In this section, I first show two problematic kinds of modal statements for modal quantifi-cational language. Roughly speaking, one kind of modal statements involve quantifica-tion over all possible individuals simpliciter or under the scope of some modal operator, and the other kind of modal statements involve comparatives between the actual and the possible. By ‘modal statement’, I mean statements about metaphysical modality.17 Modalists are expected to be able to provide the metaphysical semantics of every modal statement. However, the two problematic kinds cannot be formulated in terms of modal quantificational language, which reflects the problem about modal statements.
Then I discuss how to deal with the problem about modal statements, through which, I first explicate Forbes’s treatment and then provide my treatment which is supplementary to Forbes’s. The aim is to show how to formulate those problematic kinds in a language
17Of course, there are, if understood in a broad sense, other kinds of modal statements, e.g., statements about essential property, disposition, or causation. Even if these kinds of modal statements might be problematic for modalism in other debates, they are not in the debate over modalism and anti-modalism.
For ‘being a possible world’ is not essential to analyses of these kinds of modal statements according to anti-modalism. For every analysis of some modal statement of these kinds available for anti-modalism, there is some parallel analysis available for modalism.
containing modal operators but no ‘being a possible world’ and ‘it is true in w that’.
3.1 Two Problematic Kinds of Modal Statements
We start from the first kind of modal statements I mentions above, which are modal statements involving quantification over all possible individuals simpliciter or under the scope of some modal operator. Roughly understood in terms of possible world discourse,18 statements of this kind are statements which, in a certain world, quantify over individ-uals among many possible worlds. Therefore, statements of this kind can also be called statements about cross-world quantification.19 This terminology ‘cross-world quantifica-tion’ and this understanding would be revealed further if we know how anti-modalists formulate statements about cross-world quantification in terms of a language containing
‘being a possible world’ and ‘it is true in w that’.
Consider the following two cases:
(1) There could have been some individual other than there actually are.20 (2) The rich could have all been poor.21
In the loose sense of ‘there is’, (1) simply means that there are some merely possible indi-viduals, individuals which does not exist but possibly exists. Then it is clear that, in (1), there is quantification over all possible individuals simpliciter. Some possible individual does not exist simpliciter. So (1) is a statement about cross-world quantification. Again, in the loose sense of ‘every’, the intended meaning of (2) here is that possibly, every possible individual which is actually rich is poor. So (2) involves quantification over all possible individuals under the scope of some modal operator. (2) is as well a statement about cross-world quantification.22
18The issue about the usage of possible world discourse will be discussed in the next section, which shows that it is legitimate to make classification in terms of possible world discourse even if the subject matter is metaphysical modality.
19The terminology ‘cross-world quantification’ comes from Kocurek (2016).
20The example is from Hazen (1976).
21The example is from Cresswell (1990).
22From the perspective of the debate over actualism and possibilism, one might think that the two statements can be formulated in terms of possibilist quantifier, so modalism should or could be combined with possibilism. However, the treatment entangles the debate over modalism and anti-modalism with the debate over actualism and possibilism, which is what Fine does in Fine (1977). Unlike Fine’s concern,
The two cases cannot be formulated in terms of modal quantificational language. One might try to formulate (1) in terms of modal quantificational language as follow:
(3) ♦∃x∀y¬(x does not actually exist)
(3) is not a correct formulation of (1). As we have stipulated, the only modal terms in modal quantificational language are ‘’ and ‘♦’. Therefore, ‘actually’ in (3) is a non-modal term, because of which, ‘actually’ cannot make the statement ‘x does not exist’
under the scope of ‘♦’ a statement about what is the case simpliciter rather than about what is possibly the case. But for (3) to be a correct formulation of (1), ‘actually’ has to function in the above way. ‘Actually’ as a non-modal term is merely redundant. So (3) is equivalent to that it is possible that something does not exist, which is false.
One might also try to formulate (2) in terms of modal quantificational language as follow:
(4) ∀x(x is rich → ♦(x is poor))
Both (2) and (4) seem true. However, (4) is true simply because every rich individual is possibly poor. But (2) is true because it is possible that every actual rich individual exists and is poor. Suppose there are only three rich individuals, a, b, and c. For (2) to be true, it has to be possible that a, b, and c all exist and are poor. But for (4) to be true, what is required is only that it is possible that a is poor, it is possible that b is poor, and it is possible that c is poor. Therefore, (4) is not a correct formulation of (2) in modal quantificational language either.
There is no problem to formulate (1) and (2) in terms of a language containing ‘being a possible world’ and ‘it is true in w that’. The language in question can be first augmented with one rigid designator, @, which denotes the actual world. Then one can formulate (1) in terms of a language containing ‘being a possible world’ and ‘it is true in w that’
as follow:
(5) ∃w∃wx(w is a possible world ∧ it is true in @ that x does not exist).23
the concern of this paper is merely the former. Moreover, the entanglement would make the discussion more complex. Therefore, in order to focus on the debate over actualism and possibilism, I assume that actualism is true and would not consider possibilism. That is also why I add the phrases “in the loose sense of ‘there is’” and “in the loose sense of ‘every’”.
23The ‘∃w’ in this formulation is an actualist quantifier relative to w. That is, ‘∃w’ quantifies only
One can formulate (2) as well:
(6) ∃w1∀w∀wx((w1 is a possible world ∧ w is a possible world) ∧ (it is true in @ that x is rich → it is true in w1 that x is poor).
Now it seems that a language containing ‘being a possible world’ and ‘it is true in w that’, unlike modal quantificational language, has no problem dealing with statements about cross-world quantification.
Then consider the second problematic kind of modal statements, statements involv-ing comparatives between the actual and the possible. More precisely, they are modal statements which involve comparatives between one individual as the actual way it is and another individual as the possible way it is. It should be noted that the two things might be identical. Again, we roughly understand this kind of modal statements in terms of possible world discourse. They are statements which compare some individual with respect to one world to another individual with respect to another world. Therefore, they can be called statements about cross-world predication. Statements about cross-world predication compare two individuals with respect to a pair of worlds rather than merely one world.
Consider the two following statements about cross-world predication:
(7) The table could have been the same colour as that chair actually does.
(8) John could have been taller than he actually is.
(7) compares the table as the actual way it is and the chair as the possible way it is. (8) compares John as the actual way he is and John as the possible way he is. For simplicity, we denote the table and the chair in (7) by t and c respectively, and John in (8) by j.
The two statements cannot be formulated in terms of modal quantificational language as well.
One might try to formulate (7) in terms of modal quantificational language as follow:
(9) ♦(t has the same colour as c does).
individuals which are actual relative to w. I take this symbolism because I want to disentangle the debate over modalism and anti-modalism from the one over actualism and possiblism. If you find any difficulty understanding the formulation, ignore the subscript of the quantifier but bear in mind that the debate over actualism and possiblism is irrelevant here.
For, as I have noted, ‘actually’ in modal quantificational language can only be taken as redundant. However, (9) is not a correct formulation of (7) in modal quantificational language. (9) is true. But (9) is true simply because it is possible that t and c have the same colour. The colour possibly shared by t and c need not be the same colour as c actually has. On the contrary, the truth of (7) does not require that it is possible that t and c have the same colour, but only that t possibly has the colour which c actually has. (7) and (9) have different truth condition. If one use the same way to formulate (8) in terms of modal quantificational language as above, the problem would be clearer.
Consider the following statement:
(10) ♦(j is taller than j).
(10) is necessarily false. For one cannot be taller than itself!
The problem arises because, in modal quantificational language, no device can be used to hold fix the characters possessed by some individual in the actual situation. That is, when considering what is possible, the actual characters of some individual cannot be held fixed. Put this idea in terms of a meta-language of modal quantificational language.
Even if some predicate ‘P ’ is in fact satisfied by an individual a. We cannot expect that
‘P a’ is true under all scopes of modal operators. For example, even if Trump satisfied the predicate ‘being the president of America’, ‘Trump is the president of America’ is still false under the scope of ‘’. However, in formulations of statements about cross-world predication, we need a device which makes sure that for some predicate ‘P ’ actually satisfied by a, ‘P a’ is true under every scope of modal operators.
The same problem might as well arise in a language containing ‘being a possible world’
and ‘it is true in w that’. For a language containing ‘being a possible world’ and ‘it is true in w that’ might have the same feature that the actual characters of some individual cannot be held fixed.
However, the same problem does not arise if the set of predicates in the language in question is extended in a certain way. The set of predicate Φ should be extended in such a way that for every n-place predicate ‘Pn’ in Φ (n≥2), we add a corresponding 2n-place predicate ‘P2n’ into Φ. Those additional argument-places in the added predicates are only
occupied by terms standing for possible worlds. For example, for the 2-place predicate
‘being larger than’, we add a corresponding 4-place predicate ‘x in w1 being larger than y in w2’. With those additional argument-places for worlds, we can hold fix the character one individual possesses with respect to any world we want. The statement ‘Allan in w1 is larger than Mary in w2’ then compares Allen as he is in w1 and Mary as she is in w2 by holding fixed Allen’s characters in w1 and Mary’s characters in w2.
Now the formulations of (7) and (8) is possible in a language containing ‘being a possible world’, ‘it is true in w that’ and the extended set of predicates. (7) can be formulated as follow:
(11) ∃w(w is a possible world ∧ t in w has the same colour as c in @ does).
(11) means that there is some possible world w such that t as it is in w has the same colour as c does as it is in the actual world. (8) as well can be formulated as follow:
(12) ∃w(w is a possible world ∧ j in w is taller than j in @).
(12) means that there is some possible world w such that j as he is in w is taller than j as he is in the actual world. So, again, there is no problem for a language containing
‘being a possible world’ and ‘it is true in w that’.
Therefore, the problem about modal statements can be understood as follow:
The Problem about Modal Statements. Modal quantificational language, which contain no ‘being a possible world’ and ‘it is true in w that’, does not have enough expressive power to provide the metaphysical semantics of statements about cross-world quantification and about cross-world predication.
3.2 Treatments of the Problem
Forbes’s treatment of cross-world quantification is to introduce the actuality operator
‘A’ and the pair of the indexed modal operator ‘i’ (or ‘♦i’) and the indexed actuality operator ‘Ai’. ‘A’ means ‘actually’. ‘A’ just functions as ‘actually’ in (1). It is true that φ if and only if Aφ. But it should be noted that it is not necessary that φ if and only if Aφ.
Crossley and Humberstone have constructed a logic system for ‘A’, which includes the following five axioms:24
(A1) A(Aφ→φ)
(A2) A(φ→ψ)→(Aφ→Aψ) (A3) Aφ≡¬A¬φ
(A4) φ→Aφ (A5) Aφ→Aφ
The most important axiom might be (A5), indicating the rigidity of ‘A’. That is, what is actually true could not vary. There could be another logic system for the pair of the modal operator ‘i’ and the actuality operator ‘Ai’, which includes five parallel axioms:
(B1) iAi(Aiφ→φ)
(B2) i(Ai(φ→ψ)→(Aiφ→Aiψ)) (B3) i(Aiφ≡¬Ai¬φ)
(B4) i(φ→Aiφ) (B5) i(Aiφ→Aiφ)
(B5) as well indicates the rigidity of the indexed actuality operator Ai. Under the scope of some modal operator indexed by i, what is actuallyi true could not vary.
The following passage from Stephanou’s article is helpful for understanding the func-tion of ‘A’ and ‘Ai’.
“This function of [‘A’ and ‘Ai’] can be captured in a rule. Each occurrence of [‘A’ or ‘Ai’] neutralizes all modal operators in whose scope it lies, up to the operator (if any) to which it is bound; it neutralizes them in the sense that when we reach the clause it governs, we are talking as if those operators were not present. That is the rule.” (2010, p.172)
That is, ‘A’ and ‘Ai’ are device helping us to say what is true simply under the scope of no modal operator or the corresponding indexed modal operator respectively.
Understood in terms of possible world discourse, ‘A’ is a device which fixes the ref-erence to the actual world. If the statement under the scope of some ‘♦’ is used to say
24See Crossley and Humberstone (1977).
what is true in some possible world, and the statement under the scope of some ‘’ is used to say what is true in every possible world, then the statement under the scope of
‘A’ is used to say what is true in the actual world. The statement under the scope of ‘Ai’ which is under the scope of ‘♦i’ is used to say what is true in the world which is denoted by ‘♦i’.25
With ‘A’ and the pair of ‘i’ and ‘Ai’ at hand, formulations of statements about cross-world quantification have no problem. (7) can be formulated as follow:
(13) ♦∃xA(x does not exist).
With ‘A’, we can talk about what is actually the case in (13), which is essential to the correct formulation of (1). (2) can be formulated as follow:
(14) ♦1A∀x(x is rich → A1(x is poor)).
Again, because of ‘A1’ in (14), we can talk about what is the case under the scope of ‘♦1’ in (14), which makes it possible to formulate (2).
Forbes’s treatment of cross-world quantification is satisfactory even if the treatment might require fundamentality of ‘A’ and the pair of ‘i’ and ‘Ai’. For anti-modalists as well require fundamentality of one more notion, ‘it is true in w1/w2 that’, for cross-world predication. To increase the number of fundamental notions is not a real problem.
Forbes introduces new operators again for cross-world predication. The operators introduced by Forbes are term-binding operator ‘Ax’ and ‘Axi’. The term-binding oper-ators have the same feature of rigidity as their non-term-binding counterparts. But they are also bound to some term position. Put this idea in rough claims. The term-binding operators, on one hand, rigidify what is true simpliciter or what is true under the scope of some modal operator. On the other hand, they are bound to some term position.
Therefore, the term-binding operators can hold fixed how something is simpliciter or
25There is a question whether it is legitimate to explain ‘A’ and the pair of ‘i’ and ‘Ai’ in terms of possible world discourse. The question arises from Melia’s objections. I think it is. For the explanation is merely epistemic. It should not be expected that a metaphysically fundamental notion must be more comprehensible than a metaphysically nonfundamental notion. For example, the notion of ‘being a table’
is more comprehensible than the notion of ‘being a particle’. However, the notion of ‘being a particle’
might be metaphysically fundamental, but the notion of ‘being a table’ is definitely not. For Melia’s objections, see Melia (1992).
However, the explanation without possible world discourse is still available. Forbes himself explains these notions in terms of plural quantification. For Forbes’s explanation, see Forbes (1989, p.90-102).
how something is under the scope of some modal operator. ‘Ax’ and ‘Axi’ mean ‘as x actually is’ and ‘as x actuallyi is’ respectively. Therefore, ‘♦1Ax1Ay(Rxy)’ means that it is possible1 that x as it actually1 is stands the relation of R to y as it actually is.
Then (7) and (8) can be formulated. (7) can be formulated as follow:
(15) ♦1At1Ac(t has the same colour as c does).
(8) as well can be formulated as follow:
(16) ♦1Ax1Ay(x is taller than y ∧ (x = j ∧ y = j).
It is because ‘Axi’ and ‘Ax’ are term-binding terms that (8) has to be formulated in such a indirect way. It seems that, with the term-binding operators, there is no problem about cross-world predication.
However, Forbes’s treatment of cross-world predication could be more satisfactory. It seems that there is close connection between cross-world quantification and cross-world predication. If so, it seems that the metaphysical semantics of statements of the two kinds might have the same constituents. That is, the notions needed by the metaphysical semantics of statement about cross-world quantification might be just the notions needed by the metaphysical semantics of statements about cross-world predication, and vice versa. If a language F can provide the metaphysical semantics of statements about cross-world quantification, F appears to be as well able to provide the metaphysical semantics of statements about cross-world predication, and vice versa. So, it would be good that the two kinds of statements could be formulated in a unified way.26
Even if the term-binding operators and their non-term-binding counterparts have the same feature of rigidity, the two sets of notions are still wholly distinct. The explanation of one in terms of the other is still not available. The treatments could be more satisfactory if the explanation of one in terms of the other is available. In the rest of the section, I would provide an explanation of the term-binding ones in terms of the non-term-binding ones, which makes the treatment more satisfactory.
Consider the distinctions Lewis draws among relations. For Lewis, there are three kinds of relations: internal relations, external relations, and extrinsic relations. Internal
26Kocurek has the same idea. See Kocurek (2016, p.704).
relations are relations which supervene on the intrinsic natures of the relata, e.g., the relation of ‘having the same colour’. The intrinsic natures of the relata either necessitate the instantiation of the relation of ‘having the same colour’ or necessitate the uninstantia-tion of the relauninstantia-tion. External relauninstantia-tions are relauninstantia-tions which are not internal but supervene on the natures of the relata taken together, e.g., spatio relations. The instantiation of distance relations would be determined by the natures of the relata taken together. But the instantiation of distance relations would not be determined by the natures of the relata taken separately. Extrinsic relations are neither internal nor external, e.g., the relation of ‘having the same father’.27
Consider a statement about cross-world predication ‘♦1Aa1AbRab’. Suppose the se-mantic value of ‘R’ in the statement is an internal relation. How could we know whether the statement is true? Fist, consider the normal case. That is, consider the statement
‘Rab’. Suppose that we cannot directly know whether the internal relation R holds be-tween a and b. How else could we know whether R holds bebe-tween a and b? There is one
‘Rab’. Suppose that we cannot directly know whether the internal relation R holds be-tween a and b. How else could we know whether R holds bebe-tween a and b? There is one