Finally, it will be necessary to allow for states of affairs that involve quantification.22 In ana- lytic metaphysics, it has for a long time been the dominant view that ‘existence’ is univocal.23 But there has also been some opposition to this doctrine: on the one hand from those who think that there may well be multiple equally legitimate ways of carving the world into indi- vidual entities,24 and on the other hand from those who think that there may well be multiple kinds, or ‘modes’, of existence.25
Although it will not be necessary for the purposes of this dissertation to take a stance on these issues, I do at least want to leave room for the possibility that there may be more than
22As far as I am aware, quantification-involving states of affairs have first been recognized by Russell (1985,
p. 103), who admitted both ‘existence facts’ and ‘generality facts’. The latter have more recently also been admitted by Armstrong (1997) under the label of (obtaining) ‘totality states of affairs’.
23Cf., e.g., Quine (1948; 1960, pp. 131, 242), and van Inwagen (1998).
24The most prominent recent proponent of this form of opposition is Eli Hirsch (2002; 2005), whose view
has important predecessors in the work of Hilary Putnam (e.g., 1987; 1994) and Rudolf Carnap (1950). For a discussion of the differences between Hirsch and Carnap, see Hirsch (2008). A related line of thought, which goes back to Frege (1884), has to do with the possibility of giving deflationary truth-conditions for sentences that prima facie carry ontological commitment to certain sorts of entity. On this issue, see in particular the work of Agust´ın Rayo (e.g., 2008; MSa).
25See, e.g., McDaniel (2009; 2010), Turner (2010). Both McDaniel and Turner point out that their view has
one philosophically respectable interpretation of the existential quantifier. For this reason, I shall take the unusual step of treating the existential quantifier ‘∃’ as a denoting expression that denotes whatever sense of ‘existence’ is assigned to it by the relevant interpretation. Just what sort of thing these senses of ‘existence’ are will here be left open. On the metaphysically most conservative view, they are simply (equivalence classes of) concepts. In this case, it would perhaps be more appropriate to say that the existential quantifier expresses, rather than denotes, a sense of ‘existence’; but nothing important depends on this choice.
As for universal quantifiers, these will be treated not as denoting expressions, but rather as abbreviatory devices. Thus, if E is an existential quantifier denoting a certain sense of ‘exis- tence’, then a corresponding universal quantifier A may be introduced by stipulating that for every formula ϕ and κ-many pairwise distinct variables v1, v2, . . ., the formula pAv1, v2, . . . ϕq abbreviates p¬Ev1, v2, . . . ¬ ϕq. The universal quantifier that corresponds to ‘∃’ will here, of course, be the familiar symbol ‘∀’.
In order to ensure that the present framework allows for quantification-involving states of affairs, I will adopt the following two principles:
(Q1) For any sense of ‘existence’ S and any κ-ary attribute A, there is exactly one state of affairs to the effect that there exist, in the sense S, some entity or entities that (jointly) instantiate A.26 This state of affairs obtains if and only if there are such entities. (Q2) For any existential quantifier E, the expression pE v1, v2, . . .
| {z }
κ-many
ϕq will be a formula if and only if ϕ is a formula and v1, v2, . . . are pairwise distinct variables. In that case, the expression will have a denotation if and only if E denotes a sense of ‘existence’ E and pλv1, v2, . . . ϕq is a λ-expression that denotes an attribute A; in which case the expression will denote the state of affairs that there exist, in the sense E, some entity or entities that (jointly) instantiate the attribute A.
With (Q1) and (Q2), the list of metaphysical assumptions and semantic stipulations that constitute the present framework is almost complete. It only remains to add two further
assumptions concerning the identity conditions of states of affairs and thus, indirectly, also of attributes. Their formulation, however, requires some additional groundwork, which will occupy us in the first two sections of the next chapter. The assumptions themselves will then be formulated in §3.3.
Chapter 3
Foundations II: Logical Modality
In the previous chapter, we have introduced a system of symbolic expressions for the purpose of referring to attributes and states of affairs. In addition, we have listed some metaphysical assumptions as to what attributes and states of affairs there are. However, it still remains to specify necessary and sufficient conditions for the identity of states of affairs. This lacuna will be filled in §3.3. In preparation for this, I shall need to introduce the concept of necessitation, which will in turn be based on that of entailment. Moreover, I will introduce concepts of ‘logical’ possibility and necessity, as applied to states of affairs. This work constitutes one part of the basis for the elucidation of de re modal discourse that I will propose in §9.3. The other part will be supplied by the concept of essentiality that will be developed in chapters 4 to 7.
3.1
Formulas and ‘Formal Languages’
Several of the principles introduced in the previous chapter (in particular: (I1), (N2), (C2), and (Q2)) state sufficient conditions under which a symbolic expression counts as a formula, and thereby provide part of a definition of what it means to be a formula in the present framework. To complete the definition, I now only have to add a stipulation to the effect that no further expressions are to count as formulas:
(F) An expression is a formula only if it is a formula according to the above principles, and is ‘built up’ in a finite number of steps from atomic constituents.
Atomic constituents include, in particular: all variables, constants, existential quantifiers, logical connectives, and delimiters (i.e., commas and parentheses). For the purposes of the following, abbreviations of formulas will also be treated as formulas, even though they do not strictly count as such according to the present definition.
Just what expressions are to count as variables, constants, or existential quantifiers has so far been left relatively vague. This has made for welcome flexibility, as it has here allowed us – and will continue to allow – to introduce new constants and variables more or less ad libitum. For the purposes of the following, however, it will be convenient to work with a conception of language under which every given language has fixed classes of (respectively) variables, constants, and existential quantifiers. Within the context of the present framework, I shall refer to these languages simply as formal languages. Each formal language L is fully determined by three pairwise disjoint classes of atomic symbols:
• Var L is the class of L’s variables. It is a proper class, i.e., it has as many elements as there are ordinals.
• Const L is the class of L’s constants. It contains at least one symbol, viz., ‘I’.
• Quant L is the class of L’s existential quantifiers. It contains at least one symbol, viz., ‘∃’.
Further, every formal language L is itself a proper class of expressions – viz., precisely of those expressions that count as terms under the definitions of the previous chapter, and under the assumption that all and only the members of Var L are variables, all and only the members of Const L are constants, and all and only the members of Quant L are existential quantifiers.