changed the underlying truth... just the way we refer to it. 11 in base three is 4 in base ten, so we've said exactly the same thing.

If you want to disprove any of these statements, you are going to have to drag all these definitions and assumptions out into the light. And because many of them are implied instead of overtly stated, this can sometimes be quite a bit of work. But the work can often pay off!

For example, I can think of situations where 2+2=1. If you put four 'fighting fish' into a bowl, they will fight until only one is left. And here you can also see what definition I have changed: I'm not talking about a hypothetical two non-objects, I'm talking about very specific objects. And likewise by '=' I'm not talking about an instantaneous process, but one that occurs over a period of time. So 2+2=1 is very true... given my definitions. And given those definitions it is also completely untrue that 2+2=4. It just doesn't happen.

So some things ARE impossible. But to distinguish between the possible and the impossible, you have to look at the entire picture, not just some corner of it. Consider everything you are regularly assuming to be also true and see if they still hold. It takes practice, but you can probably get the hang of it.

Hope that helps!