The Cosmological Argument Using Math

The cosmological argument to prove the existence of God has been my favourite proof of its kind since the first time it was presented to me. It was simple enough to understand, it felt rigorous enough to be convincing, and it left a lot of room open to extract implications and corollaries that were interesting and useful.

Still, the fact that the argument was presented in natural language, rather than the language of symbolic logic, always left me with an unsettled feeling: shouldn’t this proof, like all others that are watertight and irrefutable, be possible to express in mathematical terms? I made some shallow attempts to codify it myself, but I never managed to do it in a way that left me satisfied.

But recently I came across the book Just The Arguments: 100 of the Most Important Arguments in Western Philosophy, edited by Michael Bruce and Steven Barbone (Wiley-Blackwell, 2011), and my mind was blown. I should state immediately that I neither accept nor endorse all of the subjective decisions made by the editors, whether it be the choice of arguments to include or the opinions expressed in the commentaries of various arguments, but that aside– this book is an incredible reference that not only lists and explains many of the common discussions that take place in Western philosophy, but — critically — presents them neatly and precisely in structured premise-conclusion format that pares down the prose to its essential underlying content. It still uses English words, but it is just one step away from being the symbolic formulation I crave.

So, of course I will take that one step further and convert it to math language. That’s what we’re here for today.

Now, the book presents the cosmological argument I am familiar with in at least three different ways. The first two are both listed under Aquinas’ Five Ways for proving the existence of God: one uses the concept of “efficient causes” and the other uses the framework of “possibility and necessity”. Then there is the proof that is actually called The Contingency Cosmological Argument, which was the most interesting to me since it takes a major departure from what I ever understood as the cosmological argument. It’s not necessarily more convincing to me than the other other versions, but because it was so different, I found it the most fascinating, and that’s what I am focusing on below.

Side point: there is something about the conversion of natural language into the language of symbolic logic (or mathematical expressions in general) that allows redundancies, complications, and errors to truly reveal themselves. (Perhaps that’s one of the reasons I like this process so much.) In my attempts to perform this conversion on various arguments I found in the book, I found that they were not as convincing — or at least, as optimal — as their textual versions made them feel. In fact, I’m a lot less impressed by the argument below now, after having converted it to symbols, although it is still valid (assuming the premises are true) and hence still bears weight.

Anyway, let’s get into it.

The Contingency Cosmological Argument

According to Mark T. Nelson, the author of the corresponding chapter of Just the Arguments, the following argument is attributed originally to Samuel Clarke (1675-1729) and revived by William Rowe (b. 1931). It uses the following terminology:

• Contingent being: a being such that it is logically possible for it to exist, but also logically possible for it to not exist
• Dependent being: a being whose existence is explained by the causal activity of other beings
• Self-existent being: a being whose existence is explained by itself (ie. its own nature)
• Positive fact: a fact that, by being obtained, entails the existence of at least one contingent being

I am not a huge fan of the vague term “explanation”, but we can take it to mean something like “justification” or “reason to believe”. Let’s introduce some notation to make these concepts easier to work with:

S := the set of all entities that exist or have ever existed
E(x, y) := “x (or the causal activity of x) is an explanation for y (or for the existence of y)”
C(x) := “x is a contingent being”, as defined above
T(e, y) := “The explanation e is in terms of y“, as in, y is one of the beings that plays a role in how e is expressed

Then we can define the following, for x ∈ S:

D(x) := “x is a dependent being” = ∃e ∈ S: (e ≠ x and E(e, x))
N(x) := “x is a self-existent being” = E(x, x)
𝒫 := Set of all positive facts = {P : P → ∃x: C(x)}

We will also make use of the following statement, which also can be taken as a premise:

Q := “There exists a dependent being” = “∃x ∈ S: D(x)”

We are also presented the Principle of Sufficient Reason, which is intended to be something that people should be willing to accept:

PSR1: For every being that exists or ever existed, there is an explanation of the existence of that being.
PSR2: For every positive fact, there is an explanation of that fact.

Finally, time for the actual proof. The premises here are numbered using the prefix P, and some explanation for each is provided at the bottom. Conclusions, including intermediate ones, are numbered with prefix C, and are immediately followed by their logical justification. (I slightly reordered the premises and conclusions from the way they were presented in the book, in a way that I thought increased clarity; they have been correspondingly renumbered.)

Here we go:

P1: (∀x ∈ S: D(x)) ⋁ (∃x ∈ S: N(x))

P2: (∀x ∈ S: D(x)) → (∀e: (E(e, Q) → (∀y ∈ S: (T(e, y) → D(y)))))

P3: Q ∈ 𝒫

P4: ∀P ∈ 𝒫: ∃e: E(e, P)

C1: ∃e*: E(e*, Q) (instantiation, P3, P4)

C2: (∀x ∈ S: D(x)) → (E(e*, Q) → ∀y ∈ S: (T(e*, y) → D(y))) (instantiation, P2, C1)

P5: ~∃e: (E(e, Q) ∧ ∀y ∈ S: (T(e, y) → D(y)))

C3: ~∀x ∈ S: D(x) (modus tollens, C2, P5)

C4: ∃x ∈ S: N(x) (disjunctive syllogism, P1, C3)

(Therefore, God exists!! We did it!!)

The premises in plainer language:

• P1: Either every being is a dependent being, or there is at least one being that is self-existent. (Note: this is equivalent to saying that every existing being is explained either by itself or by something other than itself; it means that there is no being that exists without some kind of explanation. Hence P1 is the same as PRSR1.)
• P2: If every being is a dependent being, then any explanation for the statement “There exists a dependent being” can only be in terms of dependent beings. (This is clear, since if all there is are dependent beings, that’s all that can be used in any explanation for any statement.)
• P3: The statement “There exists a dependent being” is such that, if true, would imply that a contingent being exists. (Undoubtedly true.)
• P4: For every positive fact, there is an explanation of that fact. (This is exactly PSR2.)
• P5: It is not possible for an explanation for the fact that “at least one dependent being exists” can only be in terms of dependent beings. (Hidden in this premise is the entire argument about infinite regression and cross dependence being impossible when it comes to causes for existence, it seems.)

All this leads to the final conclusion, C4, which states that it must be the case that a self-existent being exists. That is, there must exist some being (at least one) whose existence is explained by itself only, rather than being explained by (as being dependent upon) other beings.

We can choose to call this being God.