fruitless formalism? « Nate Charlow

fruitless formalism?

February 8, 2009 in Language, Philosophy, Semantics

Maria Aloni (“Free Choice, Modals and Imperatives,” Natural Language Semantics 15) gives a semantic account of free choice effects when possibility modals scope over disjunctions in terms of salient alternatives. She defines an alternative function ${\it alt}$ mapping from a higher-order extension of a first-order modal language into sets of formulas. Free choice and no-choice interpretations of may(φ or ψ) are assigned these intermediate logical forms, respectively.

${\it may}(\exists p (p \wedge (p = \phi \vee p=\psi)))$ (Read: something is the case, either φ or ψ.)
${\it may}(\exists p (p \wedge p = \phi \vee \psi))$ (Read: something is the case, namely: φ or ψ.)

A disjunction whose logical form is rendered as $\exists p (p \wedge (p = \phi \vee p=\psi))$ induces $\phi$ and $\psi$ as alternatives, while one whose logical form is rendered as $\exists p (p \wedge p = \phi \vee \psi)$ induces only $\phi\vee\psi$ as an alternative.

may is treated as a complex modal operator that takes a formula of the language and yields a formula expressing that each of the formula’s alternatives are possible:

${\it may}(\phi) := [\lozenge]({\it alt}(\phi))$
$\ [\lozenge] (\phi_1 , ... , \phi_n) := \lozenge \phi_1 \wedge ... \wedge \lozenge \phi_n$

So: the final account of free choice permission appears to be this: free choice and no choice interpretations are assigned the following logical forms.

$\ [\lozenge]({\it alt}(\exists p (p \wedge (p = \phi \vee p=\psi))))$
$\ [\lozenge]({\it alt}(\exists p (p \wedge p = \phi \vee \psi)))$

I can’t really see any theoretical reason to posit such opulent logical forms (or such an opulent object language).

Presumably features of context (focus marking and the like) determine which LF gets associated with a sentence of the form may(φ or ψ). But the LF’s themselves do no apparent explanatory work–they’re used because the formal apparatus for generating alternatives from formulas (which I didn’t get into in this post) uses propositional variables and their assignment functions. Contexts tell us which LF to use, in view of the formal apparatus for generating alternatives. We use one LF when context tells us we want genuine alternatives, another when context tells us we don’t.

Context, that is to say, is doing all the interesting explanatory work here. The formal apparatus itself seems to be idle–formale gratia formalis. Ultimately I think we could replace Aloni’s account with something like the following, without losing anything important. Depending on features of context (which alternatives context makes salient), may(φ or ψ) receives one of the following LF’s.

$\lozenge\phi\wedge\lozenge\psi$
$\lozenge(\phi\vee\psi)$

An alternative version. Let $I_c$ be an interpretation function for the language. Depending on features of the context $c$ , $I_c({\it alt}_c(\phi\vee\psi))=\{\{v: v \models \phi\},\{v: v \models \psi\}\}$ or $I_c({\it alt}_c(\phi\vee\psi)))=\{\{v: v \models \phi\vee\psi\}\}$ , while may(φ or ψ) always receives the following LF.

$\ [\lozenge]({\it alt}_c(\phi\vee\psi))$

Truth conditions for this LF would be given as follows: $\ [\lozenge]({\it alt}_c(\phi\vee\psi))$ is true at $\langle c,w \rangle$ iff for each alternative $A$ in $I_c({\it alt}_c(\phi\vee\psi))$ , $A$ contains at least one w-accessible world.

Either way, I think we preserve the basic insight of Aloni’s work, lose nothing else important, but get rid of some really clunky formalism and an unwieldy object language.

2 Replies

Craige Roberts

November 16, 2009 at 1:53 pm

I think you’re probably onto something here. I read some of Aloni’s older stuff and was put off by the obtuse formalism. I have strong feelings about keeping the formal tools simple so that the insights shine through. Otherwise, the reader has to guess whether it’s worth her trouble to figure it all out. I guess I have struggled through too many formal frameworks in my time only to figure out in the end that they’re ill-motivated. So it’s getting harder to be patient.

And maybe more important: it’s my belief that when a theory is essentially complex (which you’re suggesting is not the case with Maria’s), something is usually missing. But because of the complexity, it’s difficult to untangle it all and see what the real predictions are. So it behooves the theorist to simplify, to assure the reader that complexity doesn’t hide problems and to permit one to see the predictions.

Sometimes formalism is said to be motivated by other assumptions in the framework, about phenomena orthogonal to what’s at issue; but then I like to follow the method of fragments, abstracting away from what isn’t relevant to get at the core.

OK, off my soapbox.

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1 aloni anticipated by aqvist « de crapulas edormiendo Pingback on Jul 10th, 2009 at 4:58 pm
[...] Aloni does make use of a special logic of alternatives to derive the solution, but the logic doesn’t appear to be doing any work in this case. (She also does cite Aqvist’s piece in her bibliography, but the fundamental [...]