Maydole's Ontological Argument

[9-10ths_Penguin]

Oops, it was supposed to be "If possibly P, then necessarily possibly P"

I missed the second "possibly"

S5 is:

If possibly P, then necessarily possibly P
If possibly necessarily P, then necessarily P

Ah... that helps this conversation make more sense.

I still don't agree with it, though. Unless modal logicists (is that a real word?) are using the terms "possibly" and "necessarily" in ways that don't correlate with their normal usage at all, those statements are demonstrably wrong.

Logical statements like "possibly" and "necessarily" are commutative: their order doesn't matter.

"Necessarily" implies that a thing is certain... i.e. that it has a probability of 1.
"Possibly" implies that a thing is uncertain... i.e. that it has an unknown probability (let's call them 'X' and 'Y').

This means that you could re-write S5:

If necessarily possibly P, then possibly P:

X x 1 = Y (okay so far)

and

If Possibly necessarily P, then necessarily P:

1 x X = 1 (this doesn't work)

 

[Magic Man]

I was born in Ireland. Careful with your implications. My island is the best.

It's MY island.

You're a madman.

 

[Meow Mix]

Ah... that helps this conversation make more sense.

I still don't agree with it, though. Unless modal logicists (is that a real word?) are using the terms "possibly" and "necessarily" in ways that don't correlate with their normal usage at all, those statements are demonstrably wrong.

Logical statements like "possibly" and "necessarily" are commutative: their order doesn't matter.

"Necessarily" implies that a thing is certain... i.e. that it has a probability of 1.
"Possibly" implies that a thing is uncertain... i.e. that it has an unknown probability (let's call them 'X' and 'Y').

This means that you could re-write S5:

If necessarily possibly P, then possibly P:

X x 1 = Y (okay so far)

and

If Possibly necessarily P, then necessarily P:

1 x X = 1 (this doesn't work)

Possibly and necessarily in modal logic are best thought of in terms of possible worlds semantics.

Possibly = contingent (may exist in at least one world, but may not)

Necessarily = exists in all possible worlds

The reason axiom S5 can get to "If possibly necessarily P, then necessarily P" is because in S5 modal logic you can demonstrably remove all but the last operator in a chain of operators. If something is necessarily possibly possibly necessarily necessary, then it's necessary in S5.

I don't have enough understanding of S5 modal logic to prove this, but I know it is how it works. Next semester I'll be taking a lot more pure math/logic than physics so I'm looking forward to having a better understanding of why this is the case there.

I suspect though that axiom S5, while useful for truncating ridiculous chains of modality, has its limitations in proving whether or not something exists.

 

[9-10ths_Penguin]

The reason axiom S5 can get to "If possibly necessarily P, then necessarily P" is because in S5 modal logic you can demonstrably remove all but the last operator in a chain of operators. If something is necessarily possibly possibly necessarily necessary, then it's necessary in S5.

I know that's the method, but I just don't see how it's valid.

In my understanding, if you've got a chain of operators like that, then if you have at least one "possibly", then the net result would be "possibly", regardless of order.

I don't have enough understanding of S5 modal logic to prove this, but I know it is how it works. Next semester I'll be taking a lot more pure math/logic than physics so I'm looking forward to having a better understanding of why this is the case there.

I need to learn more about modal logic myself. Some of the stuff seems so ridiculous at first glance that I think there might be some issue of terminology going on: they're just using words in completely different senses than I'm reading them.

I suspect though that axiom S5, while useful for truncating ridiculous chains of modality, has its limitations in proving whether or not something exists.

I don't dispute that it's useful to distill a large number of operators down to one, but in this case, I can't help remembering that old line: "every complex problem has a simple, easy-to-understand wrong answer."