Akiba's logic of indeterminacy

ABSTRACT

A standard approach to indeterminacy treats ‘determinately’ and ‘indeterminately’ as modal operators. Determinacy behaves like necessity; indeterminacy like contingency. This raises two questions. What is the appropriate modal system for these operators? And how should we interpret that system? Ken Akiba has developed an account of ontic indeterminacy that interprets possible worlds as worldly precisifications. He argues that this account vindicates S4 as the logic of indeterminacy. In this paper I explore one significant and surprising consequence of this view. I do this in two stages. First, I prove a technical result, which I call the Infinite Indeterminacy Theorem. Roughly put, this theorem states that, at any given precisification w, either every instance of indeterminacy admits of never-ending higher-order indeterminacy, or else there's indeterminacy in an infinite amount of distinct atomic formulas. In slogan form: either all indeterminacy is infinitely ascending or else there's infinitely wide indeterminacy. Second, I unpack the metaphysical consequences of this technical result (under its intended interpretation) and assess their plausibility. I conclude that these consequences commit Akiba's theory to highly contentious – and arguably untenable – views about the metaphysics of indeterminacy and related matters.

KEYWORDS:

• Indeterminacy
• modal logic
• S4
• ontic indeterminacy
• higher-order indeterminacy
• vagueness

Acknowledgements

A previous version of this paper was presented at the Foundations Interest Group at the Minnesota Center for Philosophy of Science in September of 2020. Thanks to the participants at that meeting for lively discussion and helpful feedback. Special thanks to Ken Akiba, Eva Enns, and anonymous referee for this journal for valuable comments on earlier presentations of these ideas.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Unless otherwise noted (e.g. in §3), I will use ‘possible worlds’ to refer to the points in a given formal (modal) model. Then there is the question of how to interpret these points in order to characterize whatever phenomenon we are interested in (e.g. indeterminacy). Specifically – and again unless otherwise noted – I am not using ‘possible worlds’ to refer to metaphysically counterfactual possibilities.

2 Unless otherwise noted, all page references herein are to Akiba (2014a).

3 Barnes and Williams do not decisively commit themselves to S5, but it is what they use for purposes of illustration in their most thorough work on the subject (2011), and I don't know of any place where they consider any system weaker than S5.

4 I say indeterminate ‘distinctness’ rather than indeterminate ‘identity’, because, as Akiba (p. 3564) points out, there's nothing wrong with Evans's proof from ‘It is indeterminate whether $a=b$’ to ‘$a\ne b$’ (at least assuming that the logic of determinacy is a normal modal logic in which the determinacy of Leibniz's Law holds). In other words, according to Akiba, Evans does successfully show that if $a=b$, then it can't be indeterminate whether $a=b$. So in that sense there can't be indeterminate identity. What Akiba takes issue with is the move (only gestured at in Evans's original argument) from ‘$a\ne b$’ to ‘it is determinate that $a\ne b$’. On Akiba's logic, ‘$a\ne b$’ and ‘it is indeterminate whether $a=b$’ are perfectly consistent. So in that sense there can be indeterminate distinctness.

5 Of course, this is compatible with their being no indeterminacy at all.

6 A few comments on clause (ii). First, the parenthetical condition is not something that Akiba explicitly states. But it is certainly something he should accept, given that he takes $R$ to be antisymmetric (as we’ll discuss momentarily). Second, (ii) should not be read simply as: some things are true at $v$ that are not true at $w$ (and not vice versa). $v$ might be a precisification of $w$ if $w$ contains $p$ and $\mathrm{\nabla p}$ while $v$ only contains $p$. Third, and relatedly, $R$ need not be monotonically expanding – $w$ may contain states of affairs not included in $v$, so long as those states of affairs are imprecise. None of these three points will be important for our purposes here.

7 Recall that the S4 axioms are complete with respect to the class of all partial orders, despite the fact that this class does not exhaust the class of all S4 frames (Blackburn, de Rijke, and Venema 2001, §4.5).

8 At least not in Akiba (2014a). Akiba (2017) uses the conditions for truth-at-a-world given by Fine (1975). It's not clear that Akiba would fully endorse these semantics, however, given that Fine's system requires stability (monotonic expansion), which Akiba rejects (see note 6). But the semantics that Akiba prefers may well be in the ballpark of Fine's. In any case, Fine's semantics satisfy our two assumptions, which we will introduce momentarily. See note 10. One last point. Akiba’s (2017) purpose is to explore the possibility of combining a modal approach to indeterminacy with a many-valued Boolean semantics for the logical connectives (see also Akiba 2014b). However, the latter is not intended as a semantics for complex formulas at a world – which is what we are concerned with presently. Instead, the many-valued Boolean semantics provides the conditions of truth simpliciter for complex formulas in terms of the concept of truth-at-a-world (in this way it is comparable to the supervaluationist's identification of truth (simpliciter) with supertruth). From this perspective the many-valued Boolean semantics that Akiba proposes is irrelevant to our present discussion. Still, it is worth noting that it too would satisfy both of our assumptions.

9 Lest modus ponens be invalidated, which there is no reason to think Akiba would accept.

10 Continuing from note 8: Fine's semantics satisfies both assumptions. First, it is specifically designed to validate all classical tautologies (among other penumbral connections). So it satisfies our first assumption. Second, though Fine's semantics is not truth-functional, it still satisfies our second assumption: the semantics for $\mathrm{\neg }$ and $\wedge$ preserve modal invariance across bisimilar worlds. See note 13 for details.

11 Because Akiba is explicit that his theory supports a reading of the accessibility relation that is not just transitive and reflexive but moreover antisymmetric, we will, as just indicated, confine ourselves to proving the theorem for the class of models with such an accessibility relation, i.e. partial orders. However, once this more narrow result is in place, it can easily be shown that the theorem applies, not just to partial orders, but to any S4 model whatsoever. The key to this generalization is the fact that any S4 model is modally equivalent to some partial order (Blackburn et al. 2001, §4.5). Technically, this fact is not all by itself sufficient for the generalization, for what needs to be shown is that any S4 model in which Modal Equivalence holds is modally equivalent to a partial order in which Modal Equivalence holds, but the remaining details are easily demonstrated. In that sense, then, our reliance on partial orders here is merely for the sake of relevance and convenience and does not ultimately represent any special limitation of the theorem (beyond S4 models generally). This is in contrast to the theorem's reliance on Modal Equivalence, which constitutes a substantial way in which the theorem depends on the specifics of Akiba's theory beyond the logic of S4.

12 Recall that atomic equivalence, in the present context, is defined in terms that allow for truth-value gaps, whereas the standard definition of a bisimulation assumes a classical semantics. So technically our definition here is a bit non-standard. But this difference does not undermine the significance of bisimulations, as discussed in note 13.

13 The standard proof of this theorem is a straightforward induction on the structure of formulas (van Bentham 2010, 28–29). The proof assumes a classical semantics for the logical connectives. However, it applies in the present context as well. The base case and the $\mathrm{\Delta }$-case are exactly as in the original proof. That just leaves the steps for $\mathrm{\neg }$ and $\wedge$. But recall our assumption (§1) about the semantics for these connectives: whatever their semantics, they preserve modal invariance of conjunctions and negations over bisimulations. So these steps hold as well, and the proof goes through. As an illustration, consider Fine’s (1975) semantics for $\mathrm{\neg }$ and $\wedge$ (see note 8). The truth conditions for $\alpha \wedge \text{β}$ are just as in classical semantics, so this step of the induction proof is exactly as in the classical case. The truth conditions for $\mathrm{\neg \alpha }$ are given by: $w\models \mathrm{\neg \alpha }$ iff , for all $v$ such that $\mathrm{Rwv}$. Now suppose for induction that the truth-value of $\alpha$ is invariant across bisimilar worlds, and suppose that $Z$ is a bisimulation between $M,w$ and $M^{\mathrm{\prime }},w^{\mathrm{\prime }}$. Then we have:

This completes the induction step for $\mathrm{\neg }$.

14 On generated submodels, see Blackburn, de Rijke, and Venema (2001, 56). The fact that $M^{\mathrm{\prime }}$ is indeed a submodel generated by $w$ relies on the fact that $R$ is both reflexive and transitive.

15 We just inferred that $w\models \mathrm{\neg }\underset{n}{\underbrace{\mathrm{\nabla }\dots \mathrm{\nabla }}}\psi$ from our assumption that . In the context of truth-value gaps this may not be warranted. However, $\underset{n}{\underbrace{\mathrm{\nabla }\dots \mathrm{\nabla }}}\psi$ can only result in a truth-value gap if it is itself indeterminate, which would contradict the leastness of $n$. So the inference is warranted in the present context.

16 In general, to say that $w$ is a root of a model $M$ is to say that $R^{\ast }\mathrm{wv}$ for all $v$ in $M$, where $R^{\ast }$ is the reflexive transitive closure of $R$. Because the particular $R$ we are working with is a partial order, if $w$ is a root of $M$ it will automatically be the (unique) root of $M$. Now, if $w$ were not the root of $M$, we could shift our attention to the submodel $M^{\ast }$ generated by $w$, of which $w$ is the root. Because modal satisfaction is invariant under generated submodels (Blackburn, de Rijke, and Venema 2001, 56), shifting focus to $M^{\ast }$ would make no difference to what formulas are true at $w$ and so would have no effect on the reasoning in the proof. That's why I say we can assume that $w$ is the root of $M$ without loss of generality.

17 There's a complication here in the context of truth-value gaps. Let $r$ be one of the atomic formulas for which $\mathrm{\nabla r}$ is not true at $w$. That doesn't immediately show that either $\mathrm{\Delta }r$ or $\mathrm{\Delta }\mathrm{\neg r}$ is true at $w$, for it leaves open the possibility that each of $\mathrm{\nabla r}$, $\mathrm{\Delta }r$, and $\mathrm{\Delta }\mathrm{\neg r}$ is neither true nor false. I’m not exactly sure what to make of this possibility. At the very least it would seem to require that each of these formulas itself be indeterminate. For instance it would seem to require that $w\models \mathrm{\nabla }\mathrm{\Delta }r$. However this implies that $w\models \mathrm{\neg }\mathrm{\Delta \Delta }r$, which in turn implies that $w\models \mathrm{\neg }\mathrm{\Delta }r$ (on S4), which means that $\mathrm{\Delta }r$ is false after all, contradicting the original supposed truth-value gap. Maybe we could suppose that $\mathrm{\nabla }\mathrm{\Delta }r$ likewise has no truth-value? This would then require that $\mathrm{\nabla \nabla }\mathrm{\Delta }r$ also has no truth-value. The reasoning here generalizes: if none of $\mathrm{\nabla r}$, $\mathrm{\Delta }r$, and $\mathrm{\Delta }\mathrm{\neg r}$ is true or false, than neither is any of $\mathrm{\nabla }\dots \mathrm{\nabla }\mathrm{\Delta }r$, $\mathrm{\nabla }\dots \mathrm{\nabla }\mathrm{\Delta }\mathrm{\neg r}$, or $\mathrm{\nabla }\dots \mathrm{\nabla r}$. Call this total state of affairs the TVG (truth-value gap) situation (with respect to $r$). I’m not exactly sure how to interpret the TVG situation. But suppose we were to amend the theorem to explicitly acknowledge it's possibility. In that case (i) would read: there are an infinite amount of atomic formulas $q_1,q_2,\dots$ such that, for each $q_i,$ either $\mathrm{\nabla }{q}_i$ is true at $w$ or the TVG situation with respect to $q_i$ holds at $w$. Certainly this is no less significant of a result than is the original formulation. Alternatively, we could always address the worry, if necessary, by (very slightly) weakening the theorem, so that (i) reads: there are an infinite amount of propositions $q_1,q_2,\dots$ such $\mathrm{\nabla }{q}_i$ is not false at $w$ for each $q_i$. Again, I think this would have virtually the same significance as the original formulation.

18 I’ll tend to put things in terms of propositions, but we could alternatively talk about states of affairs.

19 In describing some propositions as having quantificational structure or truth-functional complexity, I don't mean to take a stand on whether propositions are literally structured entities. I just mean that we can have propositions like <Everyone is happy> and <Dori is small and Dori is fierce>.

20 By this I of course don't mean to imply that the members in $\mathrm{\Delta }$ couldn't differ in truth-value across metaphysically possible worlds. $\mathrm{\Delta }$ may well contain an infinite amount of logically non-equivalent, contingent propositions, and so may admit of an infinite amount of possible truth-value assignments. My point above is confined to the actual metaphysically possible world, i.e. reality. Reality, according to Akiba, admits of worldly precisifications, and it is throughout these that each member of $\mathrm{\Delta }$, owing to its determinacy, will be constant in truth-value.

21 Talk of ‘what it is’ for so-and-so to be such-and-such is open to interpretation. Maybe it amounts to reduction; maybe it amounts to a ‘real definition;’ maybe it amounts to ‘constitution;’ maybe it just amounts to some weaker sort of characterization. The point is just that, whatever it is, it tells us something about the nature or essence of the phenomenon of interest.

22 Sometimes an account of ontic indeterminacy, especially a logical account, is construed as merely offering a way of showing that the phenomenon is intelligible, against complaints to the contrary (Lewis 1986, 212). One might think, therefore, that the aim of such a theory is not at all to tell us something necessary about ontic indeterminacy. But I think this thought is confused. To show that a supposed phenomenon is intelligible by offering a model of that phenomenon is to go some way towards showing that such a phenomenon is epistemically possible. But that is perfectly consistent with thinking that, if that phenomenon is metaphysically possible, and if the model given is accurate, then that model tells us something essential about the phenomenon. In short, a logic of ontic indeterminacy can have as its purposes both to show that the phenomenon is intelligible and to provide us with an account of how that phenomenon must behave if and when it occurs. In any case, nowhere does Akiba, as far as I am aware, describe his theory as one whose primary aim is merely to demonstrate intelligibility.

23 To be clear, I am not arguing that the necessity (given some indeterminacy) of (a) or (b) follows just from the fact that one or the other must be true at all worlds (where there is indeterminacy) in any of Akiba's models of indeterminacy. To argue that way would be to confuse what worlds represent on Akiba's models – precisifications of reality – with something else altogether, namely metaphysical possibilities. It would be to confuse two distinct modalities. The way I am arguing for the above necessity can be brought out by analogy. Suppose we have some temporal logic on which it follows that some formula $\phi$ holds at every point $t$ in every model. If we think this theory gives us an accurate picture of the way time really works, then we can conclude that $\phi$ (or whatever proposition would be expressed by $\phi$ under its standard interpretation) in fact holds at the present time. And if we moreover think that the theory gives us an accurate picture of the way time necessarily works, then we can conclude that $\phi$ is metaphysically necessary.

24 More carefully, this observation should be made in purely epistemic terms: each of these phenomena seems to represent a clear case of unclarity, where the sense of ‘(un)clarity’ here is understood as something like ‘(un)knowable in principle’. The proponent of ontic indeterminacy will interpret the unclarity as arising from ontic indeterminacy, and so, if the observation is correct, should view the case as one in which it is (ontically) determinate that it is indeterminate.

25 There is one putative case of ontic indeterminacy that might of necessity admit of infinitely ascending indeterminacy, and that is ontic indeterminate identity/distinctness. On any modal logic as strong as KT (and so on Akiba's logic in particular), $\mathrm{\nabla a}=b\models \underset{n}{\underbrace{\mathrm{\nabla }\dots \mathrm{\nabla }}}a=b$, for any $n\ge 1$ (Bacon 2018). So if the only metaphysically possible cases of indeterminacy were cases of indeterminate identity/distinctness, we’d have a way of satisfying Horn Two. But even if we accept the possibility of ontic indeterminate identity/distinctness, there seems little hope of motivating the view that it is the only possible type of ontic indeterminacy. That's not something that proponents (including Akiba) of ontic indeterminacy accept (Akiba 2004; Rosen and Smith 2004; Williams 2008a; Barnes 2010; Wilson 2013).

26 I say ‘properties’ rather than ‘predicates’ since the indeterminacy at issue is ontic.

27 Certain of these may be multidimensional. For instance, whether or not a person is bald will depend, not just on how many hairs are on their head, but also on the distribution of those hairs, and such distributions may well admit of a dense ordering. But we can still get a finite sorites series if we fix the distributional facts – individual hairs are added on one at a time, but no hair ever moves its location.