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Calling Maths experts: how can ◊□X → □X be intuitive?  "Please can anyone help answer this? Even Google is stumped!"

User is offline   Martin Howe 

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#1

Alvin Plantinga argues ◊□X → □X is not counter-intuitive and gives a possible-worlds justification for it. But all that does is *prove* it; that doesn't show how it is *intuitive* in the way that X → □◊X is. Can anyone more knowledgeable than me give me a *practical* example, with a contradiction that results if it were false?

For the curious:
Spoiler


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User is offline   Trooper Dan 

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#2

It's been about 15 years since I have thought about this stuff, but here's my quick take on an intuitive argument.

Consider the typical sorts of propositions that are necessarily true, such as mathematical theorems, simple facts of arithmetic, or truths of logic. Propositions belonging to those domains are necessary (e.g.. each mathematical proposition is either necessarily true or necessarily false -- there is nothing in between, because there are no contingencies that its truth or falsehood depends on).


So, if we are considering some mathematical proposition P, then we know that P is either necessarily true or necessarily false. But if it's possible that P is true, then it can't be necessarily false, because to assert that it is possibly true is to assert that it is not necessarily false. Therefore, if mathematical proposition P is possibly true, it is necessarily true.


The reason why this reasoning may seem like sleight of hand is that when we say "it is possible" there are two different things we can mean. We can mean it in the literal sense, as in "there are possible worlds in which it is true" or we can mean something epistemic such as "for all I know it is true."


If we are using possible in the epistemic sense, then when we assert that the mathematical proposition P is "possibly" true, then the inference that it is necessarily true fails. That is because we would be sliding from one meaning of "possible" to the other. The inference only works if we mean literally possible. But taken that way, then it is actually hard to know (in many cases) whether P is possibly true. Knowing whether P is possibly true, if P is a necessary proposition, amounts to knowing whether it is in fact true (and necessarily true).
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User is offline   Forge 

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#3

either the cat is dead, alive, or a zombie
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User is offline   Trooper Dan 

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#4

I'm sure this is coming up in relation to Plantinga's modal ontological argument for the existence of god. Our tendency to use the term "possible" in two different ways (as described in my earlier post) plays into the plausibility of this argument, I believe.

Put god aside for a moment, and let's say you ask me "Is 7903 prime?" Now let's say I don't have internet access and I'm not very good at math. You are looking for a quick answer. So, I say "Possibly." Now, what I really mean by this is "I don't know whether 7903 is prime or not." But we talk this way all the time, we say things are "possible" when what we really mean is we just don't know one way or the other -- that's the epistemic sense of "possible". In actuality, 7903 is not prime. And since every number is either prime or not prime by necessity, then it's literally impossible that 7903 be prime (i.e. there are no possible worlds where it is prime). Now notice if we apply Plantinga's modal principle, then if it's possible that P is necessarily true, it follows that P is necessarily true. So by that principle, I was wrong when I said that it was possible that 7903 is prime. In fact, I did not know whether it was possible or not -- it's the sort of thing where it's not that easy to know whether it is possible.


Enter god. Plantinga says that god by its nature would be a necessary being. That is, if god exists, then god exists in every possible world. In other words if P is the proposition that god exists, then P is a necessary proposition (it is either necessarily true or necessarily false). Now, if we are asked "Is it possible that God exists?" many of us would quickly reply "sure, it's possible". If we take that to be a claim about real possibility and not simply our state of knowledge, then by the modal principle we are committed to the existence of god, since if it's possible that a necessary being exists, then the necessary being exists. However, what I'm saying is that our answer "sure, it's possible" is misleading. In fact it is not that easy to know whether the existence of god is possible in the real sense, we only have immediate access to our epistemic state.

So, what I think is happening here is, the proposition that god (defined as a necessary being) exists seems "possible" in the sense that we just don't know whether it's true (the epistemic sense of possible). For Plantinga (or anyone) to assert that the proposition that god (defined as a necessary being) is possibly true in the literal sense of possible would be begging the question -- effectly assuming that thing that stands in need of proof.
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User is offline   Martin Howe 

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#5

Thanks very much for your insights, @Trooper Dan; your answer about the sort of things that can be necessary is what I was very roughly thinking but couldn't think up a clear enough vision of it to analyse for myself. I have no training in modal logic and my formal methods is very rusty; just an interested amateur who hates loose ends :)

BTW, this wasn't triggered by anything to do with god(s) or Plantinga's beliefs in him/her/it/them; it was purely a question of how well maths models reality and from that the idea of how anyone (whether Plantinga or someone else) could think ◊□X → □X was intuitive (as opposed to merely provable). And yes, the two kinds of "Possible" make it harder to understand.

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This post has been edited by Martin Howe: 14 May 2019 - 01:53 PM

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User is offline   Trooper Dan 

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View PostMartin Howe, on 14 May 2019 - 01:37 PM, said:

BTW, this wasn't triggered by anything to do with god(s) or Plantinga's beliefs in him/her/it/them; it was purely a question of how well maths models reality and from that the idea of how anyone (whether Plantinga or someone else) could think ◊□X → □X was intuitive (as opposed to merely provable). And yes, the two kinds of "Possible" make it harder to understand.


I would say that it's a principle that can easily be proven with an intuitive argument, but the principle by itself may not be intuitive. It really comes down to this: saying that a proposition is possibly necessary is the same thing as saying it's necessary. The argument for the principle (whether a possible worlds proof or some other kind) is what helps us to see that.

Naturally I thought you were interested in the god proof, since that's what Plantinga is most known for.
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User is offline   Martin Howe 

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#7

View PostTrooper Dan, on 15 May 2019 - 12:49 PM, said:

Naturally I thought you were interested in the god proof, since that's what Plantinga is most known for.

Ah, so that's why. LOL, not to me, I only just found out about it on Sunday reading Wikipedia :)

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