LCA – Points 2 to 4

In our previous post on Leibniz’ Cosmological Argument, we examined the first premise, the PSR. I would summarize the PSR with the claim “reality is fundamentally intelligible”, though we go into more technical detail in that previous post. To remind you, here is the full argument:

  1. Every contingent fact has an explanation.
  2. There is a contingent fact that includes all other contingent facts.
  3. Therefore, there is an explanation of this fact.
  4. This explanation must involve a necessary object.
  5. This necessary object is God.

Here we will examine points 2 to 4. These are reasonably simple, and although they are sometimes disputed there shouldn’t be anything too controversial here. Most skeptics who deny this argument will take issue with premises 1 or 5.

(2)               There is a contingent fact that includes all other contingent facts.

It is at this point that we must define what a “contingent” fact is. Briefly, a contingent fact is a fact that does not explain itself. For example, we might say “a cat exists”. Is that self-explanatory? Can we coherently ask “why does a cat exist?”. In the case of my cat, explanations may include: me and my brother asked my parents for one, and also the cat has parents, and also the cat is composed of molecules, and also the species “cat” was created by God. These are all partial explanations for the contingent fact “my cat exists”. Here, a contingent fact is any fact like this, any fact X that has an explanation Y, where Y and X are not identical.

This definition is a bit vague, and purposefully so. There are multiple conceptions of precisely how we cash out contingency, and I want my argument to be general enough to cover all of them. But if you are after a more specific definition, try this one: a contingent fact is a fact that is true in some possible worlds, but not all possible worlds. There are some worlds in which there is no cat (or no matter), and some that include a cat. Note that I am not talking about a multiverse here, some worlds contain one universe, some contain multiple, some contain none. A possible world is a way that reality could have logically possibly been, but isn’t. A possible state of affairs that could have attained, but didn’t. If this is a new idea for you, perhaps study this SEP article. We will occasionally make use of this definition during our argument, but I will attempt to stay as general as possible.

Some of you are necessitarians and reject possible worlds. I am. If that is you, then we will need a different notion of contingency. I intend to publish my thoughts on that once we have completed the entire LCA series.

The actual premise is relatively easy. If it’s true that “x is a fact” and “y is a fact”, then it’s true that “both x and y are facts”. So we can construct a true statement such as “x is a fact and y is a fact and……..” and so on until we’ve covered every contingent fact. This fact that we’ve constructed is contingent because all of the incorporated facts are contingent. If any of them could possibly be different, then so could the constructed fact. In some other literature, this fact is called the BCCF (Big Collected Contingent Fact).

There is a minor concern here that the BCCF could be self-referent. That is, the BCCF includes the BCCF . Some of you will be familiar with Russel’s Paradox, which relies on this kind of self-reference. So there are two solutions, I think either works. Either we say that this isn’t a set, and we call it a “collection” or something, and then we aren’t bothered by self-reference. Or we construct the BCCF by including every contingent fact that isn’t the BCCF, we include every other contingent fact. This is a relatively minor issue, it is in a sense just nitpicking. But it seems important to include it so people don’t get grumpy at me.


(3)               Therefore, there is an explanation of this fact.

This follows from (1) and (2). Every contingent fact has an explanation due to the PSR, so the fact we just constructed has an explanation

(4)               This explanation must involve a necessary object.

Let’s suppose it isn’t true, suppose the BCCF is explained by a contingent fact. Given the PSR, then this explanation still has to exist. We’re supposing it’s not necessary, which means it must be contingent. But if it’s contingent, then it’s part of the fact that we’re trying to explain.

This means that the explanation is its own explanation. It doesn’t depend on anything else to be true. This means it’s not contingent, and is instead necessary. So by reductio, the explanation for the constructed contingent fact must be necessary.

Some complain that I’ve equivocated between “fact” and “object” here. But this is not a big problem, like the self-reference above. Every fact refers to at least one object, and for every object X we have the fact “X exists”. We can travel between “facts” and “objects” freely. I choose the term which best primes the intuition of the person reading it.

This will conclude the middle section of the LCA. Next time, we will investigate what the necessary explanation is, we will work out what properties and attributes it must have, and end up concluding that it is God.

20 Replies to “LCA – Points 2 to 4”

  1. >This fact that we’ve constructed is contingent because all of the incorporated facts are contingent

    That’s an inference from parts to whole; How can we know it avoids the fallacy of composition?

    1. The fallacy of composition occurs when we assume that the whole has all properties that the parts have. We are not doing that here. What we are doing is proving that for this particular property, if a part has it, the whole has it.

      The proof is relatively simple. Suppose that A is contingent in the possible world sense. That is, A could have been false. In some possible world (call it P), A is false.

      Now consider the composition sentence “A and B”. It doesn’t matter if B is necessary or contingent, there is some possible world where “A and B” is false. That is, world P, since if A is false then “A and B” is false. Therefore a contingent fact composed with any other contingent fact is contingent.

      1. Ok, thanks Jon, that makes sense.

        But I think you may be equivocating here on the word contingent. On some occasions, you use it and even define it as “Here, a contingent fact is any fact like this, any fact X that has an explanation Y, where Y and X are not identical.” so a contingent is anything that has an external explanation. On other occasions like now, you say contingent in the sense that it is possible that it was different or is different in some possible worlds. How do we bridge the gap between the two different understandings of the word “contingent”? Does one entail the other? Does for instance having an external explanation entail that a fact could have possibly been different or that it is different in some possible worlds?

        Moreover, how can a necessitarian like you believe in “possible” worlds to begin with? If necessitarianism is true and there is only one way the world could be, then how can there be other possible worlds?

        Those are my two biggest issues with the Leibnizian argument and I will accept the argument if you can resolve them.

        Thank you for your patience and efforts.

        1. I am equivocating on “contingent”, and that is deliberate. I intend to draw on some of the intuition that is developed by the “possible world” intuition without actually committing myself to Lewisian possible worlds, since as you say I am a necessitarian.

          You are on the right track with the mutual implications though, and it’s easy to establish one direction. If something is contingent in the possible world (PW) sense, then in some possible world it is false. But that means it’s not self-explanatory, because self-explanatory things are always true. So it is contingent.

          The other direction is not as easy. Supposing something is not self-explanatory, there may be no possible world in which it is false. Imagine a necessary musician creating music. The musician is necessary so she exists in all possible worlds, and in all possible worlds creates music. The music is necessary in that it exists in all possible worlds, but it is not self-explanatory, since it is explained by what exists externally to it. So I seem to be in trouble there, don’t I.

          But I think I am not really in trouble. At this point, I can appeal to the freedom of God, and distinguish between different kinds of necessity. Leibniz does this as well. Suppose we have a God, who surveys all possible worlds (that is, everything He could create if He wanted to) and chooses a single possible world to create. That world is necessarily the best possible world, since God, being good, always wants to create the best possible world. Does that mean He couldn’t have created other worlds if He wanted to? No, He could have if He wanted to, He just doesn’t want to.

          In this way, we can recover a meaningful sense of a possible world: a world that God could have created if He wanted to. And yes, God creating the actual world is necessary, so there is also a sense in which there is only one possible world. Only one way the world could have been, given God’s desires. But if God had desired otherwise, any other possible world could have existed. Possibility is always considered as a compatibility with a certain set of facts. We can think of possible worlds as being worlds that are compatible with God’s power and intelligence and creativity and so on, but if we consider those and God’s will and goodness, only one world is possible.

          1. Thank you John, I think I accept the argument now after my doubts about it are resolved. Could you just shed some light on this:

            > But that means it’s not self-explanatory, because self-explanatory things are always true.

            I think I have no more doubts regarding this argument so thank you again for that.

          2. Suppose that something is self-explanatory. That means that it includes its own reason for existence. This is the same as saying that it doesn’t depend on anything else to exist. Since that reason for existence is true in every possible world (since the object doesn’t depend on anything that might fail to exist), the object exists in every possible world.

            Think of some of the things we normally call necessary, such as mathematical theorems like “The square root of two is irrational”. That can’t ever be false, since it is a tautology. All necessary things are in a sense tautological. For a necessary thing to not exist is a metaphysical impossibility.

  2. You appeal to partial explanations and I think we can use them to object to this point.

    >Let’s suppose it isn’t true, suppose the BCF is explained by a contingent fact. Given the PSR, then this explanation still has to exist. We’re supposing it’s not necessary, which means it must be contingent. But if it’s contingent, then it’s part of the fact that we’re trying to explain.

    >This means that the explanation is its own explanation. It doesn’t depend on anything else to be true. This means it’s not contingent, and is instead necessary. So by reductio, the explanation for the constructed contingent fact must be necessary.

    Let’s suppose that there is an explanation of the BCF, call it E. Let’s say that E is contingent and in this case, E is part of the BCF. If E is the explanation of BCF then I don’t think we can quite say that E explains itself. E definitely explains BCF but it does not follow that because E explains the composite, then therefore E explains the parts because then that would be a fallacy of division where we infer the properties of the parts from the properties of the composite (It would be like saying that because the entire ocean is observable by the naked eye, then its parts or the atoms are also observable by the naked eye).

    Moreover, you appeal to partial explanations and I think we can use them to refute your premise. Even if I grant that E is self-explanatory, maybe E is only a partial explanation of E. Maybe E is only a partial explanation of itself and that E has other partial explanations that do not include itself such as Z or L. If Z or L are partial explanations of E but they are not identical to E then in this case E could be self-explanatory as well as contingent in the sense that it depends on external explanations.

    I am a believer as well btw.

    1. For your first objection, it is the case the explanations are distributive. Suppose A explains “B and C”. I claim that it is the case that A explains B, and A explains C. Suppose it is the case that my wife’s love for me explains why she “cooked dinner and got me a beer”, both of which I was blessed enough to have happen to me tonight. Seems obvious to me that her love for me explains each individual action as well, that she cooked and brought me a beer.

      In general, if A contains enough information to explain and deductively entails “B and C”, we’d call A a poor explanation if it wasn’t able to explain just B alone. That is, it would not be a sufficient explanation.

      Now let’s talk about partial explanations. If E is self-explanatory, that means that E is a sufficient explanation of E. But if that’s the case, it can’t be a partial explanation of E. Sufficient explanations cannot be partial explanations, and vice versa.

      1. The first part is fair. But the second part is where the problem lies.

        >If E is self-explanatory, that means that E is a sufficient explanation of E. But if that’s the case, it can’t be a partial explanation of E. Sufficient explanations cannot be partial explanations, and vice versa.

        My problem is why should we assume that E is self-explanatory at all? I don’t think your argument quite establishes that. Your argument does establish that E could very well only be a partial explanation of E. What if E has other partial explanations in other contingent facts in the BCCF?

        1. We proved that the explanation for the BCF is self-explanatory. Because if a contingent fact C in the BCF explains the explanation of the BCF, then C is actually not contingent. If something explains it’s own explanation, then it can only be necessary

          1. Jon, I apologize if my objection is not clear. What I am saying is that E is only a partial explanation of itself. So let’s backtrack. I agree that the BCF has an explanation called E. What I am saying is that E is one of the explanations but other partial explanations of E would be other contingent explanations of E in the BCF. So E would have many partial explanations including itself as only one of them among other contingent facts in the BCF. Since it has partial explanations in other facts, then it is not self-explanatory and so is not contingent.

            Maybe clarifying what you mean by sufficient and partial explanations will help.

            Sorry if the answer is obvious and I am just not catching it.

          2. I am a bit confused here then, let’s try and straighten it out. We have a BCF, and E is a sufficient explanation of the BCF. But there is some contingent fact C in the BCF which explains a part of E. Let’s call that part of E which is explained by C “E1”, and we will call the other part of E “E2”. Call BCF’ the BCF that doesn’t include C.

            Now we have:
            – “E1 and E2” explains “the BCF’ and C”
            – C explains E1

            Now consider, what is the explanation of C? It must be either E1 or E2, or part of E1 or part of E2. And it can’t be E1, since C explains E1. So clearly, C is explained by E2.

            Now explanations are transitive. If X explains Y and Y explains Z then X explains Z. We have that E2 explains C explains E1. Therefore E2 explains E1. Since in this case E1 is contingent, this forces E1 to be part of the BCF. And so we end up with:
            – E2 explains the BCF

            Where E2 is the part of E that is not explained by C. The same can be performed for any C we take, and we end up with the bit that’s left over. As you say, E is a partial explanation for itself. So let’s consider the part of E that explains itself, and call that the necessary object. No matter how we slice it, we end up with something necessary at the end.

  3. Sorry for the late reply, I was on a family vacation.

    >We have a BCF, and E is a sufficient explanation of the BCF. But there is some contingent fact C in the BCF which explains a part of E. Let’s call that part of E which is explained by C “E1”, and we will call the other part of E “E2”. Call BCF’ the BCF that doesn’t include C.

    Now, you are engaging with my objection. What I am saying is that the part called E2, in this case, has multiple explanations. One of those explanations would be E2 among other explanations z, y, x. But as long as E2 explains the BCF, itself AND has other external explanations, then it will never be necessary simpliciter because it has an external explanation (x, y, z) **on top** of its self-explanation.

    1. Depending on how we conceive of explanations, something having explanations other than itself is not a problem. For example suppose N is necessary and C is contingent. Suppose N explains N. It is also the case that “C and N” explains N, since if X explains Y, “X and Z” explains Y no matter what Z is. And since in this case E2 is the self-explanatory part of E, (that is, E2 explains E2), then E2 is necessary.

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