Clear Language, Clear Mind

I found the original paper on the paradox. Enjoy.

en.wikipedia.org/wiki/Preface_paradox

David Makinson, The paradox of the preface

Give it a read. It is divided into 4 parts:

My Take on the Liar Paradox (Part I of IV)
My Take on the Liar Paradox (Part II of IV)
My Take on the Liar Paradox (Part III of IV)
My Take on the Liar Paradox (Part IV of IV)
All four articles combine to a total of about 8,000 words, so it will not take long for a dedicated reader to read through it.

I posted this text a long time ago in Danish and I translated it and used it on the forum. I think I forgot to post the English version here as well. I apologize for the low quality English in the text.

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I have earlier written a short article about why the stone-paradox not is a real paradox and therefore do not disprove omnipotent entities.

The paradox

The paradox is usually formulated thus:

Can an all-mighty [god, entity] create a rock so heavy that it cannot itself lift it? If it can, then it is not all-mighty and if it cannot, then it is not all-mighty. Given either outcome then the [god, entity] is not all-mighty.

Let us formulate the argument more explicit:

1. If god exists, then god is all-mighty.
2. Either god can a rock so heavy that he cannot lift it, or he cannot.
3. If god can, then he is not all-mighty.
4. If god cannot, then he is not all-mighty.
5. Given either outcome, god is not all-mighty. (2, 3, 4)
6. God does not exist (1, 5)

The problem

I claim that the problem lies with premise 3. It simply doesn’t follow that if god can create an object, which causes him to lose omnipotence, then he is not omnipotent. I will explain why later, let us now look at a similar argument which catches the problem mere clearly.

1. If god exists, then god is all-mighty.
2. Either god can remove his own all-mightyness or he cannot.
3. If god can, then he is not all-mighty.
4. If god cannot, then he is not all-mighty.
5. Given either outcome, god is not all-mighty. (2, 3, 4)
6. God does not exist. (1, 5)

It should be clear that premise 3 is false. If it in some way is not clear to you, then let me explain exactly where it goes wrong.

Actuality and potentiality – the root of the problem

All-mightyness is often defined as ‘can do everything’. This is too vague for us. Let us look at what one calls logical omnipotence (logical all-mightyness); one can do all which is logically possible. Logically possible is the weakest form of potentiality, because it just says that the thing is not self-contradictory.

If an entity can do all actions which are logically possible, then it can also create a rock so heavy that it cannot lift it. But this does not make it impotent; non-all-mighty. It is first when the rock becomes actual that the entity is no longer all-mighty. There being a possibility creation of a rock does not mean that it is created. Therefore there is a conflation of potentiality (can) and actuality (is) at the adherents to the argument.

Here’s a little paradox that I’ve come across while thinking. It’s about worldviews and knowing that at least one thing I currently believe to be true is actually false.

The argument

1.      For all x, if x is a belief in my worldview, then I hold that belief. [Premise]

2.      If for all x, if x is a belief in my worldview, then I hold that belief, then I don’t believe that my worldview contains a mistaken belief. [Premise]

3.      Thus, I don’t believe that my worldview contains a mistaken belief. [from 1, 2, MP]

4.      There is at least one proposition such that I believe it and it is false. [Premise]

5.      If there is at least one proposition such that I believe it and it is false, then my worldview contains a mistaken belief. [Premise]

6.      Thus, my worldview contains a false belief. [From 4, 5, MP]

7.      Thus, my worldview contains a mistaken belief and I don’t believe that my worldview contains a mistaken belief. [3, 6, Conj.][1]

This conclusion seems paradoxical to me. It’s not a contradiction at it stands, it’s just fishy. If I form the belief as a result of this argument, that my worldview contains a mistaken belief, then I hold two contradictory beliefs. It seems that I have to reject a premise, but I don’t find any of them weak. In fact, three of them are logical tautologies, and the last is proven by induction. I’ll discuss the premises below.

Premise one

This one may seem a little unnecessary. I had no luck formulating the argument without this premise. I think it is a logical tautology, that is, true per definition. Let me first define worldview as I use it here.

Worldview =_df The (complete) set of beliefs a person has.

Then, given the above definition, it is clear that if a belief is part of a person’s worldview, then that person holds that belief.

When I wrote this article and when I was thinking of the paradox, I noticed that it is easy to make a category error and call a worldview false. But that doesn’t make sense. ‘True’ and ‘false’ are meaningful in relation to propositions and not to beliefs. We may instead say that a worldview is mistaken which just means that the worldview contains at least one mistaken belief. A mistaken belief is a belief in a false proposition.

Common language may be broader in the use of ‘true’ and ‘false’ but here I will restrict myself for the sake of clarity of thought.

Premise two

This one is similar to the first, as it seems unnecessary and is a logical tautology.

One could argue it with a reductio. Assume that I believe that my worldview contains a mistaken belief. If I believe that, then I don’t hold the belief that is mistaken. (Since if I did, I would have two contrary beliefs.) If I don’t hold that belief, then it isn’t part of my worldview. But from the assumed we can deduce that it is part of my worldview. Thus, contradiction, and the assumption is, thus, false.

Premise three

This one is not a tautology for a change, but I think it is uncontroversial. I have a large number of beliefs, all grown-ups do, and in the past it has always been the case that a belief I had turned out to be false. Similar behavior has been observed in other humans. By induction we have good reason to believe that some of my current beliefs are false. The trouble is that I don’t know which of them it is!

Premise four

This is another tautology. By definition my worldview is the set of beliefs that I have, and if I hold a mistaken belief then it follows that my worldview contains at least one mistaken belief.

Solution?

I don’t know. I haven’t found one, if there is one.

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[1] The argument is valid in propositional logic but some of the propositions are formulated in predicate logic for extra clarity.