Quantified modal logic often plays a central role in the debate between those who believe in the existence of possibilia (possibilists) and those who deny their existence (actualists). Traditional logical and semantic theories, when applied to the subject of modality, make the derivation of the Barcan formulas possible. These theorems serve as an affront to common sense intuitions that deny the existence of objects that are merely possible, or that objects have necessary existence. Recent contributions attempt to either render the validity of the Barcan formulas irrelevant to the broader metaphysical issues, or to sate actualist intuitions in a variety of other ways. I argue that these attempts are failures. The possible worlds framework is the proper theoretical apparatus of the possibilist – but not the actualist. The actualist must find some other apparatus to underwrite their intuitions. I argue further that this is not as bad for the actualist as it sounds, since the possible worlds framework cannot itself be used as a motivation to adopt possibilism.
To make the case I will give an overview of the salient historical features of the debate, but will give special attention to the progression of the dialectic. With respect to possibilism I will consider one particular motivation in isolation – that of a conservative logical program which is often seen as a reason in and of itself to adopt possibilism. I will show why this doesn’t work. The point will be this: possibilist maneuvers against the actualist cause this particular motivation to be undermined. This doesn’t serve as a general attack on possibilism, since there are plenty of independent motivations for the view (I won’t be examining those motivations in this essay). But it does prevent the possibilist from invoking their logical conservatism against opposing metaphysical theories that don’t adopt a possible worlds framework.
With respect to actualism, I will consider their chief motivations and show how they too come to be undermined by a number of theories wishing to stay true to the conservative logical program. The conclusion will be that the possible worlds framework for modal concepts is not amenable to an actualist interpretation. However the actualist can take some heart when this result is combined with that mentioned in the paragraph immediately above since the failure to stay true to the conservative logical program is no mark against any theory whatsoever.
The best place to start is to show the conservative logical motivation for possibilism as it developed historically. The combination of two well regarded logical systems, propositional modal logic and the first order predicate calculus led to the derivation of the Barcan formulas.
(BF) (x)?Ax ? ?(x)Ax
(CBF) ?(x)Ax ? (x)?Ax
We’ll examine first the precise way in which the combination of the different logical systems encourages one to accept the validity of the Barcan formulas along with their metaphysical commitments. There are alternative ways to formulate the logic (a free logic for instance) – but the previous successes of those logics in isolation instils a philosophical conservativism that seeks to keep the combined logic as true to the originals as possible. When all is said and done, this remains the chief reason for the acceptance of the Barcan formulas.
Let us outline a proof theory for quantified modal logic so we can begin to see why this is so. First we define an alphabet which includes an infinite set of variables (x, y, z…) (where further letters are required for expositional purposes, they will be stipulated), an infinite set of constants (a, b, c…), a set of predicate n-ary variables (P, Q, R…), the logical symbols ?, ~, , , ?, ? and the grammatical symbols , and ),(.
We define the set of terms inductively such that all variables (not predicate variables) and constants are terms, and if P is a predicate variable, (t1,…,tn) are terms, then P(t1,…,tn) is a term.
We define the set of WFFs in the following way:
1) If P is an n-ary predicate variable and t1,…,tn are terms then P(t1,…,tn) is a WFF.
2) If A is a WFF and x is a variable, then (x)A is a WFF and ExA is a WFF.
3) If A and B are WFFs then, ?A, ?A, ~A and (A ? B) are WFFs.
(x) and ? are taken as primitive with ? and E being defined in the usual way.
For axioms we add all the tautologies of propositional logic and the following modal axioms depending on the system we are working within:
K: ?(A ? B) ? (?A ? ?B)
M: K + ?A ? A
S4:M + ?A ? ??A
B: M + A ? ??A
S5: M + ?A ? ??A
We include the rules of inference modus ponens plus (in): Ac / (x)Ax / (where c is a constant and does not occur in (x)Ax or any hypothesis for it) and (out): (x)Ax / Ac (where c is a constant). We also include a rule of necessitation; A / ?A.
It’s easy to see that this proof theory is just a simple combination of the two simpler logics. We haven’t added anything here that wasn’t a part of the simpler systems. All the theorems of the previous systems are provable in these systems (allowing for the different modal systems of course), although their combination obviously results in an additional set not previously provable.
Given this formulation we can remark that CBF is a theorem of all the systems listed, while BF is a theorem of systems B and S5. Given that many regard S5 as the best system to represent modal discourse, they see the derivation of BF in this system as reason enough to accept its consequences. Others don’t see the varying formulations as competing systems, but as guides in the process of distinguishing subtle differences in distinct modal notions, hence wouldn’t take this in itself as a convincing reason.
This point is not too important however given that when we come to look at the model theory, it becomes hard to see how CBF could be valid without BF being valid as well. As Linsky and Zalta point out the simplest models of QML all validate BF and if we want our proof system to be adequate to the semantics, then BF must be added as an axiom to the weaker system.
We can see here the conservative motivation in this claim. Williamson echoes the point when he says concerning the complicated nature of the completeness proofs for other kinds of models (particularly the Kripke models): “Such complications are a warning sign of philosophical error.???
It’s an interesting fact of logic that when we combine the model theories in a way analogous to that for the proof theories, the result is that the proof theory is no longer complete with respect to the semantics, as it was in the simpler logics. The simplest combination of the model theories has BF come out as valid. This then provides a very strong motivation to add BF as an axiom to the weaker systems.
To see this we must look at the “simple??? model to which Linsky and Zalta refer. A model structure is a set
The usual Tarskian notion of satisfaction is used to define truth, albeit slightly modified to handle the modal aspect of the model. A function f maps each variable to some element of D. If ‘a’ is a constant then it’s denotation with respect to a function f and a model M is simply the value ascribed to it by V. If ‘x’ is a variable, then the value of x relative to (fM) is f(x) (the value for x given by the function f). Satisfaction is then defined with respect to a model and some function f in the following way:
1) If P is an n-ary predicate letter and t1,…,tn are terms, then P(t1,…,tn) is satisfied with respect to a world w, iff
2) If A is a wff then f satisfies ~A with respect to a world w iff f doesn’t satisfy A in w.
3) If A and B are wffs, f satisfies A ? B with respect to a world w if either it satisfies B at w or fails to satisfy A.
4) If A is a wff then f satisfies (x)A with respect to a world w iff for every f`, that differs from f in at most its assignment to x, satisfies A at w.
5) If A is a wff then f satisfies ?A with respect to a world w iff for every w` such that Rww` f satisfies A at w`.
The ordering R on D will differ depending on the system we are considering. For K there is no restriction on how they must be ordered. For M, the ordering must be reflexive. For S4 it must be reflexive and transitive. For B, the ordering must be reflexive and symmetric and for S5 the ordering must be reflexive, transitive and symmetric.
A wff A is true at a world and for a model iff every f satisfies A with respect to that world. A wff is valid if it is true for all models.
We can again see how this is a simple combination. The notion of a single domain has been retained from the previous first order semantics, along with the set of possible worlds. The definition of satisfaction, truth and validity remain as they were except for the relativisation to possible worlds. The ordering R is defined exactly as it was in the simpler modal system. Of particular interest are the satisfaction conditions given for the quantified sentences. It’s because these remain identical (relativised to a possible world of course) that the actualist scruples arise – as we shall see.
Given this ‘simple’ semantics for QML it’s plain that both BF and CBF come out as valid. Consider BF first. To imagine a counter example we must set its antecedent as true at a world and its consequent false. To make the antecedent (x)?Ax true we suppose that everything in the domain has the property represented by A assigned to it in every (accessible) possible world. But if this is so, the consequent ? xAx must also be true. For it to come out false, (x)Ax would have to be false for some possible world. That is, for some constant c, Ac would have to be false at some possible world. But it can’t – the truth of (x)?Ax guarantees this. A similar process applies for CBF.
Given how this semantics plays out, it is hard to imagine how CBF could be valid without BF also being valid. But CBF is a theorem in all the systems discussed above, so it provides a strong motivation to either accept a quantified B or S5 or add BF as an axiom to the weaker systems.
As far as I am aware, this simple combinatorial procedure remains the only positive reason for the acceptance of the Barcan formulas. This doesn’t mean that it should be underestimated. Given that we aspire to a single logic that is capable of representing all our forms of discourse, to admit defeat here is to force us into the revision of past successes. The notion of validity itself is at stake here, and remains undefined for QML where we don’t have a properly specified semantics. Should it turn out that no alternative conception of the semantics for QML is available, then the choice is to either give up on the existence of a clear notion of validity for QML or to simply allow BF into the fold and accept the standard model structure as given above. The abandonment of a notion of validity for QML would be a hefty price to pay.
Nevertheless, there is no gut feeling behind our acceptance of the Barcan formulas. We don’t parse these formulas into English and immediately think them to be valid. In fact the opposite is true. Our natural inclination is to think them invalid, and it’s this intuitive reaction that gives the actualist position its strength as we’ll see below. This is an important point in the clarification of the nature of the dialectic. My aim is to be really clear about the reasons behind the acceptance of the possibilist position. If the validity of the Barcan formulas implies possibilism by necessity, then it’s important to remember how we got there.
But why do the Barcan formulas imply the possibilist position? Perhaps the easiest way to see it is to examine the following variant which is easily derivable:
(?BF) ?ExAx ? Ex?Ax
On a natural reading of this sentence it seems to say that given that it is possible that some object satisfies A, then it follows that there exists an object which possibly could have satisfied it. But which object would this be? It’s very difficult to identify an object in the actual world that would fit the bill. What’s more, as Linsky and Zalta point out an essentialism commonly held by today’s metaphysicians rules out, at least in some cases, that any other actual (spatio-temporal) object could fulfill that role. Hence it seems that the only course is to accept the existence of an object with a different ontological status, a merely possible object, or in the plural, possibilia, to fulfill the role of truth maker.
There are a number of other important considerations that contribute to this progression toward possibilism. The first is a link that is traditionally expected to hold between one’s semantics and ontology. Our semantic theory should provide, among other things, an explication of all the true sentences in our language. A true sentence, even if trivially so (a married bachelor, for instance, would be a very interesting fellow), represents some feature of the world. Hence an adequate semantic theory would represent a complete taxonomy of existing objects which serve as truth makers for the true sentences. The supposition that there must be possibilia which satisfy the Barcan formulas is motivated in part by this requirement (combined, of course, with a scepticism that an actual object can fulfill this role).
The second consideration involves the traditional understanding of quantifiers. Quine famously argued that we are committed to whatever objects we quantify over in our best theories about the world. Given that the formula above has a modal operator within the scope of the quantifier, the validity of formula commits us to the existence of the possible object. To be, as Quine put it, is to be the valuable of a variable. Another way to express this point is to refer to the common sense reading of the existential quantifier as ‘there exists an x such that???. What else could this phrase mean besides implying existence?
In actual fact, these two considerations are not distinct. The second collapses as an instance of the first. I will argue this point below. I mention them separately because this is how they are often presented in the literature (where the first is ever mentioned at all), and this separate presentation results in an interesting twist in the dialectic which I will also present below. For now it’s enough to note the additional assumptions required to force the progression from the acceptance of the validity of the Barcan formulas, to the acceptance of possibilism as a metaphysical doctrine. The denial of any of them would be enough to avoid the conclusion.
The acceptance of QML as it has been presented above results in a number of other commitments that are traditionally lumped together with possibilism. Returning to the original Barcan Formula – what this says is that if every object necessarily satisfies some sentence or property A, then it’s a necessary fact that every object does so. More interesting is the theorem that Linsky and Zalta refer to as NE:
(NE) Ex?Ey y=x
and also (?NE):
(?NE): ?Ex?Ey y=x
To get these two theorems we have to add the usual extensions for identity to the apparatus included in QML. The proof of these theorems is trivial. x=x is an axiom of QML with identity. Existential instantiation gives us: Ey y=x. The rule of necessitation is then applied along with generalization, then one more application of necessitation to get us (?NE). The first claims that everything exists necessarily. The second claims that this is itself a necessary fact. The reason for their validity is tied to the fact that we have a single domain in our model structure. This implies that every object exists in every possible world – and given the truth conditions for ? – this implies that every object exists necessarily. As Linsky and Zalta note, these two theorems are actually independent of the possibilist position, but are considered as an independent reason by actualists to abandon QML.
It’s important to take stock at this note and make some general observations about the nature of the argument as it has progressed so far. We can see that the progression has been from considerations from a logical/semantic nature to those of a metaphysical or ontological nature. The inclusion of essentialism here is not significant for it merely blocks a convenient way out for the actualist. All the positive reasons for the acceptance of the possibilist position have been logically and semantically related. As I mentioned earlier this is not the only motivation for the possibilist position, but the other motivations are not important to my argument and will not be considered.
There are authors who seem to take the failure to live up to the conservative logical program as a reason in itself to reject a theory. For example, when Linksy and Zalta consider Menzel’s account of possible worlds they argue that it “undermines the rather nice extensional characterization (of the truth conditions) of modal claims???. Likewise, when they consider the views of Salmon and Deutsch they complain that they only avoid various problems at great cost. They go on to say that: “Salmon is forced to adopt the complications of free logic, while Deutsch must abandon the rule of necessitation. The resulting systems are complicated, respectively requiring the logic of nondenoting terms (Salmon) and the logic of contexts and doubly indexed denotation and predication relations (Deutsch)…??? Here again we see that any deviation from the standard logic is seen as a failing. Doubtless they have other complaints about these theories as well but they still see this particular failing as substantive.
We can now turn to the actualist side of the debate. The motivation of the actualist proceeds from a more primitive intuitive basis than the possibilist. While the possibilist was motivated out of a conservative logical/semantic tradition, the actualist is responding directly to the apparent meaning of the questionable theorems. ?BF seems to be false to the actualist. The intuition is that there are no such things as possible objects – period. We’ve never had cause to invoke their existence before, argues the actualist. Why is it that we’re only finding out about them now? Similarly with NE, the actualist just thinks it is intuitively false that everything exists by necessity. We naturally believe of most things that they might not have existed.
But even though the starting motivation of the actualist is more primitively intuitive than the possibilist – still they need to come to the table with at least a little more than just a basic denial of the possibilist position. There has to be some other consideration that grounds the intuition. Presumably part of the concern is epistemological. We don’t have to expect of them (at this stage at least) to have a fully fledged theory of knowledge since we are talking about intuitive motivations. But it is important to recognise that at bottom their worry is that we just don’t seem to have any epistemological access to these entities, whether empirically or through apriori rationalisation and hence have no reason to suppose that they exist.
Also a principle of parsimony is at work here at well. In general, philosophers agree that one should avoid ‘multiplying things beyond necessity’. Of course, necessity is often invoked in philosophic discourse and so one would think the epistemological considerations mentioned above would provide the stronger motivation.
That the actualist has a stronger intuitive basis than the possibilist cannot be denied. In actual fact, the possibilist has no intuitive basis at all as such. As we’ve seen, they are motivated through a loyalty to a logical/semantic program that has been over a century (if not longer) in making – a century of painstaking, highly technical work. It is anything but intuitive. This is in part true of their explanatory agenda as well. They claim to be able to provide better explanations (particularly Lewis) for ordinary modal intuitions. But of course, these are up for grabs – if the actualist can explain as much then their position would be very strong indeed. Nevertheless, the possibilists can provide a coherent semantics for their position and this is no small thing in their favour. Hence the onus falls on the actualist to provide a coherent semantics which does not avail itself of possibilia. Historically this has proved very difficult to do.
The first candidate for an actualist semantic theory is provided by Kripke. He allows for distinct domains at different possible worlds. In the previous model there was just one domain, but in the Kripke model, each W is assigned its own set D of objects. U is taken to be the union of the sets D defined for each W. Intensions for the predicate letters are defined similarly as before. The members of U that don’t appear in D for a world are just not taken as part of the extension of any predicate in that world. Hence the truth conditions for the atomic formulas stipulate falsity for any that refer to objects not existing in that world. No predicates remain undefined, nor is any sentence left without a truth value. Quantification is restricted so that they range only over those objects that exist at the world of evaluation.
Given this model structure, Kripke can provide a very clear explication of the intuitions behind actualist concerns. He asks us to imagine the following model. There are two possible worlds w1 and w2. The first world has just one object, a, the second world has two objects a and b. We define our ordering R on these two worlds such that w1Rw2. We provide an intension for a predicate P such that in w1 its value is ‘a’ and in w2 its value is also ‘a’. As such Pa is true in both worlds. ?Pa is also true at w1 because Pa is true in every accessible world. Because ‘a’ is the only object at w1 it means that x?Px is true at this world also. This makes the antecedent of BF true. ? xPx comes out false because xPx is not true at w2. It’s not true because ‘b’ does not satisfy the property P. Hence the consequent of BF is false, and so is BF.
BF would be true if it were the case that ‘b’ existed at w1 and satisfied the relevant property. In fact, the traditional semantics made it compulsory that the domain be consistent across possible worlds. BF, then, represents what is called the expanding domains constraint – its truth ensures that worlds accessible to the world of evaluation must at most have the same objects (they can have less so long as they are non-empty). A similar counter example can be constructed for CBF. It expresses the related constraint such that worlds accessible to the world of evaluation must have at least the same objects (they can’t have less, but they can have more). The two constraints combined, however, result in a uniform domain across possible worlds.
We can quickly see also why NE and ?NE are not valid with a varying domains approach because objects may no longer exist necessarily in every possible world by stipulation in the model.
Given the fact that this model structure invalidates the Barcan formulas in all QML systems, including B and S5, changes in the proof theory become necessary in order to allow for its soundness and completeness. To do this, Kripke disallows the rule of generalisation to apply to formulas with free variables. Unfortunately, as Linsky and Zalta point out, the derivation of the Barcan formulas can still take place in B and S5 with constants as terms in place of the free variables. In order to prevent their derivation one has to either adopt a free logic or expel terms from the logic. This, of course, is not desirable to those wishing to remain true to the original program.
Perhaps the most significant problem for this kind of model is that the same commitments are still embedded in the meta-language. As a number of authors point out, the meta-language is a simple first order predicate calculus which quantifies over the possible worlds and their domains. But if we’re still quantifying over them in our meta-language, then the same commitments arise as they did in models where the quantification occurred at the object level. We see here that the possibilist is re-visiting Quine’s theory of ontological commitment as mentioned above.
It’s a sobering objection. It reminds us that the original semantic explication of modal operators (even in the propositional theory) – was by means of an analogy that appealed to quantification. A modal operator like ? was just a quantifier over possible worlds. The problem of commitment was alive and well in the propostional modal logic because we were, even then, quantifying over possible worlds. We were committed to the existence of those worlds even though at that point we weren’t thinking of those worlds as being constituted, at least in part, by sets of objects.
One has to wonder, then, how this discussion became about the Barcan formulas. It’s clear that the commitment occurs prior to any sort of quantification on the object level. Any atomic sentence with a modal operator prefixed to the front spells trouble for the actualist as much as the Barcan formulas do.
Perhaps the focus on Quine’s theory of ontological commitment has caused the confusion. Since most philosophers do indeed accept his requirement, everyone has been on the lookout for quantificational commitment wherever they found it. Quantification as it occurs on the object level is just easier to notice than that which goes on at the meta-level. If this is so then it’s worth while making a clarificatory point about the original intent of Quine’s theory. He offered quantification as the mark of the commitment of a theory because he had believed he been able to replace singular reference with a quantificational idiom. Indeed, he believed that quantification was the hallmark of reference, and therefore commitment. It is now widely believed that this sort of reduction cannot be achieved, thanks to Kripke – and so quantification cannot claim to be the sole indicator of commitment. The bottom line is that any theory that employs singular referring terms in its theorems is committed to those objects. Why? Because they are required to make the theorem true.
As such a broader criterion of commitment is needed. This is a prickly issue and not one I can cover adequately here. But for our purposes I want to suggest the following criterion. We are committed to whatever entities we require in the explanation of the truth of our theorems. Such a criterion is inclusive of quantification as a mark of commitment, but allows for direct reference as well. Meinongian worries are indeed a worry for this criterion given the use of terms that don’t refer in true sentences and the like – but it is not within the scope of this essay to consider them. Suffice to say, Quine never dealt with them given the inadequacy of his paraphrastic program.
The emphasis on truth-makers in our theory of commitment is just a re-emphasis of the principle mentioned above that a semantic theory should provide an explication of our ontology. Our best theory-of-the-world is our statement of what the world is. It’s our way of representing objective reality. The semantic theory which underpins this is really just an explication of what this theory means – breaking down its theorems into the component parts and providing us with the elements of which they are composed. It’s these elements which are the things to which we must commit, if we hold the theory-of-the-world to be correct. These are the truthmakers.
It’s at this point that discussion tends to break down. By this I mean that many authors give up on the constraints of the dialectic. Meeting these constraints would constitute the provision of a satisfactory actualist semantics that is compatible with the traditional logic. But many authors express scepticism that this can be achieved. One way or another they abandon the motivating forces as I described them above. Hope is lost that an actualist metaphysics can be reconciled with the traditional possible worlds logical framework. They then present the theories that are liberated from these constraints.
There is great variety in the various ways this is done. I won’t go into specific detail for each, but it’s important to note their general characteristics in order to place them in context with the overall debate. Some abandon the logical constraint and adopt variant systems. Salmon and Garson, for instance, adopt a free logic which prevents the generalization over terms without an existential assumption being satisfied first. This allows for the reintroduction of constants into the logical machinery. Hughes and Cresswell define a three valued system where atomic sentences take the value of neither true nor false at a world when defined for objects that don’t exist at that world. There are a swathe of objections levelled at these approaches, but for our purposes it’s enough to note that given they don’t address the status of the truthmakers in their theories, their attempts aren’t particularly successful. There is no point in giving up one constraint (in this case the logical), unless it gets you a long way down the track in the satisfaction of the other.
Another approach is that of Menzel who abandons the possible worlds approach and replaces them with Tarski models. The upshot of this approach, as noted above is the rejection of the possible world semantics for modality. For Linsky and Zalta this is reason enough to reject the approach. But as I will argue, if it is the case that there is no way to express actualism within the possible worlds framework, then the rejection of the framework in itself is not a reason against it. Menzel’s approach, after all, does obtain the requisite gains with respect to actualism as a result of its sacrifice. Nevertheless, if an approach can be found which respects the original constraints then it should be preferred – and a number of authors claim to have theories that have done this.
We will need to examine some of these theories in order to progress. They do so by addressing the nature of the truth makers invoked by the semantic theory. Naturally they do so in different ways, but there are also strong similarities between them. The Linsky and Zalta view introduces a kind of abstract object to serve as the truth makers, while keeping strictly to the simple QML system. As such they represent a middle of the road position. On the left is Williamson who also stays true to QML but tries to convince us of the acceptability of possibilia. On the right is Plantinga who attempts to save the Kripke models by introducing another kind of abstract object – essences.
Since Linsky and Zalta represent the middle of the road position, we’ll start with them. As mentioned above, they stay true to QML in its simplest form which means they accept the validity of the Barcan formulas. In order to avoid quantification over possible objects, they propose a kind of abstract object to serve as the truth maker. They do this by making a distinction between abstract objects which are necessarily non-concrete and those that are contingently non-concrete. Possibilia are, according to Linsky and Zalta, of the contingent kind. In the actual world they exist and have similar properties as do other abstract objects. But at other possible worlds they become concrete.
Ordinary concrete objects are reconsidered too. They are now only contingently concrete, and can become abstract at other possible worlds where they are not concrete. This is how Linsky and Zalta maintain the consistent domain across possible worlds needed to satisfy the Barcan formulas. NE is also satisfied for a similar reason. Quantification is taken to have no spatio-temporal connotations.
Linksy and Zalta also claim that their re-interpretation of possibilia allows them to avoid commitment to any non-actual objects. I do not think this is correct. It is true that the domain that they posit is only made up on either concrete or non-concrete actual objects and as such quantification in the metalanguage does not do so over non-actual objects. However the existence of the nonconcrete abstract possibilia in the actual world is contingent on the existence of its concrete version in some possible world. As such, their theory falls afoul of the broader criterion of commitment I outlined above. Non-actual possibilia are still required to make our sentences true.
Williamson presents a very similar theory but does so within a possibilist framework. Possibilia in this theory are non-spatio- temporal as in the Linsky and Zalta theory. But they are not abstract objects either. Williamson does not accept the distinction between contingent and necessarily abstract objects. They are just ‘bare possibilia’ – that serve the role of truthmakers in Williamson’s modal semantics.
Actualist intuitions get re-interpreted, so as to make their counterexamples seem harmless. As Williamson says:
The case against BF and BFC assumes that there is nothing that Wittgenstein could have fathered, that there could have been more things than there actually are, and that possibly nothing is the Inn. These assumptions will be called into question.
He calls these assumptions into question by claiming they have a spatio-temporal component to them. While there is no spatio-temporal object which could have been fathered by Wittgenstein, this does not provide a counterexample to the claim that there is a possible object that was fathered by him, because that object turns out to be a bare possibilia. Williamson has no problem with this re-interpretation of actualism since he believes actualism is a confused doctrine from the outset. By explicating their assumptions in terms of spatio-temporality, he believes he is just providing a plausible exegesis of what they originally meant.
Williamson presumably would have no problem with the charge that like Linsky and Zalta, his possibilia require for their existence, the existence of a spatio-temporal version of each possibilia in some possible world. This would be for two reasons. First, he considers his view a possibilist one, unlike Linsky and Zalta who say they their view is compatible with actualism. Secondly, Williamson sees his task as defending the validity of the Barcan formulas, and to do this, all he has to do is provide some object which actually exists to be the truthmaker. This role is played by the bare possibilia. As far as Williamson seems to be concerned – his work is done.
Nevertheless, given the dependence of bare possibilia on the existence of objects which are spatio-temporal in some possible world, there seems to me to be plenty of room for the actualist to object. If the spatio-temporality is what he meant in his initial, intuitive formulations, then he would presumably have no desire to accept those required by both Williamson’s and Linksy and Zalta’s theories.
This is just further evidence to the effect that the Barcan formulas are just not the issue when it comes to the debate between actualists and possibilists. We now have two theories which aim to make the Barcan formulas valid – one is possibilist, and one is (or at least tries to be) actualist.
The third view that we need to look at here is the unambiguously actualist theory of Alvin Plantinga. Like Linksy and Zalta, he uses a kind of abstract object to serve as the possibilia, haecceities or essences. An essence is a property that each individual object has and has necessarily. In another possible world, no object has that essence without also being that same individual. Essences exist necessarily – in all possible worlds. But contingent objects which exemplify their essences do not. In each possible world there is a set of essences which are exemplified at that world. The union of all these sets is the set of all essences. Exemplification is similar to the property of concreteness in the Linsky/Zalta and Williamson views. It indicates that the object with that essence exists in that possible world, although it wouldn’t be correct to say that it actually exists, or even exists non-spatiotemporally. The reasons for this will be explained below.
So far things are sounding fairly similar to the Linsky and Zalta view. But things begin to diverge when we begin to consider truth conditions – particularly for sentences involving quantification. xAx, for example, is true at a world only if some member of the set of all essences is exemplified at that world while co-exemplifying the property expressed by A. Hence quantification ranges only over the exemplified essences. This is distinct from the Linsky and Zalta view where quantification ranged over all the abstract objects, regardless of whether they were concrete or not. What’s more, essences can remain unexemplified in all possible worlds. This is unlike the Linsky and Zalta view because their view requires the contingently abstract possibilia to be concrete in some possible world.
Because quantification in the Plantinga view is only over the exemplified essences, and because some essences remain unexemplified, the result is a variable set of domains of quantification and hence invalidation of the Barcan formulas. This is another point of distinction from the Linsky/Zalta and the Williamson views.
The Platinga view also does not suffer from the problem the other views do about the existence of the abstract objects depending on their concrete versions for their existence. This is because Plantinga is careful to give an account of the possible worlds themselves in terms that are acceptable to actualists. In this case, possible worlds are just another kind of abstract object – maximal states of affairs. To say that an object is exemplified at a world does not mean that that object exists spatio-temporally somewhere, it is simply expressed as being exemplified in the state of affairs that describes it as such.
Again, when we consider the Plantinga view it becomes clearer that the Barcan formulas are not the issue in the debate between actualists and possibilists. We have seen that Plantinga invalidates them by choosing to quantify only over the exemplified essences. But there seems to be nothing preventing him from changing his conception of quantification such that the unexemplified are quantified over as well. After all, the unexemplified have the same ontological status as the exemplified with respect of the fact that they are both essences that have actual existence (as essences) in the actual world. Similarly, there seems to be no reason why Linsky and Zalta couldn’t add a spatio-temporal restriction to their quantifiers to invalidate the Barcan formulas. At least, there seems to be nothing in the debate between possibilism and actualism that compel them either way. There may be independent reasons.
But if the doctrines of possibilism and actualism no longer have anything to do with whether we validate the Barcan formulas then it seems clear we’ve come a long way from the initial motivations that started the journey. In actual fact, we’ve abandoned pretty much all the initial motivations with which the dialectic began. It’s this point which underlies my argument as a whole, so I will examine it in detail.
First, let’s review the journey so far. We began by examining the motivating intuitions behind the two views in question. These motivations served to act as constraints that were expected to be satisfied by any successful theory. Possibilism was motivated out of a logico-semantic conservatism born of previous successes in other contexts. Actualism was motivated out of a concern regarding our epistemological access to possibilia, along with a desire for ontological parsimony. We noted a number of other constraints along the way – the most important of these being the connection that is expected to be preserved between one’s semantic theory and ontology.
The validation of the Barcan formulas was seen as an important part of sticking to the conservative logico-semantic program. But it was this that seemed to force us to accept possibilism and represented a failure to stick to the epistemological and epistemic constraints expressed by actualism. Now it would be fine if the dialectic led us to the position where one or more of these constraints are rejected and a particular side of the argument adopted. (This is what some authors have done, they either reject the epistemological constraint (adopting a full possibilism), the logical constraint (as seen in the case of Garson, Salmon and Hughes and Cresswell), or the semantic-ontological constraint as was noted above (as in the case of Menzel)). But it’s not fine when the positions espoused have totally lost touch with both the motivating forces that got them started in the first place. I accuse the above three positions of doing just that, even though their occupation of the middle ground was an attempt to obey all the discussed constraints.
To see why this is a problem, we need to look in detail at how these theories undermine the motivations that initially started the discussion. Consider first the logico-semantic conservatism that motivated us towards possibilism. There are two ways in which the dialectic has abandoned this as a motivation. Firstly, when it denied that Kripke models were an effective way to satisfy the actualist constraint and made truthmakers the central issue for ontological commitment they made the validation, or invalidation of the Barcan formulas irrelevant to the debate between actualism and possibilism. I’ve already shown at length how they’ve become redundant. But remember how it was the conservative desire to satisfy the Barcan formulas which initially made us lean toward possibilism? We can clearly see how the motivation has been undermined.
The reply might be that even though the Barcan formulas are no longer the issue, still the desire to stay true to the characterization of modal semantics in terms of quantification over possible worlds still has a similar, conservative motivation. It could be argued that this hasn’t yet been undermined. This brings us to the second point. The possible worlds semantics can not be used to motivate possibilism the way we thought its combination with the first order logic did. This is because possibilism is embedded right at the heart of its semantics from the outset. To be truly considered an independent movtivation, the thing to be proven can’t be assumed at the outset. The combination of the first order logic with the propositional modal logic was taken to be a genuine motivation because it was assumed that in the heart of each, there was no commitment to possibilia. It was only their combination (as expressed through the Barcan formulas) that was supposed to cause the problems. But then we learnt that this was not the case. The propositional modal logic contained the possibilist commitments at its core because of the quantification occurring at the meta level – or, said more generally, through the truth makers introduced in that semantic theory. Hence, if it is argued that quantification at the meta-level implies ontological commitment, then it cannot be argued that the propositional modal logic is conservative at all, and hence can’t serve as a motivation to adopt possibilism. A genuine meta-logical motivation would have to be something that is accepted strongly (at least by most parties) and free of the metaphysical position it is trying to demonstrate. Such a motivation would have a possible world semantics as its end product, enshrining possibilism directly.
Now the three theories under current consideration all explicitly (or implicitly) accept that quantification over the meta-language implies commitment. As such they are guilty of undermining the possibilist motivation.
But hang on, comes the reply. What about the other two? Doesn’t the fact that they’ve managed to provide an account of possibilia which satisfies actualist constraints show that the possible world semantics is not committed to non-actual objects at its core? And here is the rub. While the abstract objects posited do satisfy the actualist constraints in the sense that they are considered to be actual – they end up undermining the actualist motivations. Let’s remind ourselves what these motivations were – they were those epistemological worries we mentioned along with a desire for ontological parsimony. Let’s assume for a moment that the desire for ontological parsimony has indeed been satisfied. After all, abstracta are considered to be more parsimonious than possibilia given their abundance in other theories. But what about those epistemological concerns? Well, unfortunately, abstracta are no more epistemologically accessible than non-actual possibilia. And of course, this goes equally for Williamson’s bare possibilia. The question remains concerning how it is these abstracta act upon us to serve as the basis for our modal intuitions. Evidence for the lack of contact we have with these entities is the variety of entities on offer – each of them having different features in each of the three theories. One might respond that this is a problem for a theory of the abstract, that this should not be a problem a theory of modality should have to solve – quite. But it remains a fact that until it is, the use of abstracta as possibilia clearly undermines the original actualist motivation.
The issue in a nutshell is this. Either possible world semantics contains possibilism as an assumption and in which case can’t be used as a motivation for that position, or we posit abstracta to do the work of possibilia, which results in the undermining of the epistemological motivation for actualism. It’s catch twenty two. To adopt a position similar to those three under consideration leads us to undermine the motivations which got us started in the first place.
Let’s summarise. The progression of the dialectic, as we have seen, went like this: 1) suppose a traditional QML which gives the Barcan formulas as theorems, 2) the actualist offers counterexamples to the Barcan formulas 3) the actualist is asked for a semantics which provides a theoretical underpinning to those counterexamples and responds with the Kripke models 4) The Kripke models are shown to have the same commitment problems 5) Accounts of possibilia are re-configured to supposedly suit actualist intuitions but can be done so to either match the Kripke models or the traditional QML.
My argument is the theories that come out of point 5 – do not actually underwrite the intuitions expressed by actualists in point 2. Let’s revisit the counterexample – this one given by Williamson. As modalists and actualists we can accept that Wittgenstein could have fathered someone, but we do not accept what follows from the Barcan formulas that there is something that he could have fathered. We feel we have no means to identify which actual object this could be – and we do not accept the existence of possibilia on account of our epistemological concerns. The Kripke models are shown not to be a good theoretical basis to underwrite those intuitions, and neither are the various species of modal actualism.
I also claim that that the reasons behind point 4 undermine the possibilist motivation behind point 1. QML is not conservative with respect to possibilism. If it is to be taken as an analysis of modality then commitment is embedded in its very core.
I think the salutary course of action at this point is to reconsider the opening moves of the dialectic. I think the proper conclusion of this discussion is that QML cannot be used itself a motivation to adopt possibilism. Support for this decision is to be found in the work of David Lewis. He writes:
When I say that possible worlds help with the analysis of modality, I do not mean that they help with the metalogical ‘semantical analysis of modal logic’… For that job, we need no possible worlds, but which in truth may be anything you please… Where we need possible worlds, rather, is in applying the results of these metalogical investigations. Metalogical results, by themselves, answer no questions about the logical of modality. They gave us conditional answers only: if modal operators can be correctly analysed in so-and-so way, then they obey so-and-so system of modal logic.
The point here is that the work done on meta-logical systems is only applicable if it is decided in advance that the concept of modality is found to correspond to them. Does an individual exist in every possible world? Well, make a decision based upon whatever metaphysical considerations you can avail yourself – then pick your logic to suit. The fact that NE, for example, is a theorem of the traditional QML is not a reason whatsoever to suppose it is true.
I think the proper conclusion of this discussion is that QML cannot be used itself a motivation to adopt possibilism. This is not intended as any sort of blow against the possibilist position. There are plenty of independent reasons for being a possibilist. In summation they amount to its superior explanatory power with respect to our pre-theoretic modal intuitions. The Barcan formulas remain determinative in the sense that they force decisions on the variability of domains – but as we saw, by the time we get to those last three theories, the decision either way to satisfy them don’t seem particularly well grounded in metaphysical concerns. However, if quantification over possible worlds forces us to accept possibilia there seems no harm in taking them as valid. I think the most important conclusion here is that the possible worlds analysis is the proper semantic account for the possibilist. To remain an actualist, one must abandon the possible worlds framework.
But I don’t think this is as bad for the actualist as it sounds. As I see it, the possible worlds framework is just a theoretical account which is an expression of possibilism. It is not a reason to suppose possibilism is true. As Lewis takes pains to point out – the criterion of success of a theory is its explanatory power. He also accepts the importance of ontological parsimony, but prefers explanatory power. Should an actualist theory be found that matches Lewis’ in terms of explanation it will win hands down.
At this point in time I don’t think the actualist should lose heart or feel weighed down by expectations of conformity to the conservative logical picture. Lewis argues that the success of the possibilist picture in explaining modal intuitions is a reason to believe it’s true. But I don’t think enough time has passed for this to be true. It was natural that the account of modality in terms of quantification over possible worlds would be discovered first since it corresponds so closely to the extensional logics that preceded it. If an actualist semantics must be more complex, then it’s only natural that it would be discovered after more effort.
Post a Comment