Abstract
I argue there are some objects which do not respect the Law of the Excluded Middle (LEM), i.e., which are such that, for some property F, the disjunction Fo v ~Fo fails to be true. I call such objects “odd objects” and present three examples—fictional objects, nonsort objects, and quantum objects. I argue that each of these objects is best understood as violating LEM. I, then, discuss Jessica Wilson’s (LEM-respecting) account of metaphysical indeterminacy. I show how the indeterminacy which arises with odd objects can be accounted for on Wilson’s account. I, then, argue that my Wilson-inspired, but non-LEM-respecting, account of metaphysical indeterminacy is superior to Wilson’s in terms of costs and benefits.
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Notes
Authors differ over how to interpret LEM, see Bednarowski (1956), Bourne (2004), Cooper (1978), Geach (1956), Horn (2018), Restall (2001), Toms (1941), Woolhouse (1967), Tennant (1996), and Tye (1989). Some take it to concern the truth-value of propositions, LEMproposition: □(S v ~S), necessarily, either proposition S is true or its negation is true. Others take it to concern the instantiation of properties, LEMproperty: □(Ʉx)(Px v ~Px), necessarily, either object x instantiates property P or it's not the case that object x instantiates property P. My concern in this paper will be to argue that there are counterexamples to LEMproperty. I suspect that, modified to concern the truth-value of propositions about objects rather than the instantiation of properties by objects, these counter-examples are also counterexamples to LEMproposition. But I won't argue this here. My concern here is solely with whether there are counterexamples to LEMproperty (henceforth abbreviated to "LEM").
Other views of fiction include possibilism, Meinongianism, and fictionalism (Kroon and Voltolini 2018).
Here the A-operator stands for "ascribed to".
In order to keep my examples consistent, I've replaced van Inwagen's "Mrs. Gamp" with "Sherlock Holmes".
In order to keep my examples consistent, I've replaced Thomasson's "Hamlet" with "Sherlock Holmes".
Thomasson: "Moreover, for any property whatever, Sherlock Holmes either has that property or lacks it. What is not true is that, for any property, that property is either ascribed to Sherlock Holmes or not ascribed to him" (308). van Inwagen: Some "sentences do not describe real properties of Sherlock Holmes, but only how he is said to be according to the story" (108).
Some will argue that a disjunction can be true even when none of its disjuncts are. Given that it's standard to take a disjunction to be less fundamental than its disjuncts and to be true in virtue of the truth of (at least one of) its disjuncts, it lies with those who make this claim to say what it is—given that it's not the disjuncts—which makes the disjunction true. One can't claim it's the law of logic LEM which makes the disjunction (Bh v ~Bh) true even in the absence of any true disjuncts because whether LEM is an inviolable law of logic or only a frequent regularity is exactly what's at issue. Sometimes it's clear what grounds the truth of a disjunction, even when none of its disjuncts are determinately true, e.g., suppose Doyle wrote that the murder happened on a weekend, but did not specify on which day. Then the truth of the disjunction (Msat v Msun) is grounded (i.e., by Doyle's words) even though neither disjunct is determinately true. Sometimes, even if neither disjunct is determinately true, the disjunction itself is supertrue (i.e., true at every precisification); see Calosi and Wilson (2017) and Skow (2010) for an explanation of why this supervaluationist approach can't successfully account for the sort of deep indeterminacy that is the concern of this article. In the remainder of this article I'll take it that—if a disjunction has no true disjuncts and no other grounding story for the truth of the disjunction is told—one is justified in taking the disjunction to be not true.
See Doyle (1927).
We could be possibilists or fictionalists or anti-Realists about fictional characters, rather than Creationists.
I've focused on fictional objects because they're the most widely discussed created objects in the literature. Caplan (2004) and Salmon (2002), among others, argue that, if Creationism is our best account of fictional objects, then it's also our best account of mythical objects and of imaginary objects. The claims I've made about fictional objects being LEM violators (given Creationism) also apply to mythical and imaginary objects, i.e., given Creationism, mythical objects and imaginary objects are also LEM violators.
My arguments for indeterminacy in fiction should be clearly distinguished from Everett's (2005) and Schnieder and von Solodkoff's (2009) reply. Everett focuses on (1) indeterminacy concerning the identity and distinctness of fictional characters, and (2) whether such in-the-story indeterminacy entails any real world indeterminacy. Schnieder and von Solodkoff argue against the sort of indeterminacy Everett defends. I'm happy to concede to Schnieder and von Solodkoff that Everett hasn't established that there are fictional characters with indeterminate identity; the indeterminacy I'm concerned with involves not the identity of, but the properties of, fictional objects. Likewise, I'm concerned with the upshot of in-the-story indeterminacy not with whether such indeterminacy entails any outside-the-story indeterminacy; I'm happy to concede that Everett hasn't established that in-the-story indeterminacy entails real world indeterminacy. In short, the indeterminacy at issue here differs substantially from the indeterminacy Everett discuss and refutations of his views on indeterminacy in fiction are not refutations of the sort of LEM-violating indeterminacy at issue here.
Henceforth I will drop the (nontrivial de re) qualification and write only "modal properties"; I mean always (non-trivial de re) modal properties. See Goswick (2018a) for explication of what non-trivial de re modal properties are.
Sidelle, for example, notes that:
[The] conventionalist should, and should be happy to, say that what is primitively ostended is 'stuff', stuff looking, of course, just as the world looks, but devoid of modal properties, identity conditions, and all that imports. For a slogan, one might say that stuff is preobjectual…. There is [no chair] to which we are pointing when we introduce 'Ralph'—there is, as I like to say, 'stuff'. And approaching this [chair-like] stuff with a chair term, we can then determine an origin for [the chair] Ralph by seeing, roughly, when this stuff started to be 'chairish'. (1989, 54–55)
Without nonsort objects (which Sidelle calls "stuff"), there is nothing to ground the accuracy of our application of the sortal "chair". It's something about the way this bit of the world is independently of us (i.e., something about what this nonsort object is like qualitatively) which makes it appropriate to apply "chair" or "instantaneous time-slice of a chair" or "atoms arranged chairwise", etc., to it, but not to apply "elephant" or "instanteous time-slice of an elephant" or "atoms arranged elephantwise", etc., to it. See Einheuser (2011) and Goswick (2018a) for further discussion of the crucial role nonsort objects play in anti-Realist conventionalist accounts of sort objects.
See Darby (2010) and Skow (2010) for an overview of such interpretations. Darby notes that in order to get "indeterminacy, more-or-less straight from the quantum–mechanical formalism, we have used a particular reading of the eigenstate-eigenvalue link, assumed that all bases correspond equally to observable properties and that all observables are to be assigned values, assumed the assignments of values to observables are non-contextual, and adopted naive realism about the way operators correspond to observables" (Darby 239).
Note the parallel with fictional objects and nonsort objects. In the case of fictional objects, the author does the determining. When she fails to do so, LEM violations arise. In the case of the modal properties of objects, the sort does the determining. When there is no sort, LEM violations arise. According to quantum mechanics, various quantum principles and mathematical formulas model reality. When these principles and formulas show that reality has failed to determine in a particular situation (e.g., due to the superposition principle, the uncertainty principle, or quantum entanglement), LEM violations arise.
Skow (2010) argues convincingly that the uncertainty principle is metaphysical rather than merely epistemic, i.e., it's not merely that we don't know whether q3 has spin-y up at t; it's that there's no fact of the matter.
Many philosophers take quantum mechanics to provide the strongest support for the sort of deep metaphysical indeterminacy which can lead to LEM violations, e.g., "An electron can not only be here and not here, but also in any number of other states that are superpositions of here and not here. That constitutes a middle term undreamed of by Aristotle" (Polkinghorne 38, 21–22). 'Does quantum mechanics support the idea of metaphysical indeterminacy? Of course it does, suitably interpreted—the position of a particle with a determinate momentum cannot also be determinate, so it must be indeterminate" (Darby 243). “Quantum mechanics and metaphysical indeterminacy are deeply connected. Indeed, some take quantum MI to represent the best case for thinking that there is or could be properly metaphysical indeterminacy” (Calosi and Wilson 23). See also Bokulich (2014) and Wolf (2015).
See Dirac (1930), Hughes (1989), and Polkinghorne (2002) for overviews of differing interpretations of quantum mechanics. See Priest (1989) for explicit argumentation that quantum mechanics does not give rise to LEM violations:
“The ‘particle’ [in the double-slit experiment] has no position independently of being observed on the screen. Indeed, there is no particle in reality, just a \( \psi \) state determined by the light-source and mask…. The real situation [in the Bell experiment] is described, quite literally, by a certain \( \psi \) function. There is but a single state, and spin-at-point-A and spin-at-point-B are two of its observer-relative (Newtonian) properties. Thus, the Bell experiment may force us to give up the intrinsic nature of Newtonian properties, but it does not force us to give up [LEM]” (Priest 34).
For myself, I'm convinced the LEM violations arise due to the existence of fictional objects and nonsort objects. I understand the varying philosophical accounts of fiction and of modality. I believe our best theories of each include LEM violating objects; so I believe there are exceptions to LEM. Although indeterminacy arising due to quantum mechanics is a favorite example of philosophers, I'm much less certain regarding whether LEM violations arise here. Certainly, there are interpretations of quantum mechanics on which they do. But, not being a physicist, I'm unable to understand these interpretations in the same way I understand the varying philosophical views of fiction and modality. I'm thus less certain regarding whether our best theory of quantum mechanics really does include LEM violating objects.
Note that, on Wilson's account, if an object has multiple determinates (at the same level and same time) of a determinable, it will always have them in a relativized fashion, e.g., the iridescent feather is not both entirely red and entirely blue; rather it's entirely red relative to angle of viewing v1 and entirely blue relative to angle of viewing v2.
In general, gappy metaphysical indeterminacy understood according to Wilson's determinable/determinate model can arise when what grounds the determinable differs from what grounds the determinate. If x grounds both the determinable and the determinate (e.g., reflecting lights waves in pattern p grounds both x's being red and x's being scarlet), then indeterminacy will not arise. If what grounds the determinable differs from what grounds the determinate, then indeterminacy will not arise so long as the grounding bases of both the determinable and the determinate are present. But, if (i) what grounds the determinable differs from what grounds the determinate, (ii) the determinable's grounder is present, and (iii) the determinate's grounder is not present, then gappy indeterminacy will arise.
See Wilson (2016) for arguments that some cases of indeterminacy that prima facie may seem not to fit the determinable/determinate model of metaphysical indeterminacy really do fit the model.
Consider whether, when presented with cases like Bh v ~Bh, one (a) takes the negated disjunct to be true, or (b) takes the disjunction to be undetermined. I suspect which option one finds intuitive hinges on how one views matters Doyle didn't discuss. If one views Doyle's non-discussion of Holmes's blood type as Doyle didn't say Sherlock Holmes has blood type B, one's inclined to think ~Bh is true. If one views Doyle's non-discussion of Holmes's blood as Doyle didn't comment on whether Sherlock Holmes has blood type B, one's inclined to think the disjunction is undetermined.
See Wilson (2012) for arguments that this understanding is errant.
One could say that within the story height properties work differently than height properties. So, e.g., although not being over 6' tall entails being exactly 6' tall or shorter, it's not the case that not being over 6' tall within the story entails being exactly 6' tall or shorter within the story. Arguably, it's less of a stray from our intuitive understanding of fiction (and from the Creationist account of fiction) to say that fictional objects are LEM violators. After all, the problem doesn't just arise for within the story height properties—it arises for every undetermined within the story property. It's a stretch to say all these within the story properties work differently than do their outside the story counterparts. If that's the case, why even think that e.g., height within the story concerns height at all (rather than, e.g., mass or temperature).
For another example, consider the picture of Irene Adler which Doyle says (in "A Study in Scarlett') is on the desk in Holmes's study. Adler's picture has the determinable being on the desk within the story, but lacks any determinates (being at location 1 on the desk within the story, being at location 2 on the desk within the story, etc.) of this determinable. According to a Wilsonian Creationist, for every location on the desk within the story, it's not the case that the picture is at that location within the story. Yet it is the case that the picture is on the desk within the story. This strains credulity. Better to say, as I say, that, for every location on the desk within the story, it's undetermined whether the picture is at that location within the story.
Given the connection between LEMproposition and the Principle of Noncontradiction-proposition (henceforth, PNCpropostion) in classical logic—namely, that if LEMproposition is false, then so is PNCpropostion—one might think that by allowing objects to violate LEMproperty, I'm already committed to embracing contradictions. This is incorrect. First, I don't argue that LEMproposition is false; I argue that there are counter-examples to LEMproperty and that LEMproperty is sometimes undetermined. Exactly what this shows regarding LEMproposition is a question yet to be investigated. It would be surprising if showing that LEMproperty is undetermined entailed that PNCpropostion is false. Second, many non-classical logics reject the classical connection between LEMproposition and PNCpropostion. So, even if LEMproposition is false, one can't automatically assume that PNCpropostion is false.
Arguably, this is less troubling than is violating the connection between e.g., not having hair on one's head and being bald. One might argue that what quantum mechanics shows is that certain properties we might intuitively have expected to be connected (e.g., having up-spin iff not having down-spin) are, in fact, not connected.
See Skow (2010) for the related discussion of how certain phenomena which occur at the quantum level—e.g., wave-particle duality, incompatible observables, etc.—can impact property connections, viz., "The Kochen-Specker theorem shows that there are not complete precisifications of reality which respect the dependencies among properties in orthodox quantum mechanics" (Skow 858).
I take the upshot of indeterminacy to be that some objects violate LEM. One could instead take the upshot of indeterminacy to be that there are far fewer exhaustive incompatible properties than we initially thought there were, e.g., neither being alive and being dead nor being necessarily F and being possibly not F are really exhaustive incompatibles. What does Schrodinger's cat really tell us? Does it tell us something about the cat, e.g., that it's in an odd circumstance, a circumstance such that it's neither alive nor dead? Or does it tell us something about properties in general—namely, that being alive and being dead aren't really exhaustive incompatibles? Arguably, modifying our general account of properties to account for odd objects (or, objects in odd circumstances) is a more radical move than is denying LEM. Space constraints prevent me from giving this issue the attention it deserves. (Thanks to an anonymous referee for pressing me on this point.)
Quantum mechanics is often heralded as a real (i.e., independent of human representations) case of metaphysical indeterminacy. Whether this is the case depends on how we take quantum mechanics. If we see it as a descriptive theory whose target is the real (i.e., independent of human representations) world, then this is correct. If, on the other hand, we see quantum mechanics as a model which aims at accurately representing reality, but which may fail in certain respects (just as, e.g., a biography aims to accurately portray its subject, but may fail in certain respects), it's less obvious that quantum mechanics provides an example of "real" metaphysical indeterminacy, i.e., because saying the model fails to determine F doesn't mean the object the model models fails to have F, it just means that its having F is determined by something the model is unable to capture. For what it's worth, I don't think the distinction between being independent of human representations and being dependent on human representations is the right place to locate whether an indeterminacy is real or merely representational. What matters for real indeterminacy is not what the source of the indeterminacy is, but whether it's resolvable. An indeterminate newspaper account of what Jane wore to the ball is accidentally indeterminate—there's a determinate fact to be captured; the newspaper just fails to capture it. Whereas, an indeterminate novel whose author is dead is unresolvably indeterminate. It's not an incomplete representation of something which isn't indeterminate; it's a complete representation of something which is indeterminate.
An anonymous referee points out that Wilson could simply reply that, in cases of metaphysical indeterminacy (which are admittedly already "odd"), it would be entirely natural if the usual property entailments weren't in place. I find it more natural to claim that LEM is violated. However, my goal here isn't to argue that violating LEM is the only option. My goal is simply to show that there are costs to respecting LEM. In particular, one must violate the intuitive connection between exhaustive incompatible properties. I leave it to the reader to decide which violation is the more egregious.
Metaphysics has long been taken to be hostage to classical logic. Before blithely accepting such a situation we should, at the very least, examine whether logical monism with classical logic as the one true logic is true or whether logical pluralism is true. If the latter, the fact that classical logic doesn't allow for a metaphysical theory which quantifies over objects which violate LEM isn't a reason to reject the metaphysical theory; it's a reason to represent it using non-classical logic. A reviewer worries that violating LEM requires "introducing a third truth value or degrees of truth [and that there] are well-known concerns and costs associated with such accounts". If logical pluralism is correct, wholesale revision isn't required. Classical logic works really well most of the time, so we can keep using it most of the time. It isn't a good logic for modelling odd objects, so we should use some other logic(s) to model them. See Priest (2001) for an overview of the various options. See McSweeney (2019) for arguments that we shouldn't automatically assume classical logic is the one true logic and, thus, shouldn't automatically assume that our metaphysical theorizing should be hostage to classical logic. See Barnes and Williams (2011), Williams (2008), and Williamson (1994) for additional accounts of the interaction between LEM, vagueness, and indeterminacy.
An indeterminacy is mere mundane vagueness if that which determines the determinate is present, but it's vague exactly how it determinates it. For example, that which grounds the number of hairs on your head (i.e., your head and what it's like) exists; it's just unclear exactly what counts as a hair on your head. Cases of deep metaphysical indeterminacy arise only when that which determines the determinate is absent (gappy indeterminacy) or when that which determines the determinate overdetermines (glutty indeterminacy). See Wilson (2013).
This paper owes a huge debt to Jessica Wilson. Although Wilson doesn’t endorse LEM violations, she does think worlds can be complete without being precise and that the actual world is such a world. Wilson (2013) and Wilson (2016) gave me a framework for developing my own views and convinced me that it wasn’t crazy to think our world is simply imprecise. Thanks are also due to attendees at the Australian Metaphysics Conference (April 2018) and to members of the virtual Metaphysics WIP Reading Circle (July 2018) who read and provided feedback on an earlier version of this paper, to Paul Teller for extensive written comments and discussion, to members of the Logic Group at the University of Melbourne for introducing me to logical pluralism and helping me realize that if I could think it, there was some logic that could represent it, and to Boris Kment for being a fellow LEM skeptic.
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Goswick, D. Odd Objects: LEM Violations and Indeterminacy. Erkenn 86, 1615–1633 (2021). https://doi.org/10.1007/s10670-019-00173-8
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DOI: https://doi.org/10.1007/s10670-019-00173-8