When moving between standard modal logic and quantified modal logic, the naïve conversion produces a rather scary set of results, namely, The Barcan Formula (BCF) and The Converse Barcan Formula (CBF): $$ \begin{align*} \Diamond \exists \alpha \phi \to \exists \alpha \Diamond \phi &\equiv \forall \alpha \Box \phi \to \Box \forall \alpha \phi \ \ \ \ (\textbf{BF}) \\\ \exists \alpha \Diamond \phi \to \Diamond \exists \alpha \phi &\equiv \Box \forall \alpha \phi \to \forall \alpha \Box \phi \ \ \ \ (\textbf{CBF}) \end{align*} $$ One of the issues with these results is that they can be used to prove $\Box\forall\alpha \Box \exists \beta (\alpha = \beta)$; ’everything exists necessarily!’ In order to derive both BF and CBF you need S5 so of course I feel like we should just reject S5 and continue about our lives as if nothing was wrong, but sadly, this response doesn’t seem adequate given that I have to argue both ways in at least 500 words each way, so with great regret, I shall look at the best arguments for accepting the Barcan Formulas, or adapting variable domains to get around the above metaphysical repercussions.
The best way to approach the debate is to look at the more general case, actualism (those who will reject the Barcan Formula) and possibilism (those who will accept it). Actualists believe in something superficially intuitive, namely, that all things that exists are actual. So if I were to consider $\forall x Px$, we can be sure that the bound variable $x$ will never name an impossible man eating bunny from mars, because such thing do not exist. Possibilists, however believe that things can exists in a contingently non-actual way (mere possibilia). This seems very counter intuitive, but consider again, $\forall x P x$. What if the denotation of $Px$ is ‘$x$ can only be found in our universe’. In actualist philosophy, this a tautology. This seems just as counter intuitive as the possibilist’s claim. Consider why can’t $x$ ever represent a dodo? They are known to have existed but we know (to great certainty) that they no longer exists in our universe; there is no actual dodo around. The reason that the actualism debate is a more general case of the Barcan Formula debate is because, consider BF, it seems clear that in the expression $\exists \alpha \Diamond \phi$, $\alpha$ names either a real object, or some mere possibilia. In the next two sections, I will write my favourite arguments for actualism and for possibilism as to indirectly answer the question whether we should accept or reject the Barcan Formulas.
Case for actualism
The best argument for actualism is that the introduction of mere possibilia is a flawed, hefty and unjustifiable ontological claim. I will deal with all of these separately.
In order to outline the flawed nature of the ontology on mere possibilia, I will introduce two counter examples. Traditionally, Wittgenstein’s child is used to demonstrate the madness of mere possibilia, but there are far better counter examples given by Tim Williamson (a possibilist simply accepts the mad result of the Wittgenstein’s child argument as true and nothing is gained on either side). Firstly, we will consider the predicate $P$ where $|P\alpha| = x$ is not in the set of everything there is. $\Diamond \exists x Px$. This is equivalent to saying ‘maybe there could have been more than there is’. But we can use BF and MP to conclude $\Diamond \exists x Px \models_{S5} \exists x \Diamond Px $, or in English, ‘There exists something that possibly isn’t in the set of everything there is.’ Surely this is a semantic contradiction? Now we will look at the predicate ‘Nothing is $x$’. I prefer to formalise this as $Px = \lambda x. \neg \exists y (x = y)$. Consider the sentence ‘There is something that could have been nothing.’ We formalise this as $\exists x \Diamond Px$. We use CBF and MP and we get $\exists x \Diamond Px \models_{S5} \Diamond \exists x Px$, or in English, ‘Possibly there is something that is nothing.’ Not only does this seems completely semantically contradictory, but also, we intuitively want $\Box \forall x \exists y (x = y)$ to be true (necessarily everything is something) but: $$ \begin{align*} \Diamond \exists x Px &\equiv_{\text{def }P} \Diamond \exists x (\lambda x.\neg \exists y \ \ x=y)x \\\ &\equiv_{\text{def }\lambda} \Diamond \exists x \neg \exists y \ \ x=y \\\ &\equiv_{\text{def } \Diamond} \neg \Box \neg \exists x \neg \exists y \ \ x=y \\\ &\equiv_{\text{def } \exists} \neg \Box \neg \neg \forall x \neg \neg \exists y \ \ x=y \\\ &\equiv_{\text{def } \neg \neg} \neg \Box \forall x \exists y \ \ x=y \\\ \end{align*} $$ That truth would lead to logical inconsistency!
The hefty unjustified ontological claim argument is simply a rehash of the above arguments. Maybe, you do not mind that everything is not necessarily something? Maybe you are fine with acknowledging the existence of things which are not in the set of everything there is? But those claims are, even if they could be true, are not justified and seem very counterintuitive. Consider the dodo example from §1. A possibilist would say that $\forall x Px$ where $|Px|$ means ‘x can be found in the universe’ is false because there are mere possibilia that are only contingently actual. This is because there could exist something that cannot be found in our universe ($\Diamond \exists x \neg Px$) which implies by BC that there exists something that possibly can’t be found in our universe ($\exists x \Diamond \neg Px$). This $x$ is mere possibilia. So consider the dodo. Maybe they didn’t die out? So it is possible that $x$ could point to Jerry, one of the many dodos prospering on a world accessible to ours. By the possibilist account, Jerry exists in our world but in a non-actual sense, even though Jerry was never born in our world and will never be born… seems sketchy to me… Now consider a few months ago, before Jerry was born into a loving family (or conceived for that matter). According to the possibilist, Jerry still exists. Birth (or conception) was simply the act of becoming actual. Death, in turn will be the opposite. Jerry was even present at the beginning of the universe in a non-actual form, as were we all and much of what there is presently. Although this might be possible that mere possibilia exist, it isn’t probable, it isn’t even intuitive and there is no justification! This isn’t a strong argument for actualism, but it certainly show that possibilism isn’t justified either.
Case for possibilism
The best reasons for possibilism is that mere possibilia do not necessarily make ontological claims over and above numbers, and variable domain quantified modal logic massively reduces the set of sentences expressible, as well as, almost ironically, removes the ability for the logic (designed to reason about both objects and contingencies) to reason about contingent objects. After all this, I will finally look at Lewis’s possibilist(ish) theory. But first, looking at the arguments above, there are no contradictions unless you adopt the initially more intuitive actualist metaphysical theory. A possibilist would have no issue with any of the results.
Firstly, the ontological claim about the existence of numbers. Although some people are skeptical to the existence of numbers because they don’t occupy any physical space, the general consensus is that they do exists, and the consensus with actualists is that they also actually exists. I will now ask you to cast your mind back to when Wittgenstein was alive. What was the probability that he could have had a child? Maybe it was 0.7, maybe it was 0.00000001, maybe it was even 0. Either way, those that accept that numbers exists, must accept that those probabilities exist. Now, we don’t know what the probability was, but we can say that $\mathbb{P}(\text{Wittgenstien had a child}) = p$. And we don’t doubt that $p$ exists. So, lets define the following function:
$$ \begin{alignat*}{2} A &: [0,1] && \to {0,1} \\\ A &: x && \mapsto \begin{cases} 1 & x = 1 \\\ 0 & x \not = 1 \end{cases} \end{alignat*} $$
This is a possible was to define the predicate ‘is actual’. Only things that definitely exist(ed) are actual according to an actualist so only things where $p=1$ are actual, but given that they do not deny the existence of $p$ when $0 \leq p < 1$, why can’t they accept that $p$ is the mere possibilia?
The next argument is that the variable domain correction of modal quantified logic renders quantifiers meaningless. This is where all non-actual objects are hidden in the domain of a world and considered in a purely formalist fashion. This makes BF unprovable in quantified modal logic. But this makes $\forall$ meaningless as the objects iterated over are different in every possible world. What if I want to consider all objects and contingent objects? This might be a scenario that the actualist doesn’t believe in but consider the very legitimate proposition:
“It’s possible that the pope could have had a son and, if this were true, he could have also grown up to be the next pope meaning that the catholic celibacy rules actually guard the church from becoming a tyrannical, infallible monarchy.”
I provided a real world example to show that the proposition is “real world” even though we will only formalise the start. “It’s possible that the pope could have had a son and, if this were true, he could have also grown up to be the next pope” could be formalised $\Diamond \exists x (Sxp \wedge \Diamond Px)$ where $Sxy = $ ‘$x$ is the son of $y$’ and $Px = $ ‘$x$ is going to be the pope.’ If we ignore the conjunction there is no problem for the actualist. The question is essentially about whether a possible world exists, not whether a possible son exists. But when we introduce the conjunction, valuation becomes tricky for the actualist. We have our $x$ (set by the existential quantifier) and now we are asserting a further possibility about the very object $x$. We need $x$ to reference something so that we can track the underlying object from possible world to possible world. I think this might be easier to think about as an analogy. In the C programming language we have things called pointers. A pointer holds a reference to an object in memory and can be used to access the object. For example:
int i = 10; // i is an actual integer
int j = 10; // j is an actual integer
int *p = &i; // p is a pointer and points to the reference of i
int *q = &j; // q is a pointer and points to the reference of i
p == q; // This is FALSE! p doesn't equal q because one points to i and the other j
*p == *q // This is true! what p and q points to has the same value
int *r = null // r is a pointer that points to nothing; a null pointer
The actualist sees the above formalisation as doing (continent) logic with (possibly null) pointers (p,q,r) whereas the possibilist does all logic with concrete values (i,j). This issue is that every null pointer is equal as they all point to nothing. So if we are asking a question such as ‘is null going to be the next pope?’ any answer makes no sense! It doesn’t matter if we give null a name such as r, its still nothing. It needs to point to an object for that question to make sense and in this case, that object is purely contingent, something the actualist doesn’t believe in. So even variable domains assert in some way the existence of contingent, non-actual objects.
But there is something more worrying about the variable domain account for actualists. Earlier I said the following:
" If we ignore the conjunction there is no problem for the actualist. The question is essentially about whether a possible world exists, not whether a possible son exists."
This idea of seeing whether a possible world exists is troubling. Could it be that Kripke semantics is actually assuming the genuine existence of possible worlds. Consider the following formula: $\Diamond \exists x Px \wedge \neg \exists x \Diamond Px$. Take $P$ to mean “is a god”. This means “Its possible that gods could exist, but there isn’t anything that actually could be a god.” This seems to be exactly what the actualists want to say! But the second part of the conjunct directly assumes the existence of an $a$ distinct from all actual things! It assumes mere possibilia! You can’t escape it! Or you think of modal quantifiers as directly quantifying over possible worlds as if they are real. This is the lewis approach and seems to be the only way to ensure that everything is actual in the classical sense.
Lewis
Lewis has a rathe elegant way of side-stepping the actualist/possibilist question, although his solution does require a very controversial ontological commitment, viz, possible worlds are actual. Although this does seem rather crazy, some academics working in the domain of quantum theory also believe in a many alternative worlds theory to reduce the probabilistic non-determinism of quantum mechanics to a more “meta-deterministic” account.
Classically, an object either is actual or isn’t. It is a unary predicate or attribute that an object can have. Lewis however claims that the property is in fact indexical; its truth value is predicated on the context. For example, I know there not to be a fish a chip shop anywhere near me as I am standing in a large meadow. It is, however possible (although unlikely) that someone decides to set up a fish and chip shop in this large meadow. So I could be near a fish and chip shop. Now, by BF, there must exist something that could be a nearby fish and chip shop. Now, if I were to be about a kilometre away, I would be near one. This means that the possible object in this second scenario would actually be the fish and chip shop near by. Whatever this object is, it is made actual by the context (in this case, my spatio-temporal location ) in which I utter ‘I could be near a fish and chip shop.’ So when I wasn’t near the fish and chip shop, what was the non-actual object? Well, it was the fish and chip shop in the — very real — possible world in which there actually was a fish and chip shop near me. There is no need to introduce any non-actual objects (in the classical sense). Quite elegant, but does rely on a belief in possible worlds.