First order logic and the cosmological argument

Question

The way I see it, the cosmological argument, if one takes into consideration only what has been observed in the universe, goes something like this:

1. For everything in the universe, if it has a beginning, it has a cause.
2. The universe had a beginning.
3. Therefore, the universe had a cause.

This can be written using predicate logic as follows: if U is the universe, B(x) is "x has a beginning", and C(x) is "x has a cause", then we have:

1. For all u in U, B(u) implies C(u).
2. B(U).
3. Therefore, C(U).

But this is fallacious!

For example, if U={1,{2}}, B(x) is "x is a set", and C(x) is "x has one element", we get a counterexample to the logic: U has two elements!

I believe this is the composition fallacy.

My Question:

Where have I gone wrong?

I suspect I may have misunderstood the cosmological argument.

Answer 1

The first premise is merely "Everything that begins to exist has a cause".

It does not say "everything in the universe".

Although you have the right idea in pointing out that all the things we've seen begin to exist (due to some cause) have been within the universe. This naturally leads one to wonder why we should accept that the universe itself also had a cause. Why would material effects having material causes mean that the existence of material itself has a non-material cause? We have no good reason to accept that generalisation.

Putting the above aside, it's also worth noting that this argument relies on induction (if it wants to be purely deductive, it would be a faulty generalisation / black swan fallacy). This isn't necessarily a problem, but there needs to be an acknowledgement that there is some potential for error. If this argument were otherwise sound, and concluded something more meaningful, one has to compare the likelihood of error in the premises with the likelihood of whatever conclusion has been reached.

We also have no good reason to accept that the universe began to exist. Theists like to point to the Big Bang, and claim that there was nothing before that, but actual physicists tend to not agree with that (there is a theory that there was nothing but a quantum field, but a quantum field is still something).

The argument also doesn't conclude anything particularly meaningful. The universe having a cause is still a long way from the existence of any sort of deity. The ways that people try to assign traits to that cause has even more problems than the cosmological argument.

Answer 2

You are making what is called a straw man argument in logic. A straw man argument is an argument superficially like the one you are opposing, but that is modified to be easy to knock down.

In this case, you have modified the argument to view the things in the universe in terms of set theory, which is not at all what the people making the argument had in mind. They wouldn't have viewed the universe as the set of the things in it, but the whole of the things in it. The things in the universe are not members, but parts. Take for analogy, a cat. A cat might be divided into its head, its torso, its four legs, and its tail. The cat is not a set with these seven things as its members; the cat is a whole with these things as its parts.

The logic of parthood is called mereology and there is significant literature about it. One of the issues is whether an object should be considered a part of itself. If not, then you are talking about proper parthood. If it is, then you are talking about improper parthood.

The fragment of the cosmological argument that you are dealing with must be viewed as using improper parthood, meaning the universe is a part of itself. If you want to object to this aspect of the argument, you can't just pick a formalism and say why you prefer an alternate formulation; that's a straw man. Instead, you have to address the concepts that the people who made the argument used, and explain in what sense the universe is relevantly different from its parts--relevant in terms of the argument.

Answer 3

You're conflating the universe U, and the universe of discourse. In the original cosmological argument, "everything" that has a start has a source, which means the universe itself is assumed to be inside the universe of discourse. When you added the stipulation of "everything we observe in the universe" you changed the argument in substantive ways.

That doesn't mean that there's not some merit to your objection. But the way you've expressed it is illegitimate. What you're really taking issue with is the idea we can have actual grounds for believing that everything that has a start has a source, since we haven't observed "everything."