The notion of obtainment, as it applies to states of affairs, will here be taken as primitive. To a first approximation, one might explicate this notion in terms of truth: a state of affairs obtains just in case it is expressed by a true sentence. However, this is assuming that every state of affairs is denoted by some sentence of the language in question. There may be languages for which this is the case (e.g., Lagadonian ones6), but I will here not assume that we have such a language at our disposal.
At least provisionally, I shall also take as primitive the notion of an attribute’s instantiation by a given sequence of entities, where the sequence in question may have any set-sized length greater than zero. In the special case where the sequence contains only a single entity x, I will simply speak of the attribute’s instantiation ‘by x’, rather than by the sequence that contains only x; and similarly, I will speak of an attribute’s instantiation ‘by entities’ or more specifically ‘by entities x1, x2, . . .’, where x1, x2, . . . are not necessarily pairwise distinct. Every instantiation of an attribute by (a sequence of) entities will be taken to be a state of affairs. Further, to say that some entity or entities x1, x2, . . . (jointly) instantiate a given attribute means that the instantiation of that attribute by x1, x2, . . . obtains.
While I am prepared to take almost every meaningful English predicate to denote an attribute (at least if vagueness is ignored), I will not assume that the predicates ‘obtains’ and ‘instantiates’, though meaningful, denote attributes, or any other entities. The reason for this stems from the Liar paradox: if there were a property of obtainment, there would – given our other assumptions, and given that we allow names of states of affairs to occupy subject- position – also be a state of affairs to the effect that all states of affairs that have a certain
property P do not obtain. And if here P were chosen in such a way that it is instantiated only by that state of affairs, we would have a situation where the latter obtains if and only if it doesn’t obtain. As far as I can see, a similar paradox does not arise if we assume that there exists an instantiation relation, as for instance a ternary relation I whose instantiation by entities x, y, z (in this order) is the state of affairs that x is the instantiation of y by z.7 But even so, I will here assume the existence of neither an obtainment property nor of an instantiation relation.
The notion of an attribute with a given arity can be defined on the basis of the notion of instantiation, as follows: if κ is a set-sized cardinality greater than zero, I will say that an entity x is a κ-ary attribute just in case, for some entities x1, x2, . . . (imagine here κ-many variables written; the entities themselves need not be pairwise distinct), there is a state of affairs that is the instantiation of x by x1, x2, . . . .
The following principle governs the syntax and semantics of basic formulas: (I1) For any terms a and a1, a2, . . . (κ-many variables; κ > 0), the expression
aha1, a2, . . .i
is a formula, and has a denotation if and only if a denotes an attribute A and a1, a2, . . . denote entities x1, x2, . . . such that there exists a state of affairs s that is the instantiation of A by x1, x2, . . .; in which case the expression will denote s.
Note the immediate consequence: if G is a κ-ary predicate and a1, a2, . . . are terms, then the formula pGha1, a2, . . .iq will not denote anything unless G and all the terms a1, a2, . . . have a
7If one does accept the existence of such a relation, one will arguably also have to admit instantiation
relations of higher arity, beginning with a relation whose instantiation by entities x, y, z, w is the state of affairs that x is the instantiation of y by z and w (in this order). My remarks in the text are intended to hold also for these additional relations.
denotation.8,9
In what cases is there “a state of affairs s that is the instantiation of A by x1, x2, . . .”? My general assumption here will be that there is always such a state of affairs, except where considerations about semantic or other paradoxes call for an exception. For instance, if a denotes the property of non-self-instantiation, supposing that there exists such a property, then it might be held that there is no such state of affairs as the instantiation of this property by itself.10 The assumption just alluded to can thus be stated as follows:
8The present treatment of formulas is thus similar to the evaluation of sentences in a neutral free logic
(cf. Priest 2008, §21.7). However, Priest claims that “one would have to make exceptions for the existence predicate itself” (ibid., p. 466); and supervaluationists like van Fraassen (1966) or Skyrms (1968) might wish to introduce further exceptions for the identity predicate or for predicates like pλx(Ghxi ∨ ¬Ghxi)q. By contrast, I intend to make no such exceptions. To be sure, this raises the question of what one should say about negative existentials, such as, ‘Pegasus does not exist’. Tentatively, I would respond to this along the lines of a ‘historical chain view’, as expressed, e.g., in Donnellan (1974). However, the issue may well be more difficult to settle, as one might hold that, in order for a sentence of the form ‘X does not exist’ to be meaningful, the name that takes the place of the ‘X’ will need to have some sort of descriptive meaning and not merely a history. (Consider, e.g., whether ‘The Jabberwock does not exist’ would have had any meaning if Lewis Carroll had never written the poem ‘Jabberwocky’ but instead only begun a poem entitled ‘The Jabberwock’, and had in fact never gotten beyond that title. Similar remarks might be made with respect to nomina nuda in zoological and botanical taxonomy.)
9In this and the following chapters, quotations that make use of corner-quotes should be evaluated according
to a scheme that consists of two rules. The first and main rule is as follows:
(1) All variables (except for sub- and superscripts) should be replaced by their respective referents, provided that they have a referent in the context in which the quotation is evaluated.
For example, in evaluating ‘p∀x(x ∈ S → Ghxi)q’ in a given context, ‘x’, ‘S’, and ‘G’ should all be replaced by their respective referents, provided that they are assigned any in the context in question. On the other hand, for sub- and superscripts, the relevant rule is the following:
(2) If the quotation is evaluated in a context in which the variable ξ is bound to some entity x, then all occurrences of ξ within the quoted expression should be replaced by occurrences of some constant denoting x. (Within any given context, the same constant should be used for replacing all occurrences of ξ, even if these occur in different quotations.)
When evaluating corner-quotations, this second rule should be applied before the first. For example, suppose the expression ‘pGhx1, . . . , xmiq’ is to be evaluated in a context where the variable ‘G’ is assigned a finitary
predicate G and the variable ‘m’ is assigned the arity of G. The first step would be to apply rule (2) and to replace ‘m’ by a numeral denoting the arity of G. Suppose the chosen numeral is ‘17’. In the second step, one would then apply rule (1) and replace the variables ‘G’ and ‘x1’, . . ., ‘x17’ by their respective referents.
My use of single quotes varies between scare quotes and the straightforward sort of quotation that simply yields a name of the quoted expression. In order to mark quotations of material that has already appeared elsewhere, I will continue to use double quotes. Finally, I will for the most part follow the custom of omitting all quotation marks around indented expressions. The respective context will hopefully be sufficient in such cases to disambiguate whether the omitted quotes are double, single, or corner-quotes – or whether no quotes have been omitted at all.
(I2) For any κ-ary attribute A and for any entities x1, x2, . . . (κ-many variables), there exists, barring paradoxes, an instantiation of A by x1, x2, . . . .
Of course, this is less of a principle than a policy, since it does not say how exactly the paradoxes are supposed to be avoided. I shall leave this question to be settled by a more specific metaphysics.
Semantic paradoxes constitute only one reason as to why a formula may fail to denote a state of affairs. Another reason are denotationless terms. For example, suppose we introduce a constant c as denoting whatever attribute is picked out by the predicate ‘consists of phlogiston’, or we introduce c as denoting whatever entity is referred to by ‘the present king of France’. This will plausibly leave c without a denotation, and so one might equally plausibly conclude that no basic formula that contains c will denote a state of affairs, or anything at all.11
The above principle (I1) speaks of the instantation of a given attribute by certain entities. This is not an accident, for I will assume that
(I3) For any attribute A and any entities x1, x2, . . . (κ-many variables), there exists at most one instantiation of A by x1, x2, . . . .12
I will in addition assume that the notions of attribute and state of affairs are even more closely linked, viz., by the following principle:
(I4) For any attributes A and A0, A will be identical with A0 if, for every cardinality κ and all entities x1, x2, . . . (κ-many variables): whenever there is an instantiation of A by x1, x2, . . ., there also exists an instantiation of A0 by x1, x2, . . ., and the latter is identical with the former.
11At this point it may be worth noting that, given the role that the concept of a term plays in the current
framework, this concept should plausibly not be extended to definite descriptions. The reason is that it would not be plausible to say that an expression like pGhdiq, where G denotes a property P and d is a definite description denoting some entity x, simply denotes x’s instantiation of P . For example, suppose that P is the property of being greater than 7 and d is the description, ‘the number of planets in the solar system’. Even though d denotes the number 8, it would not be plausible to regard pGhdiq as denoting the fact that 8 is greater than 7. Failure to heed this tends to open the way (via a ‘slingshot’ argument) to the unwelcome conclusion that all facts are one. (For a discussion of the philosophical significance of slingshot arguments, see Neale (1995) and Oppy (1997).)
In other words: if A and A0 are two distinct attributes, then there will be some cardinality κ and entities x1, x2, . . . (κ-many variables) such that there is an instantiation of A by x1, x2, . . ., but either no instantiation of A0 by x
1, x2, . . ., or, if there is, then the two instantiations will be two distinct states of affairs.13