Meinongianism and Cantorian Diagonalization

 Meinongianism posits that for any condition on objects, there is a unique object satisfying exactly that condition. However, for any plurality of objects, one could construct the condition of being a thought about exactly those objects. But, by Cantorian diagonalization, that means that there are more of these thoughts than objects in Meinong's Jungle. But these thoughts are within Meinong's Jungle. Therefore, Meinong's Jungle is larger than itself. Contradiction.

Squared's Razor

Our best theories of physics are not simple. Our best theories of the economy are not simple. Our best theories of psychology are not simple. Etc.

We can reason inductively that the best theory of metaphysics shouldn't be simple either. But this is in tension with the notion that simplicity is a theoretical virtue. So, we should aspire for a goldilocks metaphysical theory. One that is neither too complex, nor too simple.

A New Onto-Omniscience Argument

1) For any proposition p, knowledge of p is valuable.
2) If knowledge of a proposition is valuable, then that proposition ought to be known.
3) Ought implies can.
4) Therefore, for any proposition p, p can be known.

(4) actually implies the existence of an omniscient being. Take some proposition q that is only true in the actual world. According to (4), there is some possible world where someone knows q. But which possible world does this person who knows q occupy? Well, since the only possible world that q is true in is the actual world, this person is in the actual world! Now let r be whatever proposition you want. We can run this same argument with the conjunction of q and r, So, there is a being in the actual world which knows the conjunction of q and r.

From this point, we have two routes we can take:
-Let r be the conjunction of all true propositions. Does such a proposition exist? Probably not. But if it does, then there's a being who knows all true propositions.
-Argue on the basis of parsimony that we should posit one being that knows all truths, rather than infinity beings that each knows a single proposition.

Pascal's Wager and the Ultimate Good

 A worry about Pascal's Wager that I've had for a long time is that there are different sizes of infinities.

Let f(R) be the cardinality of the set of pleasurable experiences promised by religion R. It seems that for any religion A, once could make up some religion B such that f(A) < f(B). Now, my credence in B will probably be less than that of A, but credences are all real numbers as far as I'm concerned. So, as long as my credences are nonzero, if f(A) < f(B), then I should always favor religion B to A. But, if for any religion, there is another religion I should follow rather than that one, then I have no way to choose which religion to follow.

This worry has never been huge in my mind, since for any any infinite set of pleasurable experiences, I believe that Christianity is my best chance at attaining a set of pleasurable experiences comparable to that set. But this solution depends on my specific credences being lined up in a certain way. For other distributions of credences, one could easily be led to decision-theoretic gridlock. Here's a solution that hopefully bypasses this worry:

We have the ability to consider two states of affairs and decide whether one is preferable to another.  But we haven't considered absolutely every state of affairs. (For example, until I sit down and really think about it, I am not sure whether I would rather spend an eternity at a beach or in a forest.) Maybe there exists some state of affairs that would obviously be preferable to all other states of affairs if we only considered it (but alas we haven't). Let's call such a state of affairs the Ultimate Good (UG). If we are being consistent, we should assign a nonzero credence to the proposition that the UG exists (even if we have no idea what it would look like). Now, this means that a rational agent will only pursue courses of action that have the highest chance of actualizing the UG. Any other good state of affairs -- even if infinitely desirable -- aren't worth the time of day. A rational agent goes straight for trying to actualize the UG.

An Argument for Pantheism

1) For all p, if I know p, then God knows p.

2) I know that I am Squared.

3) Therefore, God knows that He is Squared.

4) Therefore, God is Squared.

We need to be careful with how we treat indexicals, otherwise we might commit ourselves to pantheism. I reject (4), so instead of (1), I would affirm,

1') For all p, if I know p and p has no indexical, then God knows p.

God freely creates abstract objects.

Say a Libertarian free agent was offered the choice between getting a million dollars and being tortured for a week. They would always opt for the million dollars rather than being tortured because there is no reason to be tortured rather than to take the million dollars. So, under a good account of Libertarian free will, the absence of reasons to choose option A and the presence of reasons to choose option B will necessitate the agent's choosing option B. It is only when there are reasons both to choose A and to choose B that the agent's choice is indeterministic.

Let's apply this to any account where abstract objects depend on God's will. If He chooses to create the abstract objects, it seems like He doesn't have the free will to refrain from creating them (since He must create them in every possible world). But, if God necessarily has reasons to create these abstract objects and necessarily lacks reasons to refrain from creating them, then He will create them in every possible world. This is all totally compatible with His having Libertarian free will.

A paradox of non-actual persons knowing stuff

Here's a fun paradox. Say Jim is considering the following argument:

"I occupy this possible world. There have been so many indeterministic events in the past that the probability that they have all had the outcomes necessary for this world's being actual is very low. Thus, I should believe that this world isn't actual. I don't actually exist."

Jim can't see a flaw in this argument, yet he ultimately rejects it because the conclusion is crazy. Let's say that Jim's rejection of this argument is an indeterministic event. So, let's consider the possible world where Jim (or rather, the counterpart of Jim, Jim') accepts this argument. The weird thing is, Jim' is right. He isn't actual, and the reason he isn't actual is because an indeterministic event in the past had the wrong outcome. Does Jim' know that he isn't actual? Furthermore, if Jim had accepted the argument, then the argument's conclusion would have been wrong. That's weird.

A Meta-Moral Argument?

It's often assumed that without God, then there can be no morality. This assumption though, is rarely given a robust justification. The fact that this sentiment is so poorly defended and is yet widespread (even some atheists accept it!) leads me to believe that it's an intuition built into us. I wonder if this intuition can count as a moral intuition. Because a naturalist moral realist is going to want to deny as few widespread moral intuitions as possible, they should give weight this specific moral intuition. Thus, their credence in God's existence should be raised.

A paradox if God has non-occurrent knowledge

I'm toying with the notion that only some of God's knowledge is occurrent. He is not at any moment aware of the truth-value of all propositions, but can choose to become aware of the truth-value of any proposition. But this leads to a paradox. Say God will not decide to become aware of the truth-value of any proposition at time t. Thus, the proposition that "God will not decide to become aware of the truth-value of any proposition at time t" is true. But could God decide to bring that proposition into His awareness at time t? Well, there are possible worlds where God bring the truth-value of that proposition into His awareness. But in all those worlds, the proposition is false. And, in all the worlds where it is true, God is not aware of its truth. So, there is no world where this proposition is true and God is aware of its truth.

Discrete time disguised as continuous

Here's an interesting argument which I would love to see developed by someone more informed about the relevant fields.

Modeling physical systems on discrete time would require iteratively applying some function which maps a given state of the universe to its successive state. This seems prima facie to make physical systems very difficult to model. To know the state of the universe in n moments, I would need to apply this function to the present state of the universe n times. What a headache!

But what if we learn that, due to Zeno's paradoxes or something, continuous time is metaphysically impossible? That would be most unfortunate. It means that we can figure out a priori that modeling physical systems is going to be nigh impossible.

Hold up. We can model physical systems, though. What's going on here?

Well, one explanation is that God "fudged the numbers." God chose a universe whose evolution across discrete time can still be modeled mathematically at the macroscopic scale, because God wants discoverable laws of nature.

So, to flesh out this argument, we need to show that:

1) Physical systems with discrete time will usually be hard to model (barring something like a God intervening).

2) Our universe operates with discrete time.

3) Our universe is not hard to model.

What's the probability that God would create?

The prior probability that God would freely create a universe external to Himself must be some value that lies on the interval [0,1]. 

Assigning a prior probability of 0 is unacceptable for a theist. That would mean the existence of a universe external to God completely disconfirms theism.

Any value that lies within (0,1) seems like it's going to be arbitrary. Why would the value be 0.371 rather than 0.372? Some values like 0.5 or 1/e seem less arbitrary, but there's still going to be some arbitrariness, especially since we have no explanation for why the prior probability would take on such a value.

A prior probability of 1 would be surprising, since God is completely self-sufficient and has no need of a universe external to Himself. Perhaps, though, God's love gives Him very strong reasons to create beings to be in relationship with Him. This wouldn't be the fulfillment of a need, but a gift. Another point worth noting is that, even if the prior probability of God creating an external universe is 1, it may still be the case that it is possible for God to refrain from creating an external universe.

Do humans ever choose between infinity alternatives?

God would seem to (at least sometimes) make a decision between infinity different alternatives. Is that something humans ever do?

Here's one specific example: I have an itch. The harder I scratch it, the more the itchiness will be relieved. But, the harder I scratch it, the more pain I will feel from having scratched at my skin. With these factors in mind, I freely choose to scratch my itch with n newtons of force. Let's say a minimum of 1 newton of force in my scratching is required for relieving itchiness, and nothing greater than 5 newtons of force in my scratching would be merited by this particular feeling of itchiness (I have no idea if those are realistic values). If I scratch with a force closer to 1 newton, then I'm acting on the reason that I dislike pain. If I scratch with a force closer to 5 newtons, then I'm acting on the reason that I dislike itchiness. If I scratch with a force somewhere in the middle, then I'm acting partially on my dislike of itchiness, and partially on my dislike of pain. Perhaps I, without me realizing it, could actually will to scratch my itch with any force on the continuum of [1, 5] newtons. Thus, when I scratch an itch, I choose between an uncountable infinity of alternatives.

Self-exemplification and contingency in abstracta

The Argument from Sets is a really nifty argument that tries to ground the existence of sets in the mental activity of God. This is a really interesting way of looking at abstract objects, since their existence is actually contingent; they could have not existed!

An upshot of this argument is that it opens the door for grounding the contingency of other abstracta in the mind of God. For example, the property of "being an abstract object" exemplifies itself. So, some abstract objects exemplify the property of self-exemplification. But, does the property of self-exemplification exemplify itself? This is not the paradox of the property of non-self-exemplification. With that paradox, both affirming and denying that non-self-exemplification exemplifies itself entails a contradiction. However, with self-exemplification, it can either exemplify itself or not without contradiction. So which is it? And what's the explanation for why it stands in the relation to itself which it does in fact stand in? A theist can ground the property of self-exemplification in the will of God, and maintain that whether or not self-exemplification exemplifies itself is up to the free will of God. Coolio.

Are credences in between 0 and 1?

The idea behind credences is that, while I believe both that 2+2=4 and that it will be sunny tomorrow, I have far more certainty in the former than the latter. This level of certainty is measured in terms of your credence in a proposition. These are numbers that can take a value between 0 and 1, with 0 being certainty that the proposition is false, and 1 being certainty that the proposition is true.

I think this is an inaccurate way to frame credences though. Imagine a man who is happy. Then he gets happier and happier. Is there a limit to his possible happiness? It doesn't seem there is. Maybe there's a limit for his current brain, but we can also imagine his brain being regularly modified so he can continue to get happier and happier.

If we can do this with happiness, then why couldn't we do this with a feeling of certainness? If this is possible, then it doesn't seem to make sense to represent all of our feeling of certainness as values in between 0 and 1. It seem certainness should be any real number.

An alternative way of looking at this is that happiness can also be assigned values in between 0 and 1. This would lead to an interesting new kind of mathematical systemization of happiness. If p corresponds to a happiness of 0.3, and q corresponds to happiness of 0.7, what happiness would be dealt by the conjunction of p and q? Or what about their disjunction? 

What qualia are possible?

 I find the question of what possible experiences there are to be fascinating. Let me start listing some qualia: There's the taste of chocolate, the feeling of cold steel, the color red, etc. If this list were to be completed, and every qualia that could be experienced was placed on this list, what would the list look like? Would it be finitely long? Or maybe it would have countably infinite members? Or uncountably infinite members?

 Let's roll up our sleeves and try to discover what this list would look like. 

Imagine you're staring straight forward, and a large television screen takes up your entire vision. Currently the screen is completely white. Then a red circle appears on the left of the screen. Despite this, you stare straight ahead. Let's call this visual quale qᵣₗ. Then, the circle teleports to the middle of the screen. Let's call this new visual quale that you're experiencing qᵣₘ. Then the red circle turns into a purple circle without changing its location. Let's call this visual quale qₚₘ. 

What's interesting is that these qualia have varying degrees of similarity to one another. The taste of chocolate is incomparable to the feeling of cold steel, yet qᵣₗ is closer to qᵣₘ than it is to qₚₘ. Here's an explanation for this phenomenon: the three qualia are made of "building blocks." See, we can view qᵣₗ as composed of the building block "red circle" which is joined to the building block of "at location l₁." The former building block, "red circle," reappears again in the composition of qᵣₘ: "red circle" and "at location l₂."

That's interesting. These qualia are made up of the joining together of more fundamental qualitative building blocks. Let's invent a shorter notation that denotes saying that building blocks b₁ and b₂ are joined together: {b₁, b₂}. So, qᵣₘ is {red circle, at location l₂}.

Now, both qᵣₘ and qₚₘ can be decomposed into {red circle, at location l₂} and {purple circle, at location l₂} respectively. However, "red circle" and "purple circle" also share something in common. "Red circle" seems to be decomposable to the more fundamental building blocks of "red" and "circle." This latter building block is reused in "purple circle": "purple" and "circle." So, qᵣₘ is {{red, circle}, at location l₂} and qₚₘ is {{purple, circle}, at location l₂}.* The takeaway here is that some of the qualitative building blocks can be decomposed into even more fundamental building blocks. We can have quale of the form {{b₁, b₂}, b₃}.

(*Note: One may worry that our vision is "pixelated" in some way. A red circle is really reducible to a bunch of individual "vision pixels" being colored red. The takeaway here still follows, though, since the quale of seeing a circle in that case would be {{red, at location l₁}, {red, at location l₂}, ...}.)

We've already made a ton of progress on our original question. We just need to figure out what the most fundamental qualitative building blocks there are, and then figure out what rules there are for joining them together, and then we can populate our list with every possible combination of the most fundamental qualitative building blocks. 

Now, my pet theory of qualia (which I call Qualia Atomism) is that there is only one fundamental quale, which I will denote ϕ. All other quale are reducible to it. The idea is we can join ϕ to itself, and get {ϕ, ϕ}. Then we can join {ϕ, ϕ} with itself, or ϕ again to either get {{ϕ, ϕ}, {ϕ, ϕ}} or {{ϕ, ϕ}, ϕ}. The rules of how the fundamental quale can be joined to itself are supposed to roughly mirror the axioms of set theory, but instead of an empty set at the bottom of every set, we have ϕ (another difference may be that you can have the same quale in a joining multiple times, so that {ϕ, ϕ, ϕ} is different from {ϕ} in Qualia Atomism, despite that not being true in set theory). So, very quickly we can get some pretty complicated and diverse qualia. The idea is that once you build upwards enough, we get all the qualia we know and love: the smell of cotton candy, the texture of wood, etc.    

However, there's one problem. It seems that the conjunction of theism and Qualia Atomism entails an inconsistency:

1) God can think about any proposition.
2) There are as many possible qualia as there are things which God can think about.
3) There are more propositions than sets.
4) Therefore, there are more possible qualia than there are sets.
5) If Qualia Atomism is true, then there are equally many sets as there are possible qualia.
6) Therefore, Qualia Atomism is false.

The idea behind (2) is that if God is thinking about some proposition, there is something it is like to be thinking about that proposition. The idea behind (5) is that if the rules for joining qualia resemble set theory, then the cardinality of the class of sets is going to be the same as the cardinality of the class of possible qualia.

I will sketch here a way to modify Qualia Atomism in such a way which will hopefully allow us to avoid (5). 

The set theoretic universe could be extended by defining the "∈₂" relation and adding sets₂ into the universe. The idea is that the ∈₂ relation is indistinguishable from the ∈ relation when looking at sets. So, for all x and all y, x ∈₂ y if x ∈ y and both x and y are sets. However, the ∈ is undefined when we add sets₂ to our universe. What are sets₂? Well, for any class of sets c, there is a set₂ containing₂ all sets in c if there is no set which contains all sets in c. We also add set₂-building rules that mirror our ZFC set-building rules and construct a larger hierarchy on top of the existing ZFC hierarchy.

If we think of sets "corresponding" to qualia, like {{Ø}, Ø}} corresponding to {{ϕ}, ϕ} or whatnot (this would need to be tidied up), then it seems conceivable that we could have some qualia corresponding to sets₂. For example, there is no set of all sets. But there is a set₂ of all sets. So, we can have a qualia that is the joining of all qualia that correspond to sets. 

We can also build up sets₂ to sets₃, and then to sets₄ and so on. We can even get transfinite ordinals in here, like adding setsᵪ, which are built up from not just sets, but sets, sets₂, sets₃, sets₄, and so on. Then we can make setsᵪ₊₁. And so on. Our hierarchy can go as high as the ordinals do.

My thinking is that I don't see any Cantorian worries in positing that for any proposition p there will be some a such that a qualia corresponding to a setₐ is what it is like to think about p. I'm not sure if this eliminates Cantorian worries, but it will sure make them really hard to prove. And it really is a God-glorifying metaphysics, because it allows God to have some really complex thoughts!