Truthmakers and the Converse Barcan Formula

Truthmakers and the Converse Barcan Formula TIMOTHY WILLIAMSON*

Abstract

The paper criticizes the truthmaker principle that every truth is made true by something. If we interpret ‘something’ as quantifying into sentence position, we can interpret the principle as a harmless logical truth, but that is not what advocates of the principle intend. They interpret ‘something’ as quantifying into name position, and the principle as requiring the existence of truthmaking individuals. The paper argues that we have no reason to believe the principle on this interpretation. Moreover, the converse Barcan formula is inconsistent with the existence of truthmaking individuals for contingent truths. Considerations about our ability to count possible truthmaking individuals are used to argue that we should prefer the converse Barcan formula.

The truthmaker principle says that every truth is made true by something. The latter might be called a ‘fact’, a ‘state of affairs’, a ‘trope’: generically, a ‘truthmaker’. The truthmaker principle can make large differences in metaphysics, contracting the class of truths or expanding the class of things. Why accept the principle?

9 1 clarifies some aspects of the truthmaker principle. 92 argues that it is badly motivated by the underlying intuitions, because they do not warrant the metaphysical primacy that it assigns to quantification into name position over quantification into sentence position. 93 discusses the inconsistency between the truthmaker principle for contingent truths and the converse Barcan formula, and prefers the latter. The issue illustrates the philosophical power of a simple system of quantified modal logic with the Barcan formula and its converse, as favoured by Ruth Barcan Marcus.’ This paper is intended to contribute to the tradition she did so much to initiate of modal metaphysics disciplined by the rigour of modem logic.

* University of Edinburgh I Barcan 1946, 1947; Barcan Marcus 1993: 46,97, 195.

Dialectica Vol. 53, No 314 (1999)

254 Timothy Williamson

Q 1. David Armstrong voices the intuition behind the truthmaker principle in what he describes as ‘perhaps the fundamental argument’ of his recent book A World of Stares of Affuirs (1997: 1 15).2 Given a truth (in this case, that a is F), he asks:

Must there not be something about the world that makes it to be the case, that serves as an ontological ground, for this truth? (Making to be the case here, of course, is not causal making to be the case.) (ibid.)

What is the noncausal sense of ‘make to be the case’? For Armstrong, x

In the useful if theoretically misleading terminology of possible worlds, if a certain truthmaker makes a certain truth true, then there is no alter- native world where that truthmaker exists but the truth is a false proposition. (ibid.)

What about the converse? If the existence of x is strictly sufficient for P, does it follow that x makes P the case? If so, then anything whatsoever would make it the case that everything is self-identical, since it is necessary that everything is self-identical. Moreover, if P entails Q, then whatever makes P the case would also make Q the case. Thus whatever makes John a bachelor would also make him male. Some prefer a finer-grained conception of making the case.3

For present purposes we need not fix a sufficient condition for making the case; Armstrong’s necessary condition will suffice. We therefore consider the weaker claim that (necessarily) every truth is necessitated by the existence of something; a proposition strictly implies the existence of something whose existence strictly implies that proposition. We can formalize it as a schema in the language of quantified modal logic:

(TM)

Armstrong’s question expects the answer ‘Yes’.

makes P the case only if the existence of x is strictly sufficient for P:

U(A 3 3 ~ 0 ( 3 y x=y 3 A))

For other sympathetic expositions of the truthmaker principle and historical background see Fox 1987 and (in restricted form) Mulligan, Simons and Smith 1984. Fine 1982 supplies much relevant technical background.

Compare the criticisms of modal accounts of essence in Fine 1994. The two issues are closely related, since on Fine’s view the essence of x is what (noncausally) makes x what it is. He accepts that if x is essentially F then the existence of x necessitates that x is F, but he denies the converse.

Truthmakers and the Converse Barcan Formula 255

here is to be read ‘it is metaphysically necessary that’, since the intended notion of sufficiency in the truthmaker principle is metaphysical. To instanti- ate (TM), replace A by a sentence, in which x must not occur free.“ The phrase ‘truthmaker principle’ will be reserved not for (TM) but for the informal principle enunciated by Armstrong from which (TM) follows.

The initial 0 occurs in (TM) on the assumption that the truthmaker principle is intended as the claim that a truth without a truthmaker is impossible, not just nonexistent. That accords with the status apparently assumed for the truthmaker principle as a metaphysical axiom. But the arguments below would require only minor modifications if the initial 0 were deleted.

Although (TM) captures much of the truthmaker principle, no truth predicate occurs in (TM). That is consistent with its role as an axiom of metaphysics rather than semantics. In the interests of concision we continue to use a truth predicate occasionally when formulating issues about truthmaking? To render the connection with truth in (TM) explicit, we could replace both occurrences of A by a formalization of ‘the proposition that A is true’, which is generally equivalent to A.

We need a quantified modal logic to serve as a background logic for discussion of (TM). We can specify it semantically. We assume an inference is valid if it is truth-preserving throughout every Kripke model for quantified S5 with varying domains (Kripke 1963). Such a model has a set of ‘possible worlds’ at each of which each formula is true or false. Necessity is simply truth in all possible worlds: if A is true at every world in a model M, so is 0 A; if A is false at some world in M, 0 A is false at every world in M (all relative to a given assignment of values to variables). Thus no accessibility relation between worlds is required, and necessity and possibility are not themselves contingent matters. Such an S5 logic is plausible for metaphysical modality, for in explaining the difference between metaphysical possibility and practi- cal possibility we emphasize that the former, unlike the latter, does not depend on the contingencies of one’s situation. The arguments below could be modi-

Quotation marks are omitted when no serious ambiguity results. See Fox 1987: 189 for a similar formulation without semantic ascent, although also

without the initial 0.


fied to suit modal logics weaker than S5 if need be.6 Each world w has a domain of quantification dom(w), which we may conceive as the set of things that exist at that world. In such a model, the quantifiers are restricted to exis- tents. (TM) requires that for every world w at which A is true, there is something de dom(w) such that for every world w*, if & dom(w*) then A is true at w*. Thus if A is true at w and false at w*, by (TM) dom(w) and dom(w*) must be distinct, which requires varying domains. As Kripke showed, the Bar- can formula Vx 0 A x 0 V x A and its converse fail in such models; it would be inappropriate to assume them at this stage in the discussion. Although the possible worlds model theory does not give the most basic explanation of the meanings of the formulas, model-theoretic validity is plausibly taken as a pre- cise sufficient condition for validity on the intended interpretation.

We cannot strengthen (TM) by claiming that a unique object satisfies the condition on x, for if we substitute a tautology for A the result would then be equivalent to 0 3 ! x x=x (the domain of each world contains exactly one individual); 3 !x x=x is empirically false. Some might strengthen (TM) by making the existence of x necessary as well as sufficient for A to hold:

0 (Ax3 XU (3 y x=y = A))

Formally, they could then require uniqueness in (TM’):

0 (Ax3 ! XU (3 y x=y = A))

Armstrong’s combinatorial metaphysics may require him to reject the Brouwersche schema 0 [ 7 A 2 A , derivable in S5, and take S4 as his modal logic (Lewis 1999: 202). But (TM) may lack the intended modal force of the truthmaker principle when the background propositional modal logic is weaker than S5. Consider a model with two worlds w, and w2 and two individuals i, and i,, where both w, and w, are accessible from w,, only w is accessible from w, (so accessibility is nonsymmetric), dom(w,) = (i,, i2) and dom(w2) = [iJ. (TM) holds at both worlds, because i, verifies it at w, (since i, exists at no other world) and i, verifies it at w, (where 0 collapses into truth, and (TM) into A 3 3 x@ y x=y 3 A), which is equivalent to h 3 x x=x). Nevertheless, if p is false at w, and true at w2, the supposed truthmaker iz for p at w, exists at w even though p is false there, while i, makes nothing true at w2 because it does not exist there. kince accessibility in this model is reflexive and transitive but not symmetric, the theorems of S4 are all valid but the Brouwersche schema fails at w, when A = p. Correspondingly, if @ is the ‘actually’ operator, ( p 3 x@ 0 (3 y x = p A ) is false at w, when it is treated as the actual world of the model. Similar problems occur if accessibility is non- transitive. They do not undermine the argument of this paper, which assumes that (TM) is necessary for the truthmaker principle, not that it is sufficient.


We can construct models of (TM'!) by taking any set W as the set of worlds and for each WE W setting dom(w) = (X: WE X c W ) .7 Informally, in these models the domain of a world is the set of propositions true at that world, if we treat propositions as sets of worlds and being true at a world as having that world as a member; we can assign x in (TM+!) the set of worlds at which A is true. This modelling shows that (TM+!) is consistent with arbitrarily large numbers of worlds and of individuals in them given the background logic. Since (TM) and (TM') follow from (TM+!), the consistency proof extends to them. Of course (TM+!) is metaphysically quite implausible. For example, when A is a tautology, (TM+!) reduces to the theologically evocative formula 0 3 !xD3y x=y (the intersection of the domains of all the worlds has just one member: necessarily there is exactly one necessary existent): which is surely false, for if x is a necessary existent, {x) is a distinct necessary existent. Quite generally, if the existence of x necessarily implies that of {x) distinct from x and vice versa, then 3!xU (3y x=y = A) is impossible for any A. (TM+) is much less cramping; it is true in any model in which dom(w)n{X: XGW) = {X: WE XGW} for each WE W (informally: the propositions in the domain of a world are exactly those true at that world), no matter how many nonmembers of {X: X z W ) n dom(w) contains. Nevertheless, friends of truthmakers may find even (TM') too strong. For example, they may hope to dispense with dis- junctive facts; if A is the disjunction of several quite disparate disjuncts, why should its truth be equivalent to the existence of anything? On their view, a truthmaker for one of the true disjuncts suffices by itself to make the disjunction true too. The truthmaker principle should be taken to entail (TM) but not (TM+) or (TM+!).

We might try to liberalize (TM) by letting A be made true by several things collectively; arguably that makes no difference since then their sum would individually make A true (Fox 1987: 189). We might restrict (TM) to a sub- class of truths, such as the atomic ones (Mulligan, Simons and Smith 1984). These weakenings of (TM) would make little difference to the arguments below.

$2. Why should the truthmaker principle seem compelling? If a proposition is true, something must he different from a world in which it is false. In some sense of those words they express a platitude. But the truthmaker prin-

' If individuals are constituents of worlds, presumably they are not sets of worlds. Thus the models may not fit the intended interpretation of the language. That does not matter for their present use as algebraic devices in a consistency proof.


ciple assigns them a quite specific sense. For it treats ‘something’ as an individual quantifier, binding a variable in name position. The truthmaker variable x in (TM) is the subject of a first-level existence predicate defined by identity and a quantifier. In effect, (TM) connects constants in sentence position (instantiating A) with variables in name position. But why should ‘something’ in the platitude be treated as quantifying into name position? Why not treat the platitude as simply connecting the constant A in sentence position with a variable in sentence position? The result would be something like:

If A is true, something must be so that is strictly sufficient for A: but this paraphrase is to be understood in the light of (TM*), not vice versa. On the semantics to be specified below, (TM*) follows easily from this modal com- prehension principle for quantifiers that bind variables in sentence position:

(CP) follows in turn from the analogue of existential introduction for the propositional quantifier:

(El) B[A/p]A p B Here B[A/p] is the result of replacing each free occurrence of p in B by A

(when A is free for p in B). For if B is 0 (PEA), the corresponding instance of (EI) is O(A=A)>3pO(p=A); we can detach the consequent and apply necessitation to derive (CP). (TM*) is a logical truth, which hardly disquali- fies it from expressing a platitude.

Naturally, we cannot interpret propositional quantification in modal logic as quantification over truth-values, as we can in the nonmodal case, otherwise (TM*) would fail when A expressed a contingency (Hughes and Cresswell 1968: 291). But that is not a problem about propositional quantification. It merely reflects the need in modal logic to equate the semantic value of a formula with something like the set of possible worlds at which it is true rather than with its truth-value. The semantics of quantification into sentence position exploits the semantics already used for sentences.8 In the framework of

* Compare Bostock 1998: 32 on quantification into predicate position.


possible worlds semantics, an assignment for the language of propositional quantification in a model with a set W of worlds is a function 3 that assigns each propositional variable a subset of W; the set of worlds at which 3 pB is true relative to a is the union of the sets of worlds at which B is true relative to the assignments differing from a at most over p. Thus propositional quantification is treated as quantification over all subsets of W. As in 5 1, we assume an S5 treatment of 0. This semantics validates (TM*), (CP) and (EI) and even:

For at each world w, p & Vq(q3 O ( p 3 q)) is true relative to an assignment that maps p to (w) . If we could speak of truthmaking, (M) says that there must be a maximally specific truthmaker; it makes all truths true. By contrast, (CP) says that A has a minimally specific truthmaker; it makes no truth stronger than A true. (M) reverses the order of the quantifiers in a universal general- ization corresponding to the schema (TM*):

That the universal quantifier occurs inside the initial in (UTM") is insignif- icant, because the two commute: since the domain of propositional quantification is constant across worlds, the Barcan formula V p O A 3 UVpA and its converse hold. Kit Fine (1970), David Kaplan (1970) and others have studied the logic induced by this semantics. The set of valid formulas is decidable and admits a simple axiomatization.

Does the logical difference between (TM) and (TM*) make any metaphysical difference? Consider the truth that there is no golden mountain. By (TM), something is such that, necessarily, if it exists then there is no golden mountain. What is this thing whose existence prevents that of a golden mountain? It is not the sum of all actual mountains (which are not golden) or of all actual golden things (which are not mountains), for those sums could exist even if there were a golden mountain. If we say 'the absence of a golden mountain', what kind of thing is an absence? A state of affairs? Although we can try constructing a theory of such things, it is natural to suspect an illicit nominalization (compare Armstrong 1997: 196-201 with Lewis 1999: 204-6). Why try to make a noun phrase do the job already done by the sentence 'There is no golden mountain'? Since (TM*) matches the sentence A with the variable p in sentence position, it appears to raise no corresponding problem.


If the propositional quantification in (TM*) could be reduced to individual quantification, then (TM*) might offer no metaphysical advantage over (TM). The formal semantics suggests just such a reduction, for it explains quantification into sentence position in the object-language by using quantification into name position in the meta-language. By postulating a tacit truth predicate applied to each propositional variable in sentence position, we could treat the propositional quantifiers as individual quantifiers restricted to a special class of individuals, the propositions, perhaps treated as sets of possible worlds. The result would be something like:

(TM**) 0 ( A x 3 x(Proposition(x) & True(x) & 0 ( h e ( x ) x A)))

The very phrase ‘propositional quantification’ suggests just such a reduction, for to read 3 p as ‘some proposition’ is to treat it as syntactically analo- gous to ‘some person’.

Since the truthmaker principle implies (TM), propositions clearly do not serve its purpose, for they can exist even when false. If a set of worlds exists at every world, its existence necessitates only necessary truths. But the question is whether propositions serve the purpose of an intuitive conception ten- dentiously expressed in the truthmaker principle and better developed along the lines of (TM*) rather than (TM). It is not clear that they do. For one ele- ment of the underlying intuitive conception seems to be this: truth and falsity supervene on matters prior to truth and falsity (that is a possible gloss on Arm- strong’s phrase ‘about the world’). If (TM*) is merely a notational variant of (TM**), it does not capture that intuition, for the antecedent of the consequent A in (TM**) is True(x). (TM**) does not trace the truth of propositions back to anything intuitively prior. It does not follow that (TM**) is false given the intuition; we just cannot express the intuition by quantifying into name position when the variable in that position is restricted to propositions. But we can express the intuition with (TM*) if it is not synonymous with (TM**); perhaps quantification into sentence position is sui generis and therefore irreducible to quantification into name position. On this view, the use of the variable x in name position in (TM**) forces the introduction of the truth predicate in order to make something that can occupy sentence position; it shows nothing about the underlying conceptual structure of (TM*).

We have some reason to think that quantification into other than name position is not in general reducible to quantification into name position, for given a language with unrestricted first- and second-order quantifiers, a reduction of the second-order quantifiers to the first-order quantifiers would in effect be a

Truthmakers and the Converse Barcan Formula 26 1

one-one mapping of all subclasses of a class to members of that class, con- trary to Cantor’s Theorem; a form of Russell’s Paradox would arise.9 If we do understand the truthmaker principle along the lines of (TM*) (unreduced to (TM**)), then (TM) imports an unwarranted ontology of factlike individuals. On this view, the platitude behind the truthmaker principle is expressible only by means of quantification into sentence position.

Where does irreducible quantification into sentence position leave the model-theoretic semantics for 3p and Vp? It pairs each formula with a condition for its truth in a model formulated with quantification into name rather than sentence position (‘for somdevery subset of W’). We should therefore not regard it as expressing the intended meanings of the formulas. To express those intended meanings, we should need meta-linguistic quantification into sentence position. Analogously, on the grounds that ‘necessarily’ and ‘possibly’ are con- ceptually more basic than quantification over possible worlds, we may deny that the possible world semantics captures the intended meanings of the modal operators; to express those intended meanings, we should need modal operators in the meta-language. Nevertheless, we may be able to use the modal operators to define a derivative notion of possible world and show on that basis that truth in some [every] possible world is logically equivalent to possibility [necessity], and that the possible wolds semantics supplies an extensionally correct test of validity. Correspondingly, we may be able to show on the basis of plausible assumptions that informal validity on the intended reading of 3p [V p] is extensionally equivalent to validity on the model-theoretic account above. At the very least, the model theory provides a consistency proof for (EI), (CP), (TM*), (UTM*) and (M*) in S5. Indeed, it shows that a system for quantification into sentence position with those theorems is a conservative extension of unquantified S5, and therefore commits us to no unwanted con- sequences in the original language.

Quantification into anything but name position is always liable to be mis- understood in a natural language such as English, which cannot express it unequivocally. We can move from ‘She is charitable and her mother was not charitable’ to ‘She is something her mother was not’, but even with the latter sentence we say ‘something’. If we try to be more explicit by saying ‘She has some property her mother did not have’, the nominalization is even clearer;

The requirement that the quantifiers be unrestricted is crucial. If the first-order quantifier were restricted to a set X, the power set of X would constitute a larger domain for quantification into name position. If the first-order quantifiers were unrestricted but the second- order quantifiers did not range over all classes of individuals, there would again be no contra- diction. See Williamson 1999 for a defence of unrestricted first-order quantification.


‘property’ is a noun.1° But it does not follow that any quantification we can understand is reducible to quantification into name position. We sometimes learn new expressions by the direct method. For example, we do not understand the technical expressions of mathematics by knowing definitions of them in nonmathematical terms; such definitions would be equivocal at best, and probably incorrect if taken literally. Rather, we latch onto the mathematical meanings towards which the informal explanations merely gesture. Why should we not do the same for quantification into sentence position? Many advocates of the truthmaker principle, such as Armstrong, do not regard ordinary language as a reliable guide to truth in metaphysics.

This response may seem unsatisfying. A critic may object that quantification into sentence position is either objectual or substitutional. If objectual, it is tan- tamount to quantification into name position with variables ranging over the relevant objects; the detour through quantification into sentence position has gained us nothing. If quantification into sentence position is substitutional, then it is metaphysically neutral, and again the intuition behind the truthmaker principle is left unsatisfied. More precisely, a substitutionally interpreted ‘existential’ quantification is semantically equivalent to the disjunction (possibly infinite) of its substitution instances; (TM*) would therefore be interpreted like the trivial theorem O(A3((A &Cl(Az A)) v(B &U(BI>A)))), although with moredis- juncts. Such a formula evidently fails to capture any intuition behind the truthmaker principle; weakening it by inserting more disjuncts would not help it to do so. An analogy: although one can validate the inference from ‘Atlantis does not exist’ to ‘Something does not exist’ by interpreting the quantifier substitutionally with the name ‘Atlantis’ in its substitution class, that has no signifi- cant implications for the metaphysics of existence.

The objection assumes that all quantification is either objectual or substitutional. An objectual quantifier has objects as values; it can be explained by use of quantification into name position in the meta-language. Thus the revised criticism in effect assumes that all non-substitutional quantification can be explained by use of quantification into name position. We have been given no reason to accept that assumption. We should not assume that all quantification

lo Higginbotham 1998 and Bostock 1998 discuss issues concerning quantification into predicate position in natural language (see also Prior 1971: 34-39 on non-nominal quantification more generally). Perhaps Russell’s insistence in ‘The Philosophy of Logical Atomism’ that one can assert facts but cannot name them (1956: 187-189) is a gesture distorted by natural language at the irreducibility of sentence position to name position. If one uses ‘fact’ as a noun, the question arises why one cannot name the fact that dogs bark ‘Mary’. By contrast, ‘<<Mary. names dogs bark‘ is simply ill-formed, because ‘names’ requires a noun phrase not a sentence as its grammatical object.


is either objectual or substitutional. Why cannot (TM*) involve irreducible non-substitutional quantification into sentence position? It would be non-substitutional because it would permit 3p B to be true for some B even though its instance B[A/p] was false for every sentence A of the language. We could gesture at the reason by saying that some class of possible-worlds was not the truth-condition of any sentence in the language, but that would be a less fundamental explanation than one that itself used quantification into sentence position in the meta-language. We should not even assume that all non-substitutional quantification is interpreted in terms of assignments of values to the variables. For ‘value’ is a noun, not a sentence. To interpret quantification into sentence position in terms of assignments of values to the variables is to explain it by quantification into name position in the meta-language. If quantification into sentence position is irreducible, that explanation is incorrect. The intended semantics for quantification into sentence position will be struc- turally similar to the semantics for quantification into name position, but it will employ quantification into sentence position in the meta-language.

The critic may give up the attempt to reduce quantification into sentence position to quantification into name position, but still suspect the former of somehow engaging less directly than the latter with the world. How could the suspicion be substantiated? The critic might point out that the domain of quantification into sentence position is closed under negation and disjunction, in the sense that these principles hold: (+ Vp3rU(r= p)

(v) vPvq3rO(r= (P v 9)) The critic might think that negation and disjunction have no place in the fundamental structure of reality. One response would be to keep quantification into sentence position but interpret it to falsify (- ) and (v). In possible worlds semantics, that corresponds to restricting the quantifiers to a proper subset of the power set of the set of worlds. For example, if one restricts them to sin- gletons of worlds, then (TM*), (UTM*) and (M) still hold, although (CP), (EI), (- ) and (v) fail except in special cases. But a less concessive response is war- ranted. Why should (-) and (v) be taken to indicate any indirectness in the engagement of the quantifiers in them with the world? Analogously, that sets are closed under relative complementation and union is no reason to treat them as linguistic fictions, or quantification over them as somehow failing to engage directly with reality. Similarly, mereological closure principles for individuals do not imply that individuals are not robustly ‘in the world’. Even if (- ) and (v) have no analogues for quantification over all individuals, why should that


worry one unless one already takes quantification into name position as the standard for engagement with reality? No doubt we must beware of project- ing the structure of our language onto the world: but we do not avoid the dan- ger by taking one syntactic category with more metaphysical seriousness than another.

The irreducibility of quantification into sentence position is a coherent hypothesis. Some evidence for its truth may be found in its ability to capture in (TM*) the platitude misarticulated in the truthmaker principle, without resort to (TM) and the consequent postulation of such individuals of unobvi- ous standing as states of affairs. Of course, none of this provides positive evidence for the falsity of the truthmaker principle or (TM). The argument has been that the obviousness attributed to the truthmaker principle and (TM) rightfully belongs to (TM*). A metaphysician could still propose (TM) as a bold conjecture; it is consistent with (TM*). But one cost of accepting (TM) is that doing so forces one to be less bold in one's quantified modal logic in other respects. 93 explains why.

93. Consider the converse Barcan formula:

(BFC) O V xB (x)xV x OB (x)

Any open sentence may replace B(x). (BFC) is at least plausible, for its denial is equivalent to OV xB(x) &3 x07B(x), which entails 3xO(V zB(z) dk7 B (x)): informally, there is something that could fail to meet the condition expressed by B(x) when everything met that condition; but how could everything meet the condition if that thing did not? Postponing evaluation of this defence, let us note the implications of (BFC) for the truthmaker principle.

Let B(x) in (BFC) be 3 y x=y. Since V x3 y x=y is an ordinary nonmodal logical truth, U V x3 y x=y is a theorem of the background logic. It combines with the relevant instance of (BFC) to imply that everything necessarily exists:

(NE) V xu3 y x=y

By the background modal logic we can eliminate the first 0 in (TM) and distribute the second, which yields A 2 3 xO3 y x=y x 0 A). By nonmodal quantificational reasoning, that combines with (NE) to give b 3 xOA, from which the existential quantifier is eliminable since x does not occur free in A, resulting in A x O A . The converse is of course a theorem of the background modal logic. Thus (TM) and (BFC) together induce modal collapse. The result


is independent of the initial 0 in (TM). Even if A were made true by several things jointly, (NE) would apply to each of them, and modal collapse would still occur. Nor is the difficulty met by restricting (TM) to atomic sentences in place of A, for many atomic sentences express contingent truths. (BFC) does not allow any contingent truth at all to have a truthmaker. The existence of truthmakers for necessary truths would be small consolation indeed. One cannot combine the converse Barcan formula with any truthmaker principle worth having.

Friends of the truthmaker principle may suppose that the choice is easy. For (BFC) combines with (TM) to induce modal collapse via (BFC)'s consequence (NE), that everything necessarily exists. Cannot we dismiss (NE), and therefore (BFC), on independent grounds, without cost to the truthmaker principle? Then we must accept the consequence of denying an instance of (BFC), 3 xO(VzB(z) &,B(x)), despite its appearance of inconsistency. Since B(x) in the relevant case is 3y x=y, we must accept 3xO(Vd y z = y & 4 y z=y), which is equivalent to 3 xO13 y x=y. The Kripke semantics renders 3 xO13 y x=y true at any world unless its domain is a subset of the domain of every world. Informally, if I could have not existed, it would still have been the case that everything existed even if I had not existed, because 'everything' there means everything that would then have existed. Is not that explanation of the failure of (BFC) perfectly satisfactory?

The explanation was given in the nonmodal meta-language of the Kripke semantics; quantifiers over possible worlds replaced modal operators. The quantifiers in the object-language were explained by quantifiers in the meta- language restricted to the domains of the relevant worlds. For example, OV xFx is true at a world if and only if for every world w and every individual d~ dom(w), d is in the extension of F at w. Obviously this clause has its intended effect only if any member of the domain of any world can be assigned as a value of the meta-linguistic variable 'd' . Thus the meta-language has the resources to defined unrestricted quantifiers VU and 3u with semantic clauses like those for V and 3 except that the restriction to the domains of the worlds is dropped. The clauses for V and 3 are well-defined only if those for Vu and 3" are too. In effect, V and 3 were explained as the restrictions of Vu and by an existence predicate E, 'for the domain of a world was informally explained as containing just what exists at that world. Thus V x B(x) and 3x B(x) are equivalent to Vux(Ex 3 B(x)) and 3Ux(Ex & B(x)) respectively. We can give the identity predicate = an equally unrestricted interpretation by having its extension at each world contain <d,d> for each individual d over which the unrestricted quantifiers Vu and 3' range. The formula E x & x=y


(which implies Ey by Leibniz's Law) defines the restriction of identity to the relevant world-relative domain.

Since Vu and 3" do not have world-relative domains, the Barcan formula and its converse hold for them. This is not yet a problem for friends of the truthmaker principle, for they can insist that the quantifier 3 in (TM) be read as restricted, not as gU. Thus (TM) is understood like this:

(TM') U(AX~'X(EX & O(3Uy(Ey& x=y)~A) ) )

Since 3Uy(Ey & x=y) is equivalent to Ex, we can simplify (TM') to:

(TM") O ( A X ~ ~ X ( E X & ~ ( E X X A ) ) )

The trouble with (TM') and (TM") is that they do not capture the idea that something makes A true. For when A expresses a contingent truth, Ex but not 3uy x=y is sufficient for A, for an appropriate value d of x, according to (TM"). What makes A true is not d itself but its having the contingent property expressed by E; 0juy(x=y & 7Ex) is true when d is assigned to x. Although E is read as 'exists', the being of d in a world in which A is false means that d is not a truthmaker for A in the intended sense. The problem is not merely that (TM") is insufficient for the purposes of the truthmaker principle. It is that the principle requires (TM) to hold on an unrestricted reading of its quantifiers; for the reason just noted, (TM) on a restricted reading will not serve. But within the framework of possible worlds semantics in a nonmodal meta-language, (BFC) is valid on the unrestricted reading, so (TM) is invalid.

Friends of the truthmaker principle should therefore reject the Kripke semantics with varying domains; on the reading of the quantifiers needed for the intended interpretation of (TM), it validates (BFC) and therefore falsifies all interesting instances of (TM). They had better hold that the modal object- language can receive its intended interpretation only in a modal meta-language (for example, as in Peacocke 1978). The claim would be that, within that framework, one is no longer forced to accept (BFC) on an unrestricted reading of the quantifiers, and can therefore accept (TM) on that reading: Any appearance in the Kripke semantics of explaining the failure of (BFC) would be lost, for one would need to assume its failure in the meta-language in order to explain its failure in the object-language. Nevertheless, the friends of the truthmaker principle could deny instances of (BFC) on the intuitive grounds that many ordinary things could have failed to exist. If they could have failed to exist, why not truthmakers too?


The supposed intuitive counterexamples to (BFC) arguably depend on an equivocation in the term ‘exist’ (Williamson 1998, 1999). Consideration of our ability to count objects across possibilities strongly favour both the Barcan formula and its converse, and a conception of the modal status of objects can be given to explain how both principles can be true. Those arguments will not be repeated here. A more satisfying response to the rhetorical question ‘Why not truthmakers too?’ is to consider directly whether contingent truthmakers might be counterexamples to (BFC), as (TM) requires.

Let A in (TM) express the proposition that a given particle P is (wholly) in a given place S at a time T. Suppose that P is not actually in S at T. Since P could have been in S at T, we can infer 03 x O(3y x=y2A) from (TM). Infor- mally, there could have been a fact that P was in S at T. Moreover, we seem to have specified exactly which possible fact is at issue; there does not seem to be room for two different possible facts that P is in S at T. Thus we have singled out a unique possible fact, the possible fact that P is in S at T. Let the singular term ‘Peter’ refer to the possible fact that P is in S at T. P could also have been (wholly) in some other place S* at T, where P is not actually in S* at T. By parity of reasoning, there is the possible fact that P is in S* at T; we can use the singular term ‘Paul’ to refer to it. Now consider the set {Peter, Paul). How many members has it? Peter and Paul are not the same possible fact, for necessarily Peter exists only if P is wholly in S at T and Paul exists only if P is wholly in S* at T and therefore not wholly in S at T. Thus Peter and Paul cannot both exist; if Peter is Paul, that implies by Leibniz’s Law that Peter cannot exist; since Peter can exist, Peter is not Paul. Hence {Peter, Paul) does not have just one member. It has at least one member, for Peter E {Peter, Paul). Thus {Peter, Paul) has exactly two members. Consequently, there are at least two possible facts. Therefore, there are possible facts. By (TM), these possible facts do not exist. For Peter exists only if P is in S at T and Paul exists only if P is in S* at T since P is in neither S nor S* at T, neither Peter nor Paul exists. The quantifier in 3y x=y in (TM), read ‘x exists’, must therefore have been restricted to exclude some possible facts. But, as already argued, that quantifier in (TM) must be unrestricted. Thus (TM) has led to incoherence.

To object that ‘Peter’ is ill-defined because there is more than one possible fact that P is in S at T would not help (TM). For, if so, by (TM) there is more than one nonexistent possible fact, and the quantifier in (TM) must again have been restricted to exclude some possible facts, which yields the same incoherence. (TM) requires that there is no possible fact that P is in S at T, even though there could have been a fact that P is in S at T (this implies a failure of the Barcan formula itself). To see why the blanket denial that there are pos-


sible facts is equally unsatisfactory, it helps to state the position more abstractly.

T is a truthmaker for A if and only if it is necessary that if T exists then T makes A true (on its current meaning). Since ‘T exists’ here is simply ‘T is something’ with the quantifier unrestricted, T is a truthmaker for A only if ‘T is something’ strictly implies A. Consequently, ‘Something is a truthmaker for A’ strictly implies A. Thus if A is false nothing is a truthmaker for A. Fur- thermore, T is a possible truthmaker for A (again keeping the current meaning of A fixed) if and only if it is possibly necessary that if T exists then T makes A true. But in S5 the Brouwersche schema OOBx OB is valid, so every possible truthmaker for A is a truthmaker for A.” Thus if A is false nothing is a possible truthmaker for A, because nothing is a truthmaker for A. Conse- quently, the falsity of A completely answers the question ‘How many possible truthmakers are there for A?’. But that is wrong. For example, suppose that particle P contingently lacks spatial location at time T. Let A be the sentence ‘P is somewhere at T’. We can sensibly ask ‘How many possible truthmakers are there for A?’, in a sense in which the mere falsity of A does not answer our question. On one view, the answer is ‘One’; there is simply the possible fact that P is somewhere at T, the only possible truthmaker for A. On another view, the answer is ‘Continuum many’, because there are continuum many places, and for each place S the possible fact that P is in S at T is a distinct possible truthmaker for ‘P is somewhere at T’. Thus, granted the truthmaker principle, there is a coherent position on which A is false but there are continuum many possible truthmakers for A. But equally, granted the truthmaker principle, the falsity of A immediately entails that there are no possible truthmakers for A. We cannot reconcile these results by interpreting the last quantifier as restricted, because for reasons noted above that would not fit the intended interpretation of the truthmaker principle. Therefore, the tru-thmaker principle is incoherent.

I ’ The problem for the truthmaker principle arises even in some simple cases of sets of possible situations each of which is evidently possible from the perspective of the others, so denying the S5 axiom would not significantly help the defender of the truthmaker principle. In particular, objections to the Bourwersche schema from Armstrong’s combinatorial metaphysics that depend on worlds with fewer universals than the actual world are irrelevant, because a difference in the position of a particle need not imply a difference in the class of universals, and we can restrict possibility in our question about possible truthmakers to possibility given the actual class of universals.


Defenders of the truthmaker principle might try to reconstrue the question ‘How many possible truthmakers are there for A?’ by reconstruing its candi- date answers. The most nave idea would be that ‘There are at least n possible truthmakers for A’ means ‘Possibly there are at least n truthmakers for A’. That is clearly wrong, because it does the counting within a world rather than across worlds. For example, the different possible truthmakers for ‘P is somewhere at T’ correspond to mutually incompatible locations for P at T. A more sophisticated attempt would reconstrue ‘There are at least two possible truthmakers for A’ as something like 03x(TMA(x) &03y(x+y & TM,(y))). More generally, ‘There are at least n possible truthmakers for A’ is represented by a formula with n modal operators. But then how is ‘There are at least continuum many possible truthmakers for A represented’? An analo- gous formula would require continuum many layers of embedded occurrences of 0. Although artificial languages can be constructed with such formulas, they are surely not good representations of our understanding of English sentences. The reconstrual strategy therefore looks technically unpromising. Further- more, there is a more general implausibility in the claim that numerical quantifiers are being used in these contexts in some nonliteral way. Rather, we seem to be applying ordinary concepts of cardinality to objects of an extraordinary kind. If the argument of this paper is correct, it is a kind so extraordinary that it has no members. The truthmaker principle and (TM) are false. The truth underlying our intuitions is (TM*), which is fully consistent with the Barcan formula and its converse for quantification into both sentence and name position.

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Dialectica Vol. 53, No 314 (1999)

Truthmakers and the Converse Barcan Formula

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