Abstract
We investigate and compare two major approaches to enhancing the expressive capacities of modal languages, namely the addition of subjunctive markers on the one hand, and the addition of scope-bearing actuality operators, on the other. It turns out that the subjunctive marker approach is not only every bit as versatile as the actuality operator approach, but that it in fact outperforms its rival in the context of cross-world predication.
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Notes
These generalizations were presented at the Actuality and Subjunctivity workshop held at UC Irvine in April of 2012.
A symbol occurrence o lies within the direct scope of an actuality operator in \(\phi \) just in case there is a subformula occurrence \({\mathsf {A}}\psi \) in \(\phi \) such that o lies in \(\psi \) but not within the scope of a box in \(\psi \). In other words, o lies within the direct scope of \({\mathsf {A}}\) in \(\phi \) just in case o lies within the scope of an occurrence of \({\mathsf {A}}\), but not within the scope of any occurrence of the box that itself lies within the scope of the original \({\mathsf {A}}\)-occurrence.
It does not matter whether we require every such occurrence to lie within the scope of a naked box, or just within the scope of some box, whether naked or subjunctivized. This is because on either version of the definition subjunctivized boxes themselves must ultimately lie within the scope of a naked box to achieve subjunctive closure, and so all the material within the scope of a subjunctive box will also lie within the scope of a naked box.
For more discussion of subjunctive closure and issues arising from endowing \({\mathcal {L}}^1_{\mathsf {A}}\) with real-world truth, see Sect. 4.
For simplicity let us assume that first-order languages have only \(\lnot \) and \(\wedge \) as primitive connectives, and only \(\exists \) as a primitive quantifier, the other connectives and quantifiers being defined in the standard ways.
Where \(w\in W\), we let \(\sigma _i^w\) be the sequence that is exactly like \(\sigma \) in all places other than the ith, where \(\sigma _i^w\) has w.
It won’t do to assign the first entry in \(\sigma \) a special role, since that just means designating it as specifying the world of evaluation.
If only sentence letters could ever occur within the scopes of the \({\mathsf {A}}_i\), this problem wouldn’t arise, as we could simply stipulate that \({\mathbf {M}}\,\Vdash ^\sigma\, {\mathsf {A}}_i p\) be equivalent to \(V_{\sigma _i}(p) = 1\), which is essentially what happens in \({\mathcal {L}}^\infty _s\). But already for \({\mathbf {M}}\,\Vdash ^\sigma\, {\mathsf {A}}_i (p\wedge q)\) we must remember which world the index i points to.
As an anonymous reviewer for this journal points out, the use of the \(n(\phi )\)-th position in \(\sigma \) for simulating the world of evaluation is somewhat analogous to Stephanou’s (2001) use of \(n(\phi )\) to simulate evaluation at the actual world. Unfortunately we don’t have space to discuss the issue at any length, but it should be noted that Stephanou uses this device to dispense with the unindexed actuality operator entirely, assuming instead that truth in a model is truth relative to the sequence \(\sigma \) that has the actual world occurring in every position, so that \(n(\phi )\) points to the actual world through \(\sigma \). However, we note that on this way of eliminating the unadorned operator \({\mathsf {A}}\), one no longer has any formulas in the modal language whose truth value in a model is always independent of the world sequence \(\sigma \); in other words, eliminating the naked actuality operator also eliminates the possibility of subjunctive closure. This strikes us as an undesirable consequence.
We have tacitly corrected a couple of typographical errors in the original. Note that no equivalence numbered “(1)” occurs in Humberstone’s paper. The reference is presumably to equivalence (4.11), which reads “\({\mathsf {O}} \# (\varphi _1,\ldots ,\varphi _n) \equiv \# ({\mathsf {O}}\varphi _1,\ldots ,{\mathsf {O}}\varphi _n)\) for each n-ary Boolean connective \(\#\).”
In personal communication, Humberstone has acknowledged that the objection just discussed is based on an error.
Humberstone actually talks about the quantificational versions of the languages, but this is immaterial to the point at hand.
It might be noted that Humberstone’s objection is cited approvingly by French (2015, p. 240).
Humberstone here includes a footnote that reads: “A similar point is made in Smiley (1996) with regard to negation.” We have tacitly corrected a typographical error in the original.
This consideration would seem to apply equally to Smiley’s original version of the argument.
It is not entirely clear to us whether French (2013) fully appreciates this independence fact, for although he does acknowledge it in his Lemma 1 (p. 1691), he typically speaks of \({\mathcal {L}}^1_s\) as being endowed with real-world truth, which strikes us as misleading.
Though he didn’t call them that, d-models were essentially introduced by Hanson (2006). It has remained unclear what the conceptual benefit of d-models might be.
French also considers a variant definition on which subjunctively closed formulas are considered true in \({\mathbf {\Delta }}\) just in case they are true at @, but the objections to his argument are in each case the same, so we will not consider this variant separately.
Or rather, in varying-domain models, \(\Diamond \forall ^{\mathsf {D}}x({\mathsf {D}}Fx \rightarrow Gx)\); but we will ignore this nicety.
For a related argument, see Wehmeier (2014).
The distinction is of course also present in \({\mathcal {L}}^1_{\mathsf {A}}\), to wit, in the contrast between \({\mathsf {A}}\Diamond \phi \) and \(\Diamond \phi \), but it is somewhat obscured by the convention to apply real-world truth to the language, so that \(\Diamond \phi \), when unembedded, has the same truth conditions as \({\mathsf {A}}\Diamond \phi \). In \({\mathcal {L}}^1_s\), by contrast, with the notion of truth in a model restricted to subjunctively closed formulas, the distinction is clearly marked between \(\Diamond \phi \) and \(\Diamond ^s\phi \), with only the former being truth-eligible as a self-standing formula, and the latter only being able to occur embedded under a modal.
That is not to say that one couldn’t amend quantified \({\mathcal {L}}^1_{\mathsf {A}}\) in some other way to make the sentence in question expressible, e.g. by adding Vlach operators (thanks to an anonymous referee for the suggestion). Discussion of such an approach would lead us too far afield; suffice it to say that the actuality quantifiers appear to be the standard remedy, and that, in our view at least, Vlach operators do not have natural counterparts in natural language modal discourse.
Obviously \(\Diamond T(j,j)\) would not do either, since it would be true only if there is a world w such that John’s height in w is greater than John’s height in w, which there isn’t.
It is notationally more elegant in this setting to assign indicative predicates an explicit indicative marker rather than leaving them naked as in the simple modal environment, but this is evidently a trivial notational change.
We are here using “i” as an indicative marker. Note that, in Definition 5, we used “i” as a variable over natural numbers qua subjunctive markers. We trust that this does not cause any confusion.
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Acknowledgments
The authors wish to thank Lloyd Humberstone for reading and commenting on an earlier version of this paper. Special thanks are due to an anonymous reviewer for this journal, who provided many insightful and helpful suggestions for improvement.
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Wehmeier, K.F., Rückert, H. Still in the Mood: The Versatility of Subjunctive Markers in Modal Logic. Topoi 38, 361–377 (2019). https://doi.org/10.1007/s11245-016-9426-8
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DOI: https://doi.org/10.1007/s11245-016-9426-8