Abstract
Karen Bennett has recently argued that the views articulated by Linsky and Zalta (Philos Perspect 8:431–458, 1994) and (Philos Stud 84:283–294, 1996) and Plantinga (The nature of necessity, 1974) are not consistent with the thesis of actualism, according to which everything is actual. We present and critique her arguments. We first investigate the conceptual framework she develops to interpret the target theories. As part of this effort, we question her definition of ‘proxy actualism’. We then discuss her main arguments that the theories carry a commitment to actual entities that do not exist. We end by considering and addressing a worry that might have been the driving force behind Bennett’s claim that Linsky and Zalta’s view is not fully actualistic.
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Notes
The structural similarities between the views consist in the fact that the two views agree on the truth of certain logical formulas, even though they interpret those formulas completely differently. We’ll discuss some of these truths below, but see Nelson forthcoming for a fuller discussion.
Linsky and Zalta might have been even more explicit by saying that actualism is the conjunctive thesis that everything exists and is actual. The text following the official characterization of actualism on p. 436, however, makes clear that that is the intended reading.
CN doesn’t include a primitive existence predicate, and so to apply Bennett’s formal definition so that it accurately describes Linsky and Zalta’s view, we defined ‘x exists’ as \(\exists y(y=x).\)
The intended interpretation of I!x requires that it be excluded from the nonstandard theory of predication. An alternative is to introduce higher-level individual essences of individual essences in order to ascribe a property like being exemplified to an individual essence.
The truth conditions for this operator are as follows: \(\cal A\phi\) is true at a world w (in a model M) iff ϕ is true at the distinguished world w 0 (of M). Part of a logic of such an operator is given by the following two logical axiom schemata (Zalta 1999):
-
Axiom 1: \({\mathcal{A}}\phi \equiv \phi\)
-
Axiom 2: \({\mathcal{A}}\phi \rightarrow \square {\mathcal{A}}\phi\)
(We say “part” for reasons given in note 9. A complete logic for this operator might involve axioms that govern the interaction between actuality and other logical notions.) Note that the first axiom, but not the second, is an example of a logical truth that is not necessary. Given such an axiom, one must use a restricted Rule of Necessitation: infer \(\square \phi\) from any line ϕ of a proof as long as ϕ depends only on necessary truths.
-
Given this regimentation, it follows that ‘x is actual’ is equivalent to ‘x actually exists.’ This itself should answer any worry that the theory is committed to mere actualia!
Here is the proof that the above is a theorem. The proof has the necessity of the conclusion as a sub-proof.
1.
\(\square \exists y(y=x)\)
Theorem, simplest QML
2.
\(\square(\exists y(y=x) \rightarrow \square({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
S5 theorem: \(\square\phi \rightarrow \square (\psi \rightarrow \phi)\)
3.
\(\square({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
MP 1,2
4.
\(\forall x\square({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
UG, 3
5.
\(\square\forall x({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
Barcan Formula, 4
6.
\(\forall x({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
T Axiom, 5
Note that the logic of actuality plays no role in this proof, but consider the following proof of the nonmodal claim:
1.
\( {\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x) \)
Logic of Actuality (Axiom 1)
2.
\(\forall x({\mathcal{A}}\exists y(y=x) \rightarrow \exists y(y=x))\)
UG, 1
Note that we can’t apply the Rule of Necessitation to line 2 to derive its necessity, as such an application violates the restriction on the Rule of Necessitation mentioned above in footnote 5.
Bennet says (p. 283):
[T]he only way for such a view to accommodate the possibility of aliens is by postulating proxies, by distinguishing between two importantly different domains of quantification within the actual world. But as I have just argued here, drawing that distinction amounts to giving up on actualism.
This suggests she thinks Aliens should come out true in CN.
The derivation of the negation of Aliens begins with the fact that NE (= \(\forall x \square \exists y(y=x)\)) is a theorem of the simplest QML (Linsky and Zalta 1994, p. 435):
1.
\(\forall x \square \exists y(y=x)\)
NE
2.
\(\square \exists y(y=a)\)
UI, 1
3.
\(\exists y(y=a)\)
T Axiom, 2
4.
\({\mathcal{A}}\exists y(y=a)\)
From 3, by Logic of Actuality (Axiom 1)
5.
\(\square {\mathcal{A}}\exists y(y=a)\)
From 4, by Logic of Actuality (Axiom 2)
6.
\(\forall x\square {\mathcal{A}}\exists y(y=x)\)
UG, 5
7.
\(\neg \diamond \exists x \neg {\mathcal{A}}\exists y(y=x)\)
BF Corollary, 6
The BF Corollary cited on line 7 is:
$$ \vdash \forall x \square \phi \rightarrow \neg \diamond \exists x \neg\phi $$The proof of this is no doubt ugly, but intuitively, note that if the \(\square \) commutes with \(\forall x\) as required by the Barcan formula (BF), then it commutes with \(\neg \exists x \neg,\) so from \(\forall x \square \phi\) (i.e., \(\neg \exists x \neg \square \phi \)), it follows that \(\square \neg\exists x \neg \phi ,\) i.e., \(\neg \diamond \exists x \neg \phi .\) Finally, we prove the negation of Absentees:
1.
\(\exists y(y=x)\)
Axiom, quantification theory
2.
\(\square \exists y(y=x)\)
RN, 1
3.
\(\neg \diamond \neg \exists y(y=x)\)
Df \(\square \) , 2
4.
\(\forall x \neg \diamond \neg \exists y(y=x)\)
UG, 3
5.
\(\neg \exists x \diamond \neg \exists y(y=x)\)
QN, 4
The claim that anything that could exist actually exists is a theorem of the simplest QML enriched by the logic of actuality:
1.
\(\exists y(y=a)\)
Axiom, Quantification Theory
2.
\({\mathcal{A}}\exists y(y=a)\)
From 1, by Logic of Actuality (Axiom 1)
3.
\(\diamond \exists y (y=a) \rightarrow {\mathcal{A}}\exists y(y=a)\)
from 2, by QT: \(\phi \rightarrow (\psi \rightarrow \phi)\)
4.
\( \forall x [\diamond \exists y (y=x) \rightarrow {\mathcal{A}}\exists y(y=x)]\)
UG, 3
What carries the day is the fact that classical quantification theory is a part of the simplest QML. This shows the elegance of the simplest QML. If one were to adopt varying domains (and the logic of actuality employed in this paper), one would have to tamper with classical quantification theory by restricting the Rule of Universal Generalization.
There are other ways of formalizing ‘x is actual’ that do not have the above consequences and that will lead to a form of Meinongianism compatible with the possibilist thesis. We shall not explore these forms of Meinongianism here.
Here is the proof:
1.
\({\mathcal{A}}\exists x\neg E!x\)
anti-possibilist Meinongian axiom
2.
\(\exists x\neg E!x\)
Logic of Actuality
3.
\(\neg E!b\)
Assumption for Existential Elimination (EE)
4.
b = b
=I
5.
\(\exists y(y=b)\)
EI, 4
6.
\({\mathcal{A}} \exists y(y=b)\)
Logic of Actuality, 5
7.
\({\mathcal{A}}\exists y(y=b)\;\&\;\neg E!b\)
& I, 3,6
8.
\(\exists x[{\mathcal{A}}\exists y(y=x)\;\&\;\neg E!x]\)
EI, 7
9.
\(\exists x[{\mathcal{A}}\exists y(y=x)\;\&\;\neg E!x]\)
EE, 2, 3–8
We use EI and EE to refer to Existential Introduction and Existential Elimination.
Indeed, Meinongians would reject the formulation of those principles that we offered in Sect. 3. For those principles capture the intuition that existence is contingent only if ‘x exists’ is regimented as \(\exists y(y=x),\) which, of course, Meinongians reject.
This structural similarity has led Zalta to suggest that formal systems, like the one developed in his 1983 and 1988, which both (a) distinguish the quantifier \(\exists\) from the existence predicate E!x and (b) define the predicate ‘x is abstract’ (‘A!x’) as \(\neg \diamond E!x,\) have two fundamental interpretations: (1) a Platonic interpretation in which the quantifier ‘\(\exists\)’ is given the Quinean “existentially loaded” reading (‘there exists’) and the predicate ‘E!x’ is read ‘x is concrete’, and (2) the Meinongian interpretation in which the quantifier ‘\(\exists\)’ is given the existentially unloaded reading (‘there is’) and the predicate ‘E!’ is read ‘x exists’. Interpretation (1) is Platonic because theorems of the form \( \exists x A!x\) assert the existence of abstract objects. Interpretation (2) is Meinongian because these same theorems only assert that there are abstract objects, i.e., that there are objects that couldn’t possibly exist. There are a variety of considerations, some metaphysical, others concerning the best ways of systematizing our beliefs, that might lead one to adopt one interpretation rather than the other. But the fact is that the interpretations are inconsistent despite the structural similarities.
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Acknowledgements
The authors would like to thank Karen Bennett, Johannes Brandl, Ben Caplan, Gregory W. Fitch, and Bernard Linsky for the comments they made on the first draft of this paper.
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Nelson, M., Zalta, E.N. Bennett and “proxy actualism”. Philos Stud 142, 277–292 (2009). https://doi.org/10.1007/s11098-007-9186-9
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DOI: https://doi.org/10.1007/s11098-007-9186-9