Wednesday, May 10, 2023

The Moral Argument (A Quick Version)

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Intro



The moral argument is a family of arguments that use morality as a reason to believe in God. The version of the moral argument I’ll be discussing:
  1. If God does not exist, then objective moral wrongness does not exist.
  2. Objective moral wrongness does exist.
  3. Therefore, God exists.
But for this, it’s important to define our terms.


Moral Semantics



To roughly define what I mean by “morality” I’ll explain the sort of “ought” used in moral obligations. Some oughts are purely descriptive, e.g., when “If you want to do well in school, you ought to study” just means something like “As a matter of practical necessity, you need to study to do well in school.” Some oughts prescribe in a way that is not purely descriptive; e.g. someone saying “You shouldn’t torture infants” might be using this sort of ought, and this is the type of “ought” I’m using in my definition of morality (e.g., an action is morally wrong for someone only if they ought not to do it). Very roughly, by “moral wrongness exists” I mean that there are true facts of what people ought not do using the aforementioned not-purely-descriptive sort of “ought,” e.g., it is a fact that it is morally wrong for a man to torture infants just for fun.

An important feature of this moral semantics is that it implies that moral wrongness is non-natural; i.e., it is not part of the natural, physical world. (Facts of physics, chemistry, etc. are purely descriptive, whereas the property of moral wrongness is not purely descriptive.) Another implication of this moral semantics is that, barring the supernatural, non-natural moral properties like moral wrongness would be causally inert (since anything outside the natural world causally influencing the natural world would be supernatural) and empirically undetectable (such causally inert moral properties could not interact with photons etc.).

To illustrate, imagine a moral nihilist (who does not believe in moral wrongness) and a moral realist (who does believe in moral wrongness) observe some jerk kick a dog just for fun. The dog whimpers in pain and runs away. Both agree on all physiological and psychological facts, e.g., that the dog felt pain and suffered minor injury. The moral nihilist says, “I don’t think moral wrongness is attached to that action.” The moral realist says, “I think moral wrongness is attached to that action.” There is no empirical way to determine who is right here; both views agree on all the same empirical facts. In this sense, moral wrongness is empirically undetectable. Notice also that moral wrongness being non-natural is largely why it is invisible and empirically undetectable in this scenario.

In the context of the moral argument, by moral wrongness being objective I mean that it exists independently of human belief and perception of it.


Arguing for Moral Objectivism



Perhaps the quickest way to argue for moral objectivism is to simply point to an example of an objective moral truth, i.e., one that holds independently of human opinion. Consider the largely uncontroversial fact that it is morally wrong for a man to torture infants just for fun. That this is objectively morally wrong can be shown with this thought experiment: would it remain morally wrong if a baby torturer thought otherwise and killed everyone who didn’t agree with him? It seems that it would be, and thus we have an example of an objective moral truth. The step-by-step reasoning goes like this:
  1. In the thought experiment, something remains morally wrong even when all human opinion thinks otherwise (since the torturer killed off everyone who doesn’t agree with him);
  2. in which case the moral truth “It’s morally wrong for a man to torture infants just for fun” would be holding despite human opinion;
  3. in which case it seems we have an example of an objective moral truth (i.e., holding true independently of human opinion) thereby giving us objective morality.
If (a), (b), and (c) are all true as they seem to be, then we have an example of an objective moral truth. (For those who disagree, do you disagree with (a), (b), or (c)? If so, which one(s)?)


Arguing for the First Premise



It is a theorem of mathematics and propositional logic that “Given A, probably C” entails “Probably, if A then C.” Applied here, “Given God’s nonexistence, objective moral wrongness probably doesn’t exist” entails “Probably, if God does not exist then objective moral wrongness does not exist.” My general approach then is to assume arguendo that atheism is true and see if that leads to objective moral wrongness probably not existing. That approach can be broken down into two components:
  1. If atheism is true, we do not have good reason to believe in objective moral wrongness.
  2. If atheism is true, we have good prima facia grounds to disbelieve in objective moral wrongness.
For (4), I’ll argue that barring the supernatural, we don’t have any good reason to believe that objective moral wrongness exists. Recall that moral wrongness is empirically undetectable, but if so, how do we know about it? In practice we rely on moral intuition (intuition in the philosophical sense of the term, which is about what is immediately present to one’s consciousness and what the consciousness immediately apprehends). How does moral intuition deliver knowledge of moral truths? The theist could say that God designed our cognitive faculties (as via superintended evolution) in such away that when they are functioning properly we intuit certain moral truths, just as we intuit elementary truths of logic and arithmetic.

But for atheism, moral intuition delivering moral knowledge is problematic. Recall that objective moral wrongness is non-natural, and since anything outside the natural world causally affecting the natural world would be supernatural, then barring the supernatural such moral properties like moral wrongness are causally inert and would have no causal influence over whether our brains would produce moral intuitions, and so we'd have the same moral intuitions regardless of whether moral wrongness existed. This would seem to undercut such intuition from properly justifying our belief in morality.

To illustrate, suppose a cyborg has a metal-detecting implant that is designed to give her the intuition that a widget in her hand contains metal if and only if the widget contains metal. But suppose her metal-detecting implant malfunctions and it delivers the cybernetic metal intuition regardless of whether the widget contains metal. Then even if the widget in her hand contained metal, and she believed it contained metal solely on the basis of her cybernetic metal intuition, her true belief wouldn’t be knowledge. The same applies to brains producing moral intuition if such intuition would exist regardless of whether morality existed. Like the malfunctioning metal-detector implant, even if the belief in moral wrongness were true, this belief wouldn’t be knowledge. Moreover, if the cyborg knew that her implant would deliver the cybernetic metal intuition regardless of whether the widget contained metal, this would serve as a defeater for her belief (where a defeater is something that undermines the justification of a belief). The same applies to knowing that one’s brain would deliver the intuition of moral wrongness existing regardless of whether moral wrongness existed.

Of course, this all assumes that moral wrongness is causally inert and that no relevant supernatural intervention takes place. The atheist could get around this problem by positing us humans having supernatural clairvoyance of these invisible and non-natural properties, but these seems awfully far-fetched. It is more likely on atheism that we do not have moral knowledge.

Further analogies could be made. For example, suppose we define a “spirit” as an incorporeal conscious being; e.g., (to borrow a bit from Carl Sagan) an invisible, incorporeal dragon that has no causal influence over the physical world. Consider the following Invisible Dragon Scenario, where nearly everyone in the world believes in an invisible dragon that approves and disapproves of certain behaviors we do. The invisible dragon is of course empirically undetectable, and the only reason people believe in it is via an intuition of its existence due to a quirk of evolutionary development. If it were pointed out that people’s brains would give them the intuition for their invisible dragon beliefs regardless of whether the invisible dragon existed, this would provide a defeater for people’s invisible dragon beliefs. The same, I think, would go for moral wrongness on atheism, and I see no relevant difference between moral beliefs in what we could call the Atheism Scenario (people believe in moral wrongness on the basis of intuition, but moral wrongness is causally inert and has zero causal impact on whether our brains would give us moral intuitions) and the Invisible Dragon Scenario. Some potential rebuttals:
  • Moral supervenience is metaphysically necessary. One could say that moral wrongness associates with certain behaviors by metaphysical necessity (i.e., that it couldn’t have been otherwise), and this results in moral knowledge. This rebuttal is easily accommodated by modifying the thought experiment so that the invisible dragon is metaphysically necessary. It still seems the dragon intuition wouldn’t deliver knowledge.
  • Reliabilism. Another objection is that our moral intuitions and therefore beliefs are produced by a reliable process, and thus moral intuitions deliver knowledge. This is dubious on atheism, since it’s just as easy to conceive of evolution producing a species with very different moral codes (note the vast differences within our species regarding moral beliefs among different cultures throughout history). Even so, this can be accommodated. Imagine that the physical laws have a high probability of producing accurate intuitions about the invisible dragon, though the dragon still has zero causal impact over what intuitions would emerge. Again, it seems like the dragon intuitions would fail to deliver knowledge.
The argument from moral knowledge in a nutshell:
  1. The dragon believers in the Invisible Dragon Scenario do not have knowledge for invisible dragon beliefs.
  2. If the dragon believers in the Invisible Dragon Scenario do not have knowledge for invisible dragon beliefs, then moral realists in the Atheism Scenario do not have moral knowledge.
  3. If moral realists in the Atheism Scenario do not have moral knowledge, then on atheism we do not have good reason to believe in objective moral wrongness.
  4. Therefore, on atheism we do not have good reason to believe in objective moral wrongness.
The justification for (8) is that the Atheism Scenario is more or less the situation we are in right now if atheism is true and moral wrongness exists (again, us having supernatural powers of clairvoyance is far-fetched). The justification for (7) is that no relevant difference exists between the Atheism Scenario and the Invisible Dragon scenario such that the dragon believers have knowledge for their dragon beliefs but the moral realists do not have moral knowledge in the Atheism Scenario. Two proposed relevant differences (metaphysical necessity and reliabilism) were already discussed, those two differences being incorporated into the Invisible Dragon Scenario.

Premise (6) seems fairly obvious. Even so, one could bite the bullet and say that the dragon believers would have knowledge of the dragon (particularly in light of the reliabilist rebuttal), but for such a person I invite them to imagine them living in a world where belief in the invisible, incorporeal dragon is as common as belief in gods. Suppose you inform these dragon believers that they would have their dragon intuitions even if this invisible dragon did not exist. Wouldn’t this fact provide a defeater for their invisible dragon beliefs? It seems so, and that doesn’t seem like this should occur if the dragon believers really did have knowledge that the invisible dragon exists. Moreover, that sort of defeater also seems to apply to objective moral wrongness if atheism is true, and if that is the case then on atheism we do not have good reason to believe in objective moral wrongness.

Having justified point (4), i.e., that if atheism is true, we do not have good reason to believe in objective moral wrongness; I’ll turn my attention to point (5), that if atheism is true we have good prima facia grounds to disbelieve in objective moral wrongness. My justification for this relies on what is known in philosophy as the argument from queerness.

To illustrate the general idea behind the argument from queerness, imagine someone saying that an invisible unicorn is floating above their head. This claim is possible but not plausible and one would be prima facia justified (i.e., justified in the absence of further evidence) in disbelieving in this claim, because the invisible unicorn is “queer,” i.e., it wildly diverges from the types of things we know exist in a way to make it unlikely in the absence of evidence for it.

On atheism, objective moral wrongness likewise seems queer: it is invisible, non-natural, and we would need something like supernatural clairvoyance to know it exists. This is wildly different from the types of thing we know exist, and thus on atheism we would have prima facia justification for disbelieving in objective moral wrongness.


Conclusion



The moral argument being discussed here is this:
  1. If God does not exist, then objective moral wrongness does not exist.
  2. Objective moral wrongness does exist.
  3. Therefore, God exists.
The type of moral semantics being used here is such that the moral “ought” is that type of ought that does not have only descriptive qualities. This leads to moral wrongness being non-natural and empirically undetectable. The justification for premise (2) is a proof by example: it is objectively morally wrong for a man to torture infants just for fun, as revealed in a thought experiment in which a man who doesn’t think it’s morally wrong kills everyone who doesn’t agree with him (in that situation, it would still be morally wrong).

The justification for premise (1) is that given God’s nonexistence, it is likely that objective moral wrongness does not exist. The approach for this justification was twofold:
  1. If atheism is true, we do not have good reason to believe in objective moral wrongness. There is no relevant difference between the Invisible Dragon Scenario and the Atheism Scenario, and if that is so, then on atheism we do not have good reason to believe that objective moral wrongness exists. To oversimplify somewhat: if atheism is true then we’d have our moral intuitions of objective moral wrongness existing regardless of whether it existed, and this seems to prevent such intuition from being a good reason to believe in objective moral wrongness (recall the illustration of the cyborg and her faulty metal-detecting implant).
  2. If atheism is true, we have good prima facia grounds to disbelieve in objective moral wrongness. This was justified via the argument from queerness. If moral wrongness exists, it is non-natural and we’d need something like supernatural clairvoyance to know it exists.
Both premises of the moral argument seem justifiably true, and if the premises are true the conclusion is correct regardless of what else might be true.

Friday, June 24, 2022

Conjunction and Conditional Probability

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Intro



In this article I will prove that the probability of “Given A, probably C” is greater than or equal to the probability of “A and C are both true.” This has implications for the moral argument for the existence of God, for reasons I’ll explain next.

Background



Behold the following moral argument:
  1. If God does not exist, then objective morality does not exist.
  2. Objective morality does exist.
  3. Therefore, God exists.
In a previous article I provied via some basic mathematics and elementary propositional logic that “Given A, probably C” entails “Probably, if A then C”, e.g., “Given God’s nonexistence, objective morality probably does not exist” entails “Probably, if God does not exist then objective morality does not exist.” A lot of atheists disagreeing with the moral argument who concede that “Given God’s nonexistence, objective morality probably does not exist” are thus rationally committed to believing the first premise of the moral argument.

However, some atheists who believe objective morality does not exist say they don’t agree with the first premise despite rationally committed to doing so (or at least rationally committed to the first premise being probably true). In this article I argue that atheists who believe that objective morality does not exist are rationally committed to believing Given God’s nonexistence, objective morality probably doesn’t exist which in turn implies If God does not exist, then objective morality does not exist is probably true.

More precisely, I will use the power of mathematics to show that anyone who believes God does not exist and objective morality does not exist is rationally committed to believing Given God’s nonexistence, objective morality probably does not exist.

The Proof



For brevity’s sake, I’ll use a bit of symbolic logic, where ∧ represents “and” and ¬ represents “not” as part of my abbreviations:

G =God exists.
¬G =God does not exist.
M =Objective morality exists.
¬M =Objective morality does not exist.
¬G ∧ ¬M =God does not exist and objective morality does not exist.
P(¬G ∧ ¬M) =The probability that ¬G and ¬M are both true, i.e., the probability that God does not exist and objective morality does not exist is true.
P(¬M|¬G) =The probability of ¬M given ¬G, i.e. the probability of objective morality not existing given God’s nonexistence.


One assumption I will make is the person who believes that ¬G ∧ ¬M is true also believes that ¬G ∧ ¬M is at least probably true. Generally speaking, to be rationally consistent, if you are to believe that some proposition P is true, you should also believe that P is at least probably true.

To generalize, I’ll use two propositions A and C. The goal is to prove this is true:
P(C|A) ≥ P(A ∧ C )
To start with, note the following equation familiar to many high school graduates and intelligent middle school students:
P(A ∧ C) = P(C|A) × P(A)
What is the highest P(A ∧ C) can be if we hold P(C|A) constant? Well, we would want P(A) to be as high as it can be, which is 1. Thus, in finding an upper limit for P(A ∧ C), this inequality is true:
P(A ∧ C) ≤ P(C|A) × 1
    ⇔ P(A ∧ C) ≤ P(C|A)
After finding this upper limit for P(A ∧ C), there’s just one more step:
P(A ∧ C) ≤ P(C|A)
    ⇔ P(C|A) ≥ P(A ∧ C)
Since A and C were of course arbitrary placeholders, we can use all sorts of propositions, including ¬G and ¬M. So for example this is true:
P(¬M|¬G) ≥ P(¬G ∧ ¬M)


Conclusion



Thus anyone who believes God does not exist and objective morality does not exist is rationally committed to believing Given God’s nonexistence, it is unlikely that objective morality’s existence, since P(¬M|¬G) is going to be greater than or equal to the probability of ¬G ∧ ¬M.

Since God does not exist and objective morality does not exist being probably true entails Given God’s nonexistence, it is unlikely that objective morality’s existence, which in turn entails that If God does not exist, then objective morality does not exist, one who believes God does not exist and objective morality does not exist should also believe that the first premise of the moral argument (If God does not exist, then objective morality does not exist) is at least probably true.

Tuesday, September 14, 2021

Can an Infinite Sequence of Coinflips Come Up All Heads?

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Intro



If there is an infinite sequence of coinflips (for simplicity, assume that heads or tails are the only possible outcomes for each coinflip, and that this is a fair coin), is it possible for the coinflip to come up heads? I think so, but how do we prove it?

Background



This issue bears relevance to the Eternal Society Argument against an infinite past. Roughly (in the paper the Eternal Society Paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).

Now imagine the Eternal Society has the following Annual Coin-Flipping Tradition: each year they flip a coin. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise they do not do the chant for that year.

The coinflips are probabilistically independent events, so any particular infinite permutation of coinflips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year for the Eternal Society practicing the aforementioned Annual Coin-Flipping Tradition. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.

Again, the coinflips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past for the Eternal Society engaging in the Annual Coin-Flipping Tradition. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.

Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:
  1. If an infinite past were possible, an Eternal Society would be possible.
  2. If an Eternal Society were possible, then scenario S1 would be possible.
  3. If S1 would be possible, then S2 would be possible.
  4. S2 is not possible.
  5. Therefore, an infinite past is not possible.
The Eternal Society Paradox Argument Against an Infinite Past is a deductively valid argument—the conclusion (line 5) follows logically and inescapably from the premises (lines 1-4). A sound argument is a valid argument with all true premises, so the only way the argument can fail to be sound is with a false premise.

One could deny premise (1) particularly since that seems to be the most vulnerable premise, but as the Eternal Society Paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).

An Objection



One objection I’ve seen is that it’s just not possible for a coin to come up heads infinitely many times, such that S1 is possible (since both heads and tails coming up infinitely many times is possible) but S2 is not (it’s impossible for an infinite sequence of heads to happen). The reasoning behind this may stem from a confusion of the law of large numbers or thinking that if the probability of heads is 50% for each trial, an infinite sequence of such trials leads to a probability of 0, and a probability of 0 means that it’s impossible (this is not true; any infinite particular permutation of coinflips where both heads or tails has a “probability of 0” as the number of trials goes to infinity; so this “probability of 0” reasoning would imply that no outcome is possible for an infinite sequence of fair coinflips). Regardless, can we mathematically prove that an infinite sequence of heads is possible under the conditions of the Eternal Society Paradox? We can.

The Proof



To define some terms (with the caveat that different sources may define these terms slightly differently): in probability a random experiment like flipping a fair coin is called a trial. In the case of an individual coinflip for the scenario used here, the outcomes are binary meaning that there are only two possible outcomes (heads or tails). Mutually independent trials are where the probability of given trial’s outcome is unaffected by the outcomes of other trials, including any combination of the outcomes of the other trials. A sample space is the set of all possible outcomes of a random experiment.

In the case of an infinite sequence of coinflips, each trial has exactly two possible outcomes, each trial is mutually independent (in the sense that the probability of the outcome is unaffected by the outcomes of other trials), and the probability is the same in each trial. Some additional assumptions to make the reasoning a bit easier to follow if nothing else: each candidate outcome for a trial (heads or tails) has a nonzero probability, and any outcome with a nonzero probability is possible. Let H represent “heads” and T represent “tails.”

Recall that for each coinflip, both H and T have nonzero probabilities and thus both are possible. So the fact that the probability of given trial’s outcome is unaffected by the outcomes of other trials entails that which outcomes are possible for each trial is also unaffected by the outcomes of other trials.

So for each trial (and thus each sequence element in the infinite sequence of coinflips):

    (a)  both outcomes (H or T) are possible; and
    (b)  which outcome is possible is unaffected by the outcomes of other trials.


Given the aforementioned conditions, since every element in the sequence has the property of heads or tails both being possible values regardless of the values of the other sequence elements, this permits any infinite sequence of the binary values.

To illustrate, suppose we have a fair coin and the coinflips are mutually independent in the sense I described earlier, and there are an infinite number of coinflips. Can the first coinflip be heads? Yes, since for each sequence element, H or T is a possible outcome, which would include the first coinflip (confer (a)). Given that the first coinflip is heads, can the second coinflip also be heads? Yes, since the possible outcomes of the second coinflip is unaffected by the outcomes of other coinflips (confer (b)); hence both H and T can be used here. If the first two coinflips are heads, can the third one be also? Yes, because the possible outcomes of the third coinflip is unaffected by the outcomes of other coinflips (confer (b)), hence both H and T can be used here, and so on ad infinitum for all of the remaining sequence elements. And since the outcomes of heads is arbitrary here (e.g., we could just as well have used T, H, T for the first three sequence elements, and then choose whatever binary sequence we wish for the remaining sequence elements) this can be generalized so that the sample space consists of all infinite binary sequences.

But since the set of all possible outcomes consists of all infinite binary sequences, this would include a sequence of coinflips coming up H for each trial.