15.1.2 Non-indexical reductions of actuality
One could argue that actual truth is not a variety of truth, or a way of being true, but simply truth simpliciter, truth without qualification. Potential or possible truth, in contrast, is not a kind of truth, but only the possibility of or potentiality for truth. So, one might argue that the thought behind Actual-Truth-Defined Anti-Indexicalism is a kind of reductionism that reduces actuality to the non-modal notion of truth (simpliciter). That is, the view is better expressed as Truth-Defined Anti-Indexicalism:
15.2T.3 Truth-Defined Anti-Indexicalism. A possible world is actual if and only if it corresponds to the class of true (simpliciter) propositions.
Though Truth-Defined Anti-Indexicalism is a form of reductionism about actuality, it is not Modal Indexicalism because it denies that all possible worlds are metaphysically equal. It is a form of Anti-Indexical reductionism. Plantinga (1974), for instance, seems committed to something like Truth-Defined Anti-Indexicalism, though he formulates his view in terms of states of affairs rather than propositions. He says that obtaining and truth are fundamental notions. Possible obtaining and possible truth, and actual obtaining and actual truth are to be understood in terms of obtaining and truth simpliciter (Plantinga 1974: 48–49). Adams (1974) identifies the possible worlds with classes of propositions and takes the actual world to be the class containing all and only true propositions. At least two Magical Abstractionists, then, are committed to a form of reductionism about actuality that looks very much like Truth-Defined Anti-Indexicalism.
Those who accept some version of the correspondence theory of truth, in particular, those who embrace Truthmaker Theory (2.1T/2.1A.1T/2.1A.1A.1T) in one form or another, would have good reason to be unsatisfied with taking truth as a fundamental, monadic property of propositions. They would instead want to think of truth as a property possessed by a proposition in virtue of the obtaining of the binary relation of truthmaking between something that exists and the true proposition.
We could then define the actual world as the one that contains the existence of absolutely everything, and according to which nothing further exists. A non-actual world can fail to be actual either by not including the existence of everything or by including the existence of something that doesn't exist, or both. Worlds that don't include everything are quantificationally deficient. Worlds that contain something extra are not reality-bounded.
Def D15.1 Quantificational Deficiency. A world w is quantificationally deficient if and only if something would not have existed, had w been actual.
Def D15.2 Reality-Boundedness. A world w is reality-bounded if and only if there are some things that would have been (collectively) the totality of all that would have existed, had w been actual.
To be clear about what we're suggesting here, we need to look back to the discussion of Actualism (12.1T) and Anti-Actualism (12.1A). We're not saying that a quantificationally deficient world w contains things that both would have and would not have existed according to w. Rather, we're saying that something (in effect, quantifying over the things in the actual world) would not have existed had w been actual. Similarly, we are not saying that in a non-reality-bounded world w there is a class that contains more than everything (in world w). Rather, we're saying that, had w been actual, there would have been things that would not be identical to anything (in the actual world). We have to understand the ‘something’ and ‘no things’ in Def 15.1 and Def 15.2 as quantifying over all of reality.
Using these definitions, we can define actuality in terms of existence:
15.2T.4 Existence-Defined Anti-Indexicalism. The actual world is that unique possible world that is reality-bounded and not quantificationally deficient.
Actualists should favor Existence-Defined Anti-Indexicalism, while Anti-Actualists must reject it, since Anti-Actualists believe that some things don't exist (see Section 12.1). Existence-Defined Anti-Indexicalism also presupposes that there are no merely possible worlds agreeing exactly with the actual world with respect to what exists. This will be true only if we adopt Classical Truthmaker Theory (2.1T), rather than Non-Classical Truthmaker Theory (2.1A.1T) or Truth Supervenes on Being (2.1A.1A.1T).
What can Actualists say about worlds that are not reality-bounded? Again, these are worlds that contain the existence of everything, but that, in addition, contain the existence of things that don't actually exist, like twenty-first-century dodos or children of Pope John Paul II. Actualists cannot say that some things exist according to those worlds that do not exist in the actual world. For in saying that, Actualists would commit themselves to the existence of things that do not exist in the actual world, and this is exactly what they refuse to do! Instead, they must say that, if those worlds were actual, there would have been things in existence that do not in fact exist.
Anti-Actualists, it turns out, can use existence as the basis for actuality. They could suppose that actual existence is existence simpliciter, while merely possible existence is merely the possibility of existence. The actual world could be that world according to which exactly those things exist that exist simpliciter. This is Existence-Simpliciter-Defined Anti-Indexicalism
15.2T.5 Existence-Simpliciter-Defined Anti-Indexicalism. The actual world is that unique world w such that, for every x, x exists according to w if and only if x exists simpliciter.
Even if some things don't exist, we can still identify the actual world by finding the world that ascribes existence to exactly the right things (the things that really exist, that exist simpliciter). On this view, to exist possibly but not actually is not a kind of existence. The only way to exist is to exist actually. Merely possibly existing isn't a kind of existing, it's merely the attribute of the sort of thing that might have existed but doesn't. However, there is some plausibility to the idea that merely possible existence is a kind of existence. If so, actual existence would also be a kind of existence, and so we could not simply identify existence simpliciter with actual existence.
Nonetheless, we might take actual existence to be the focal meaning of ‘existence’. On this view, the word ‘exists’ has a number of different but analogous meanings, just as the word ‘health’ does. We describe many things as healthy, from people to diets to urine. These different uses have different but interrelated meanings, and they are interrelated because they are anchored in a single, focal meaning, namely, that of being a healthy human being or organism of some kind. A diet is healthy because it promotes health in this focal sense; urine is healthy because it indicates health in the focal sense. Similarly, we might suppose that only actually existing things exist in the focal sense of the word. Merely possible things exist in some perfectly appropriate sense of ‘exist’, but not in the focal meaning of ‘exist’.
What makes actual existence more fundamental than merely possible existence? Many philosophers, beginning with Aristotle, who introduced the distinction between the actual and the possible, have taken it to be obvious that the merely potential is always dependent on or grounded in what is actual, including the powers and tendencies of actual things.
Priority of the Actual. There are no metaphysically fundamental truths about what is merely potentially the case. All truths about mere potentiality depend metaphysically upon truths about what is actually the case.
If the Priority of the Actual is correct, then a better approach for the Anti-Indexicalists looking for a reductive account of actuality would be to appeal to the class of fundamental truths. Given the Priority of the Actual, we could define the actual world as that world containing all fundamental truths.
15.2T.6 Fundamental-Truth-Defined Anti-Indexicalism. The actual world is that unique world w such that every fundamental truth is true in w.
This view of actuality has the advantage that it will work for Abstractionists whether or not they are Actualists. Further, given that Fundamental-Truth-Defined Anti-Indexicalism is motivated by the Priority of the Actual, it is plausible that Fundamental-Truth-Defined Anti-Indexical
ists will want to be reductionists about not just actuality, but modality more generally. If we want to ground facts about actuality in fundamental truths, and facts about possibility (a kind of potentiality) in what is actually the case, then we should ground facts about possibility and necessity in fundamental truths as well.
Whether Fundamental-Truth-Defined Anti-Indexicalism is plausible depends on the account it can give of the truth of propositions of the form ‘Possibly p’. Suppose that it is actually the case that p is possibly true. Fundamental-Truth-Defined Anti-Indexicalists must then argue that the possible truth of p is somehow grounded in the actual existence of something, something whose existence is contained by the actual world. In fact, reductionists about actuality who are committed to Truthmaker Theory must say that the possibility of every possible world, other than the actual one, is grounded in some fundamental truth contained within the actual world, and that the actual world is unique in this respect. No other world contains every truth that is fundamental.
Here again, Fundamental-Truth-Defined Anti-Indexicalists must appeal to what is fundamental simpliciter. Any possible world could have been actual, and so any world could have been fundamental. However, only one world is in fact fundamental, in the sense that every fact about other worlds is grounded in some truthmaker that exists according to the actual world. The actual world is (in fact) unique in this respect.
The most plausible approach for Fundamental-Truth-Defined Anti-Indexicalists to take grounds all propositions about mere possibilities in truths about the powers and tendencies of actual things. We'll take this issue up in Section 15.3 below, in which we develop an Aristotelian theory of modality.
15.1.3 A dilemma for Magical Abstractionism
David Lewis (1986a) raised a powerful objection to Magical Abstractionism. If propositions represent ways that things are, then the way things are “selects” certain propositions. The proposition that grass is green represents that grass is green, and since grass really is green, the way things are (which includes the fact that grass is green) selects the proposition that grass is green. Magical Abstractionists, given the Uniqueness of the Actual World, maintain that one possible world is uniquely selected by the universe (where the universe is here understood very broadly so as to include not just us and our physical surroundings, but any immaterial and abstract objects as well). The world that is selected in this way is thereby the actual world. Selection occurs because the actual world represents just what the universe is like. Had things been different, the universe would have selected a different world. So far, so good. Now, consider whether selection an internal or an external relation. That is, is which world is selected determined by the intrinsic characters of the universe and the actual world (in which case selection is internal, see Def 2.2) or is it not so determined (in which case selection is external)? Lewis believed both answers are problematic and, therefore, that Magical Abstractionism faces a dilemma.
Suppose that selection is an external relation. Thus, whether it holds between the universe and a world is not a function of the intrinsic features of the way things are and of possible worlds. Lewis thinks that, if this view of selection is true, then the selection relation must be “magical”. What did he mean by this? Everyone agrees that worlds, and propositions more generally, have their representational features essentially. The actual world, Alpha, could not represent things as being a different way than it in fact represents them as being any more than the proposition that grass is green could fail to represent that grass is green. (This is just Essentiality of World-Content from Section 14.2.) This point is almost too obvious to state. It would be strange indeed to think of the proposition that grass is green representing that grass is blue, or worse, that there are exactly 17 electrons in the universe. One would be right to wonder what a person meant by “represent” if they were to make such a suggestion! So, what a world represents is essential to it. But selection is just the dual of representation; the two go hand in hand. Propositions are selected if and only if what they represent in fact occurs. Thus, a certain world must be selected, given that things are are a certain way, and given the representational features of worlds. What is the problem? According to Lewis, the problem is that every uncontroversially external relation is contingent. Consider spatial relations, a paradigmatically external relation. Just given the fact that two things stand in a certain spatial relation does not guarantee that they stand in that spatial relation. A necessary connection that isn't anchored in the intrinsic natures of the two relata would be a brute, inexplicable necessity. Given that there are literally infinitely many worlds, not to mention propositions generally, the view that selection is external requires infinitely many brute, inexplicable necessities. This is a wild violation of Ockham's Razor (PMeth 1.2 and 1.3). Lewis goes a bit further, and suggests that if selection is both necessary and external, then the relation would be magical, because there is nothing to explain why it obtains between this pair of things and any other pair. Lewis's is insisting that no relation can be both external and necessary, in the above way.
Suppose, then, that selection is internal. And it seems that selection is an internal relation. Indeed, that is how we introduced it. Given the way things are, given the intrinsic character of the universe, Alpha could not fail to be selected. And had the universe been different, Alpha couldn't have been selected. We might think of it this way. Given that worlds must represent what they in fact represent, selection must be an internal relation. But if selection is internal, then it's a relation that holds in virtue of the intrinsic characteristics of worlds and the intrinsic characteristics of the universe. So worlds must exemplify a rich variety of intrinsic features. Presumably, these intrinsic features are just representational features. To simplify, consider the proposition that grass is green. This proposition represents that grass is green, which is to say, it has the representational property of representing that grass is green. In virtue of this representational feature, this proposition is selected if and only if grass is green. Magical Abstractionists, given that they say nothing more about these representational features, have here “danced around a tiny circle”, according to Lewis (1986a: 178). We thought we were getting an account of the selection relation, and Magical Abstractionists have offered us representational features. But in order to understand what these representational features are, Magical Abstractionists point us back to selection. We cannot, therefore, come to understand what these representational features are without understanding the selection relation, and we cannot understand the selection relation without understanding what these representational features are. Thus, we have no way to truly understand the selection relation. If we are able to grasp it, Lewis says, we must have magical powers. Whether the selection relation is internal or external, Magical Abstractionism involves magic. And philosophical views oughtn't involve magic.
Lewis's objection presupposes a certain view about how it is possible to think about properties and relations: the Acquaintance Model. According to the Acquaintance Model, it is possible to think about a property or relation only if we are acquainted with the property in our experience, only if our experience presents things to us as having this property or relation.
Acquaintance Model. It is possible to have thoughts about a property P only if one's experience sometimes includes the appearance that some things have P.
Lewis takes it as obvious that our experience never includes any appearances involving abstract possible worlds.
Even if we were to accept the Acquaintance Model, its relevance to this case depends on how we are to understand the notion of experience. If experience is limited to something like sensory experience, then it is certainly true that we never experience mere possibilities as such. However, it seems plausible to say that we also have intellectual experiences and that in those experiences certain abstract entities, like numbers, pure sets or possible worlds, do appear to be a certain way to us. If so, Magical Abstractionists can argue that a possible world c
ould appear to us to represent the existence of golden mountains or talking donkeys, either by virtue of its intrinsic character or by virtue of an external representation relation.
In addition, Peter van Inwagen (2001: 238–242) offers a tu quoque response to Lewis (a response that shows that Lewis's overall theory is subject to the same objection Lewis lodges against Magical Abstractionism). According to van Inwagen, Lewis is also committed to the reality of “magical” representation. The problem is that Lewis accepts standard set theory. Consider the unit set or singleton whose one and only member is the Eiffel Tower. This set represents the Eiffel Tower, since by its very intrinsic nature it is such as to “select” the Eiffel Tower and only the Eiffel Tower as a member. However, it is highly implausible to think that sets represent their members in anything but a primitive way. Sets are supposed to be simple, structureless things. The set membership relation is either magical, or we require magical powers to grasp it. Either way, Lewis is subject to the same dilemma that he puts to Magical Abstractionists.
15.2 Structural Abstractionism
If one is dissatisfied with Magical Abstractionism's silence about the way that worlds represent but wants to avoid Concretism, it seems that the best account of how worlds represent will have something to do with the way that worlds are “put together”. In other words, one will want to attribute to worlds a certain structure, where that structure will explain how it is that different worlds represent different things. This is Structural Abstractionism:
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