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The Intensional Many - Conservativity Reclaimed

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Abstract

Following on Westerståhl’s argument that many is not Conservative [9], I propose an intensional account of Conservativity as well as intensional versions of EXT and Isomorphism closure. I show that an intensional reading of many can easily possess all three of these, and provide a formal statement and proof that they are indeed proper intensionalizations. It is then discussed to what extent these intensionalized properties apply to various existing readings of many.

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Article 22 July 2014

Jeffrey Sanford Russell, John Hawthorne & Lara Buchak

Notes

  1. The set of worlds W serves as a basis from which to derive intensional standards that are not (heavily) dependent on the interpretations in any one world. The idea here is not that W would include every logical possibility, but rather that it is made up of worlds which are much like the actual world except (possibly) for the issues at hand, for which they will by and large correspond to our expectations and the things we consider normal and plausible

  2. In more traditional intensional semantics, the thing we call a structure above is referred to as a model, and Q m XY would be expressed as \(S \models QXY[m]\).

  3. This assumption may sometimes be undesirable, but keep in mind that this reading is merely an illustrative example. We shall see in Section 2.6 that there is a broad general form such that any reading of that form will possess Conservativity and other key properties. Thus, for certain infinite W the average could be generalized using series summation \(\left (\lim \limits _{n\rightarrow \infty }\frac {1}{n}\sum _{i=1}^{n} \frac {| \mathbf {Y}^{w_{i}}\cap \mathbf {X}^{w_{i}}|}{| \mathbf {X}^{w_{i}}|} \right )\) or integration \(\left ( \int _{W} h(w) \frac {|\mathbf {Y}^{w}\cap \mathbf {X}^{w}|}{| \mathbf {X}^{w}|}\mathrm {d}w, where\, \int _{W}h(w)\mathrm {d}w=1\right )\), or be replaced by an intensional standard based on a probability function on W, a subset of particularly ‘normal’ or normative worlds, or some other notion (see also Section 3). For any of these, the desirable properties remain attainable.

  4. To get around division by zero, we may harmlessly use \(\frac {0}{0}=1\).

  5. Given how much here revolves around intensions and extensions, to do otherwise could invite confusion.

  6. As a first thought it might look desirable to go much further and that f be an actual isomorphism; i.e. that f[X m] = X m′ holds for all properties. However, one can always define, say, a property X for which X m and X m′ do not even have the same number of elements. Thus, making such a broad demand would guarantee that no such f exists for any structure, rendering the whole thing worthless. Therefore we are forced to work only with those properties which work well with f (for at least one f ).

  7. Though note that strictly speaking, ‘other’ here would include Scandinavian itself.

  8. Admittedly this decision is a crucial step, and making a different choice here might potentially lead to a different outcome. Still, the choice seems appropriate enough and no alternative that actually gives a different outcome comes to mind.

  9. In addition to two absolute readings which we are not interested in here.

  10. Lappin uses “situation” where we would use “world” or “model”.

References

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Acknowledgments

The research in this paper is supported by a grant from NWO as part of the Vagueness – and how to be precise enough project (project NWO 360-20-202). I would like to thank Johan van Benthem for his helpful comments.

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Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands

    Harald Andreas Bastiaanse

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  1. Harald Andreas Bastiaanse
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Correspondence to Harald Andreas Bastiaanse.

Appendix: Reductions

Appendix: Reductions

1.1 A.1 Lifting Theorem

Definition 5

A non-intensional quantifier Q is a function which when given a domain M and two sets U, V ⊆ M gives an evaluation of true or false. We will write \(Q_{M}UV\) to denote that this evaluation is true.

For a non-intensional quantifier Q, define its intensional lift Q∗ as follows:

$$ \mathbf {Q}^{*}_{m}\mathbf {X}\mathbf {Y} \Leftrightarrow Q_{D(m)}\mathbf {X}^{m}\mathbf {Y}^{m}$$

Also, for any set U in domain D (m), the lift l m (U) is the set of properties X for which X m = U.

Theorem 3

Where Q is a non-intensional quantifier and Q∗ is its lift:

  • Q∗  satisfies Intensional Conservativity if and only if Q is Conservative

  • Q∗  satisfies Intensional EXT if and only if Q satisfies EXT∗, where EXT∗ is like EXT but applies for any M, M′ such that A, B ⊆ M, A, B ⊆ M′

  • Q∗  satisfies Intensional Isomorphism closure if and only if Q satisfies Isomorphism closure

Proof

Conservativity is the easiest. First assume Q∗ satisfies Intensional Conservativity. For a given set M, let m be a model with D(m) = M. Then

$$\begin{array}{lll}Q_{M}UV &\Leftrightarrow \exists \mathbf{X}\in l_{m}(U), \mathbf{Y}\in l_{m}(V):\mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} & \mathit{by\,\, construction}\\ &\Leftrightarrow \exists \mathbf{X}\in l_{m}(U), \mathbf{Y}\in l_{m}(V):\mathbf{Q}^{*}_{m}(\mathbf{X})(\mathbf{X}\wedge \mathbf{Y}) & \mathit{by\,\, Intensional\,\, Conservativity} \\ &\Leftrightarrow \exists \mathbf{X}\in l_{m}(U), \mathbf{Z}\in l_{m}(U \cap V):\mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Z} & \mathit{see\, below}\\ &\Leftrightarrow Q_{M}U(U\cap V) & \mathit{by\,\, definition} \end{array} $$

For the third step , note that

$$(\mathbf {X}\wedge \mathbf {Y})^{m}=\mathbf {X}^{m}\cap \mathbf {Y}^{m}=U\cap V$$

Therefore \((\mathbf {X}\wedge \mathbf {Y})\in l_{m}(U\cap V)\).

Next, assume that Q is (regularly) Conservative, m is some model with D(m) = M and X and Y are properties. Then

$$\begin{array}{lll} \mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} &\Leftrightarrow Q_{M}\mathbf{X}^{m}\mathbf{Y}^{m} &\mathit{by\,\, definition} \\ &\Leftrightarrow Q_{M}\mathbf{X}^{m}(\mathbf{X}^{m}\cap \mathbf{Y}^{m}) &\mathit{by\,\, Conservativity} \\ &\Leftrightarrow Q_{M}\mathbf{X}^{m}(\mathbf{X}\wedge \mathbf{Y})^{m} &\mathit{by\,\, definition} \\ &\Leftrightarrow \mathbf{Q}^{*}_{m}\mathbf{X}(\mathbf{X}\wedge \mathbf{Y}) &\mathit{by\,\, definition} \end{array}$$

For EXT∗, let \(U,V\subseteq M, M'\), \(D(m)=M\), \(D(m')=M'\) and let \(\mathbf {X}, \mathbf {Y}\) be such that \(\mathbf {X}^{m}=U=\mathbf {X}^{m'}\), \(\mathbf {Y}^{m}=V=\mathbf {Y}^{m'}\).

First assume \(\mathbf {Q}^{*}\) satisfies Intensional EXT. Then

$$\begin{array}{@{}rcl@{}} Q_{M}UV &\Leftrightarrow \mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} & \mathit{by\,\,definition} \\ &{\kern1pt}\Leftrightarrow \mathbf{Q}^{*}_{m'}\mathbf{X}\mathbf{Y} & \mathit{by \,\,Intensional \,\,EXT} \\ &{\kern4pt}\Leftrightarrow Q_{M'}UV & \mathit{by\,\,definition} \end{array} $$

For the other direction, assume Q satisfies EXT∗. Then

$$\begin{array}{lll} \mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} &\Leftrightarrow Q_{M} \mathbf{X}^{m} \mathbf{Y}^{m} &\mathit{by\,\, definition} \\ &\Leftrightarrow Q_{M'} \mathbf{X}^{m'} \mathbf{Y}^{m'} &\mathit{by\,\, EXT}^{*} \\ &\Leftrightarrow \mathbf{Q}^{*}_{m'}\mathbf{X}\mathbf{Y} &\mathit{by\,\, definition} \end{array}$$

For Isomorphism closure, let f be a bijection from \(D(m)\) to \(D(m')\) and let \(\mathbf {X}^{m'}=f[\mathbf {X}^{m}], \mathbf {Y}^{m'}=f[\mathbf {Y}^{m}]\).

First assume that Q satisfies Isomorphism closure. This yields

$$\begin{array}{lll} \mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} &\Leftrightarrow Q_{D(m)} \mathbf{X}^{m}\mathbf{Y}^{m} &\mathit{by \,\,definition} \\ &\Leftrightarrow Q_{D(m')} f[\mathbf{X}^{m}]f[\mathbf{Y}^{m}] &\mathit{by\,\, Isomorphism\,\, Closure} \\ &\Leftrightarrow Q_{D(m')} \mathbf{X}^{m'}\mathbf{Y}^{m'} &\mathit{by\,\, condition} \\ &\Leftrightarrow \mathbf{Q}^{*}_{m'}AB &\mathit{by\,\, definition} \end{array}$$

The other direction is almost trivial: where \(U, V\subseteq D\), pick a structure with \(D(m)=M\), \(\mathbf {X}^{m}=U, \mathbf {Y}^{m}=V\) and assume \(\mathbf {Q}^{*}\) satisfies Intensional Isomorphism closure to obtain

$$\begin{array}{@{}rcl@{}} Q_{D(m)}UV &\Leftrightarrow \mathbf{Q}^{*}_{m}\mathbf{X}\mathbf{Y} &\mathit{by\,\, definition}\\ &\Leftrightarrow \mathbf{Q}^{*}_{m'}\mathbf{X}\mathbf{Y} &\mathit{by\,\, Intensional \,\,ISOM}\\ &{\kern13pt}\Leftrightarrow Q_{D(m')}UV &\mathit{by\,\, definition} \end{array} $$

□

1.2 A.2 Extensional Intensional Quantifiers

It is a matter of some interest to see under which conditions a given intensional quantifier can be interpreted as a lift of a non-intensional one. As one might expect, the answer is that this is so iff the truth value in a given model depends only on that model and the local extensions there. The following two propositions demonstrate this.

Proposition 1

If an intensional quantifier Q is such that \(\mathbf {Q}_{m}\mathbf {X}\mathbf {Y}\) is a function of \(\mathbf {X}^{m}, \mathbf {Y}^{m}\) and \(D(m)\), then there is a non-intensional quantifier \(Q^{2}\) such that \(\mathbf {Q}_{m}\mathbf {X}\mathbf {Y} \Leftrightarrow (Q^{2})^{*}_{m} \mathbf {X}\mathbf {Y}\).

Proof

For the proof, define

$$Q^{2}_{M}UV \Leftrightarrow \forall m' \mathit {with\,\, domain }\;\;M: \forall \mathbf {X}\in l_{m'}(U), \mathbf {Y}\in l_{m'}(V): \mathbf {Q}_{m'}\mathbf {X}\mathbf {Y} $$

This gives

$$\begin{array}{@{}rcl@{}} (Q^{2})^{*}_{m}\mathbf{X}\mathbf{Y} &\Leftrightarrow &Q^{2}_{M}\mathbf{X}^{m}\mathbf{Y}^{m} \\ &\Leftrightarrow &\forall m' \mathit{with\,\, domain }\;\;M \\ & & \forall \mathbf{X}'\in l_{m'}(\mathbf{X}^{m}), \mathbf{Y}'\in l_{m'}(\mathbf{Y}^{m}): \mathbf{Q}_{m'}\mathbf{X}'\mathbf{Y}' \\ &\Leftrightarrow &\forall \mathbf{X}'\in l_{m}(\mathbf{X}^{m}), \mathbf{Y}'\in l_{m}(\mathbf{Y}^{m}): \mathbf{Q}_{m}\mathbf{X}'\mathbf{Y}' \\ &\Leftrightarrow & \mathbf{Q}_{m}\mathbf{X}\mathbf{Y} \end{array} $$

(In the most important step, we may eliminate “\(\forall m' \mathit {with\,\,domain }\;\;M\)” because \(\mathbf {Q}_{m}'\mathbf {X}'\mathbf {Y}'\) depends only on the domain and the extensions there and the latter have already been fixed by quantifying over \(l_{m'}(\mathbf {X}^{m})\), \(l_{m'}(\mathbf {Y}^{m})\). Similarly, the next universal quantification may be eliminated because by definition all \(\mathbf {X}'\in l_{m}(\mathbf {X}^{m})\) have the same extension in m as \(\mathbf {X}\) (and the same for \(\mathbf {Y}\))).

This covers one direction The other direction is covered by the proposition below, which is trivial enough to require no further proof.

Proposition 2

For any lift \(\mathbf {Q}^{*}\) of a non-intensional quantifier Q, \(\mathbf {Q}^{*}_{m}\mathbf {X}\mathbf {Y}\) is a function of \(\mathbf {X}^{m}, \mathbf {Y}^{m}\) and \(D(m)\).

As mentioned before, good readings of ‘many’ (certainly any reading that avoids the problem mentioned in the introduction while still being Intensionally Conservative) will not be interpretable as a lift of this kind. Such readings will necessarily depend on information beyond what can be drawn from the local extensions and domain, and hence will not be interpretable as a function of only these. □

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Bastiaanse, H.A. The Intensional Many - Conservativity Reclaimed. J Philos Logic 43, 883–901 (2014). https://doi.org/10.1007/s10992-013-9301-7

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