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  There is, however, at least one way around these difficulties. We could suppose that all facts are totality facts, and that each totality fact is associated with some fundamental property (the sort of thing that we will identify as a “universal” in Chapter 7). Thus, every fact will have the following form:

  The instances of U are (exactly): the x's.

  Or, equivalently:

  The instances of U are exactly the members of class C.

  So, if being a raven and being black are two fundamental properties, there will be totality facts for ravens and for black things. However, there need be no totality fact for the whole universe, or for complicated, non-fundamental properties (like being a black raven). In addition, there need be no atomic facts at all. If Edgar is a particular raven, then the totality fact for ravens will be a truthmaker for the proposition that Edgar is a raven (and similarly for all other atomic predications of ravenhood), since Edgar will be a member of the class of ravens. Now, we won't have any necessary connections between distinct truthmakers. Each totality fact will be logically independent of all the other totality facts. Totality facts will be truthmakers both for positive predications (like ‘Edgar is a raven’) and for negative ones (like ‘Fido is not a raven’). More precisely, the truthmaker for the proposition that Fido is not a raven will consist of the totality fact for ravenhood along with Fido itself. A truthmaker for the proposition that some things are not ravens will be the totality fact for ravenhood, together with one or more particular things that aren't ravens.

  There is now only one necessary connection left: each universal must have only one totality fact. There cannot be two different sets each of which includes all of a universal's instances. However, this isn't a brute necessity, since we could suppose that it is something to do with the nature of universals that explains why no universal can have two different totality facts.8

  We have arrived to a version of Truthmaker Maximalism: Totality Fact Maximalism.

  Def D2.4 Totality Fact. A totality fact is a connection between a fundamental property (or universal) U and a class of entities C that is a strict truthmaker for the proposition that the class C contains all of the instances of U.

  2.1T.3 Totality Fact Maximalism. Every true proposition has a truthmaker, which includes one or more totality facts, possibly together with one or more ordinary existing things. Each universal is associated with at most one totality fact.

  What do we do with the problem of negative existentials? The true proposition that there is no golden mountain isn't a problem: it will be made true (jointly) by the totality fact for mountains and the totality fact for golden things (assuming that these are both fundamental properties). Similarly, the true proposition that all ravens are black will be made true jointly by the totality facts for ravens and for black things. What about a simple negative existential, like the proposition that there are no unicorns? If we assume that being a unicorn is a fundamental property, this could be a problem. The totality fact for unicorns will “say”: the instance of unicornity are: ______. What can we use to fill in this blank? There are no unicorns to put there!

  There are at least six options here, none of which is entirely satisfactory:

  The totality fact for an uninstantiated universal connects the universal to the empty set. This would mean that all totality facts are fundamentally about sets, which are abstract, mathematical objects. This doesn't seem right: the proposition that all ravens are black shouldn't require positing a set, so neither should the proposition that no unicorns exist.

  We could assume that each universal is self-instantiating. So, the universal UNICORN is a unicorn. But this seems to give the wrong answer to the question, How many unicorns are there? (There would be one, rather than zero.)

  There are special negative facts, in addition to totality facts, one negative fact for each uninstantiated universal. These special facts would have to have necessary connections with other totality facts, in the sense that when a universal has an associated negative fact, it cannot have a totality fact (and vice versa).

  There are no uninstantiated universals. So, UNICORN doesn't really exist (as a fundamental, natural property), and so there is no true proposition of the form ‘there are no unicorns’. Many philosophers, following a popular interpretation of Aristotle's theory of properties, have embraced this view. However, it produces some inconveniences, as we shall see when we consider non-existent things in Chapters 12 and 15.

  We could hypothesize that there is an uninstantiated-universal totality fact. This involves treating the property of being a first-order universal with no particular instances as having its own, higher-order universal U*.9 This uninstantiated-universal universal will have its own totality fact of the form: These are all the uninstantiated first-order10 universals: U1, U2, … If T* is this totality fact, then it will be a truthmaker for the non-existence of unicorns, since UNICORN will be among the universals contained in T*. This seems to be a viable solution, although it requires us to complicate our story by adding at least one higher-order universal (a universal with other universals, and not particulars, as its instances) to our theory. In addition, there would have to be necessary connections between the totality facts for second- and first-order universals.

  We could make an exception Maximalism for true propositions asserting that a universal is uninstantiated. These true propositions have no truthmaker at all. Fortunately, these propositions are relatively rare. All ordinary true propositions will still have truthmakers. Still, making an exception undermines the claim that truth is absolutely natural and unified.

  So far, we have been focusing on a special case of negative existentials: those that simply deny that anything in the world has a certain natural property (or universal). Let's look briefly at a more general case, that of universal generalization. To say that everything is Φ, where Φ is some complex property, is equivalent to saying that nothing is not-Φ. However, we can't dodge this problem, as we did in option 4 above, by simply supposing that there is no not-Φ universal, since the proposition that everything is Φ will still exist, even though there is no not-Φ universal. For example, suppose it is true that everything is material or spiritual. This means that nothing is neither material nor spiritual. Since the universals MATERIAL and SPIRITUAL exist, this true proposition exists and requires a truthmaker.

  Similarly, option 3 (special negative facts for uninstantiated universals) won't work in this case, unless we are willing to add special negative facts for every uninstantiated property, no matter how complex. This would result both in a large number of fundamental facts and a very large number of necessary connections between distinct facts, the very things we are trying to avoid.

  Consider a relatively simple universal generalization, such as the truth that all ravens are black. This truth is made true by the combination of the totality fact for ravens and the totality fact for black things (assuming that RAVEN and BLACK correspond to simple universals), since the totality of black things includes the totality of ravens. In fact, for any true generalization of the form ‘all ravens are Φ’, the corresponding truthmaker will consist of the totality fact for RAVEN, plus a truthmaker for each proposition of the form ‘x is Φ’, where x is a raven. More generally, any true universal generalization whose antecedent clause is a conjunction or disjunction of universals will have a truthmaker of the same kind. We can call these the positively bounded generalizations.

  Thus, the only difficult cases are those without antecedents (such as ‘everything is concrete or abstract’) and those whose antecedents that are not positively bounded, such as ‘all non-ravens are beautiful’. Here again, we will have to resort either to the uninstantiated-universal totality fact (option 5), or to the supposition that there are no uninstantiated universals (option 4), or else make an exception to the truthmaker principle for such cases (option 6).

  If we take a variant of option 5, we will need to add still one more totality fact: the Universal Totality Fact, one that lis
ts all of the universals. We can now assemble a global totality fact G, consisting of the totality of all universals, the totality of all uninstantiated universals, and the totality facts for each of the instantiated universals. The union of all of the particulars showing up in the totality facts for instantiated universals will be the complete set of all actual particulars.

  Alternatively, if we suppose that there cannot be any uninstantiated universals (just as there cannot be any particulars that do not instantiate anything), then we could instead make do with a first-order universal totality: a totality fact that includes all of the first-order universals. This universal totality, together with all of the totality facts for the first-order universals themselves, would define for us a universal domain of particular objects. Every particular will belong to at least one totality fact, and the universal totality fact ensures that every universal has been surveyed.

  If we were to take option 6 instead, we could say that unbounded universal generalizations are true, not by having a truthmaker, but by virtue of the non-existence of a fact that would make them false. However, it seems that most of the universal truths in which we are interested in science and ordinary life are positively bounded generalizations. Thus, we could still maintain that all “ordinary” truths have truthmakers.

  2.5 Alternatives to Truthmaker Maximalism

  Given the difficulties with Truthmaker Maximalism, philosophers have proposed three alternative truthmaker theories that do not require classical truthmakers for every truth:

  Atomic Truthmaker Theory. Only logically atomic or simple propositions have (classical) truthmakers.

  Spectral Truthmaker Theory. Logically atomic propositions have spectral truthmakers: entities that make the atomic truth true, not by simply existing, but by existing and having an intrinsic character of a certain kind. (Josh Parsons 1999).

  Truth Supervenes on Being. The truths of the world are fixed by fixing which things exist, and what natural properties and relations those things have (Lewis 2001).

  2.5.1 Atomic Truthmaker Theory

  The two worries we discussed in the last section (concerning logically complex truths and negative existentials) motivate the idea that we ought to restrict our truthmaker theory in a way that upsets Maximalism. Maximalism demands that every truth, even ones that seem to be far from fundamental, have a unique classical truthmaker. Both of the worries above suggest that this is wrong. One might, in response to these worries, simply restrict Maximalism in the most minimal ways possible. For example, you might keep Maximalism except for denying that negative existentials have unique truthmakers. This would accommodate the second worry. You might keep Maximalism except for denying that non-fundamental truths have unique truthmakers. This would accommodate the first worry. In the face of other worries along similar lines, one can just keep restricting one's Maximalism in the most minimal ways possible to avoid the worries.

  If we were to make all these restrictions, the result would plausibly be Atomic Truthmaker Theory.

  2.1T.4 Atomic Truthmaker Theory. Every atomic (simple, positive) truth has a (classical) truthmaker.

  If only atomic truths have truthmakers, how can we account for the truth of complex propositions? Conjunctions (propositions involving ‘and’) and disjunctions (propositions involving ‘or’) pose no real problem. If p and q are simple truths, then we can explain why the conjunction ‘p and q’ is true: it is true by virtue of each conjunct's having a truthmaker. Similarly, the disjunction ‘p or q’, if it is true, is true by virtue of one or the other of its disjuncts having a truthmaker.

  What about negations? If p is a simple proposition, and Not-p is true, what account can we give of its truth? It won't be true by virtue of having a truthmaker. If Not-p is true, then p is false, and so p does not have a truthmaker. This gives us our answer: the ground of the truth of Not-p is to be found in the absence of a truthmaker for p. Not-p is true, not because it has a truthmaker, but because it doesn't have a falsity-maker.

  One interesting fact that follows from Atomic Truthmaker Theory is that the set consisting of the property of being a true complex proposition weakly supervenes on the set consisting of the property of being a true atomic proposition.11

  Def D2.4 Weak Supervenience. A set of properties A weakly supervenes on a set of properties B if and only if it is impossible for any two worlds to agree on which things have which B-properties but to disagree about which things have which A-properties. That is, two situations that are indiscernible in respect of the B-properties must also be indiscernible in respect of the A-properties.

  An important special case of weak supervenience is that in which a set containing a single property, say {F}, supervenes on a set containing another single property, {G}. In that case, we shall say, for simplicity's sake, that F supervenes on G. This means that whether anything is F or not-F is determined by the set of things that are G and the set of things that are not-G.

  For example, you might think that the extension of the property of being a true proposition about clouds is completely determined by the extension of the property of being a true proposition about the location of water molecules in the atmosphere. It is plausible that one couldn't get a difference in truths about clouds without a difference in truths about the location of water molecules in the atmosphere. Once you've settled the truths about the water molecules, you've settled the truths about clouds. Similarly, if one thought that one couldn't get a mental difference without a brain difference, then one thinks that mental properties weakly supervene on brain properties.

  If we know what the set of truthmakers for positive atomic truths is, then we know what the set of positive atomic truths is. Once we know what the set of positive atomic truths is, we know what the set of negative truths is (and similarly for the sets of all the logically complex truths). A negated proposition Not-p belongs to the set of negative truths just in case its positive counterpart p does not belong to the set of positive truths. You can't have a difference in the set of negative truths without a corresponding difference in the set of positive truths, and you can't have a difference in the set of positive truths without a difference in the set of truthmakers. Thus, the property of being a negative truth weakly supervenes on the property of being an existing truthmaker. Once we know which possible truthmakers exist, we know all there is to know about which negative propositions are true.

  If we move from Truthmaker Maximalism to Atomic Truthmaker Theory, do the five arguments for truthmakers still apply?

  1. Catching cheaters. Trenton Merricks (2007) argues that Atomic Truthmaker Theory cannot be used as a weapon against metaphysical cheaters. Atomic Truthmaker Theorists admit that some truths lack truthmakers: all complex truths, including especially negative truths. Thus, Atomic Truthmaker Theorists are themselves guilty of cheating metaphysically. How can they complain when, for example, Presentists deny that past-tensed truths have truthmakers?

  Here is a possible response to Merricks. According to the Atomic Truthmaker Theory, the extension of the property of being a complex truth (being a true negation, conjunction, disjunction, and so on) is completely determined by the extension of the property of being an existing truthmaker (even though only positive, atomic truths have truthmakers). Once you have the information about which possible truthmakers exist, you can determine which propositions (including negative ones) are true. In contrast, assuming Presentism (see Section 20.4), there is little or no reason for thinking that the set of past-tensed truths is determined in this way by the set of truths about what truthmakers exist now. This lack of supervenience of the properties of past- or future-tensed truth on the actual properties of what really exists seems more problematic than the mere absence of truthmakers for every truth. Thus, Presentism seems guilty of somewhat worse cheating than Atomic Truthmaker Theory.

  However, this difference doesn't seem so impressive once we realize that the definition of supervenience has built into it the assumption that negative facts are unproblematic: the
set of negative truths is determined by the facts about which atomic positive propositions are and are not true. By the same token, Presentists can point out that the set of past truths is determined by the set of facts about which propositions are and were and will be true. There doesn't seem to be any difference here. Just as Atomic Truthmaker Theorists help themselves to negation, Presentists can help themselves to the past and future tenses. Ultimately, Atomic Truthmaker Theorists likely cannot appeal to the catching cheaters argument.

  2. Distinguishing ontological and ideological differences. Atomic Truthmaker Theorists can claim that this argument still works, despite the restriction to atomic truths. Two theories differ ontologically if they differ in what truthmakers do and do not exist.

  3. Correspondence Theory. Atomic Truthmaker Theorists must give up, to some extent, their commitment to the Correspondence Theory of Truth, at least in the form we initially described. It is no longer the case that every truth corresponds to some truthmaker. Instead, we could say that every positive atomic truth corresponds to reality, and that the class of all other truths weakly supervenes on the class of atomic truths. This means that truth is a disjunctive or complex property, made up of two or more quite different components. This would come close to supporting deflationism.