With sufficiently robust assumptions about how causation works, it is possible to formulate various Cosmological Arguments (there's more than one) in a strictly deductive way. But in our age there isn't enough of a consensus about how causation works, or even whether it really exists, to create an argument with broad popular appeal. Some people even say that Science has refuted the idea of causality and replaced it with other concepts.
Instead I'm going to go the long way around and focus on the concept of explanations. It seems utterly clear to me that Science is in the business of finding explanations for certain phenomena. If it can't do that, it loses any claim to make sense out of the world. If you want your theory to explain even one single thing about the world, then that requires you to give an account of what circumstances explain that thing. Then we say that the thing happened because of those circumstances. And as you can see, that word “because” has the word cause sitting inside of it. Grammar itself teaches us that Science requires some notion of causation.
For example, we say things like “the planets all rotate around the sun in the same direction because the solar system formed from a rotating disk of gas”, or “the heat capacity of matter is finite because it is made out of atoms”, or “electrons attract protons because they have negative charge”, or “energy is conserved because the laws of nature are invariant under time translation.” It is not obvious that these various senses of “because” all correspond to the exact same concept, but we can take almost any of them as examples for what I am going to say. (Except perhaps for the “because” of pure logical deduction from definitions, i.e. “cold is the absence of heat because that is what the word cold means; 2+2 = 4 because 4 is twice two. But pure logic is not enough to explain everything that happens, since avoidance of strict logical inconsistency is a rather weak constraint.)
Aristotle famously said that there are four different types of causes: material, formal, efficient, and final. As a classification of the types of "because" answers we give to various questions, this seems fairly reasonable. But some people would argue that the physical sciences only use some of these types of explanations. In particular, it is disputed whether final causes, which express purposes—e.g. that the sun exists in order to produce light, or that the animals eat in order to receive nourishment—really exist in reality outside of our own minds. This question will pop up later when we discuss Ethics, not to mention the Fine-Tuning Argument. For the time being, let's consider less controversial notions of causation, without assuming Aristotle’s division to be correct.
In order to side-step some possible misunderstandings about what I mean by an explanation, let me make it clear that I am using the term explanation in a broad sense, so that:
A. Explanations can be Nondeterministic. An explanation does not need to be deterministic, in the sense that the effect invariably follows from the cause. It is enough if the explanation produces a framework, in which it makes sense that the cause might be produced. For example, Quantum Mechanics is a nondeterministic theory, in the sense that if you start with a given initial condition, and you know the laws of physics, there is generally more than one possible final outcome, and you can only predict their probabilities. So long as the probability of what happens is not so low as to indicate that the theory is probably wrong, I'm going to count this type of thing as an explanation of the experimental outcome. That's because it seems that the universe is not deterministic, and if it isn't, then probabilistic explanation is the best that we can do.
For example, if a radioactive atom decays, you can't predict exactly when it will decay, but you can still explain why it can decay with reference to the forces and particles in the nucleus. It's not like the decay occurs in an explanatory void. So I think in a quantum mechanical theory, we need to generalize our notions of "explanation" and/or "causation" to allow for such nondeterministic explanations. Just because you can't predict exactly what happens, doesn't mean there isn't a set of circumstances which causes whatever does happens to happen. It's just that there's more than one possible outcome that set of circumstances could have produced. That's different from something happening without any causes at all.
B. Explanations can be Unknown. Similarly, there obviously exist some phenomena which do have an explanation, but we don't know what it is yet. In that case, I want the term “explanation” to refer to the actual reason why the thing occurs, and not merely to those explanations which are currently known to our limited human minds. Maybe we will never find out the full explanation for something, but that doesn't mean it doesn't exist. This point is important because Cosmological Reasoning, as I understand it, involves the attempt to push back the concept of explanation back to its ultimate fundamentals, to see what a fundamental explanation would have to look like in principle. For example, a Naturalist doesn't necessarily need to know what the most fundamental theory of physics should look like, to propose that if we did know it fully, it would then explain everything else.
On the other hand:
C. Explanations are not just Regularities. I reject the Humean view that causality is just a name for the constant conjunction of events. In the end, I think this boils down to a renunciation of the possibility of explaining anything. “Rocks have always fallen down before” is simply not an explanation for why the next rock falls down. It doesn't have the right form to be an explanation. It may make it rational to believe that the next rock will fall down, but the reason it does so is that we believe that there are underlying causes (in this case, gravitational attraction to the Earth) which remain roughly the same for each rock. Conversely, there are many types of regularities (e.g. as of 2014, a woman has never been U.S. President) which may be unbroken in our experience, but that doesn't make those recurrences Laws of Nature. Hence, when we seek to explain things, we ought to demand more than just the occurrence of regularities.
This seems to me to be elementary common sense, and I don't think the mathematization of physics (which merely describes these regularities, in increasing generality and detail) should change this conclusion. But even if you accept the Regularity View, you still can't do Science at all unless you try to figure out how to describe these regularities with the most fundamental and deep laws possible. On any reasonable view of causation, we can ask whether this process of explaining things finally terminates; and that will be the topic of the next section.