Conceptual alternatives

Abstract

Things we can say, and the ways in which we can say them, compete with one another. And this has consequences: words we decide not to pronounce have critical effects on the messages we end up conveying. For instance, in saying Chris is a good teacher, we may convey that Chris is not an amazing teacher. How this happens is an unsolvable problem, unless a theory of alternatives indicates what counts, among all the things that have not been pronounced. It is sometimes assumed, explicitly or implicitly, that any word counts, as long as that word could have replaced one that was actually pronounced. We review arguments for going beyond this powerful idea. In doing so, we argue that the level of words is not the right (or at least not the only) level of analysis for alternatives. Instead, we capitalize on recent conceptual and associated methodological advances within the study of the so-called “language of thought” to reopen the problem from a new perspective. Specifically, we provide theoretical and experimental arguments that the relation between alternatives and words may be indirect, and that alternatives are not merely linguistic objects in the traditional sense. Rather, we propose that competition in language is significantly determined by general reasoning preferences, or thought preferences (preferences that may have forged the lexicons of modern languages in the first place, as argued elsewhere). We propose that such non-linguistic preferences can be measured and that these measures can be used to explain linguistic competition, non-linguistically, and more in depth.

Appendix: Statistical analysis

1.1 General description and notations

Semi-formally, our measure is based on three parameters for each quantifier pair p. First, $\overline{\alpha _p}$ represents our best estimate of the true rate of ‘yes’ responses in the no cases, and $\overline{\beta _p}$ represents our best estimate of the true rate of ‘yes’ responses in the yes cases. In an ideal world, we should find $\overline{\alpha _p}=0$ and $\overline{\beta _p}=1$.⁸ Critically, $\overline{\gamma _p}$ represents the preference between the weak and the strong quantifier, estimated as the position of the responses to the target types not as an absolute rate, but as a proportion with $\overline{\alpha _p}$ and $\overline{\beta _p}$ as extreme points. In pseudo-formula, this means that the actual response in the target type is predicted by $\overline{\alpha _p} + \overline{\gamma _p} \cdot (\overline{\beta _p} - \overline{\alpha _p})$. Concretely, a maximal preference for the weaker quantifier in p would correspond to $\overline{\gamma _p} = 100\%$, i.e. an expected response rate in the target case as high as in the yes cases: $\overline{\alpha _p} + 100\% \cdot (\overline{\beta _p} - \overline{\alpha _p}) = \overline{\beta _p}$. Conversely, an extreme preference for the stronger quantifier would correspond to $\overline{\gamma _p} = 0\%$, i.e. an expected response rate in the target case as low as in the no cases: $\overline{\alpha _p} + 0\% \cdot (\overline{\beta _p} - \overline{\alpha _p}) = \overline{\alpha _p}$. Finally, a neutral preference corresponds to $\overline{\gamma _p} = 50\%$ and an expected response rate exactly in between $\overline{\alpha _p}$ and $\overline{\beta _p}$: $\overline{\alpha _p} + 50\% \cdot (\overline{\beta _p} - \overline{\alpha _p}) = (\overline{\alpha _p} + \overline{\beta _p})/2$.

To explain, imagine an ideal world in which participants always infer one of the two rules based on the quantifiers in the pair, and never any other rule. Then the yes items (where both rules are satisfied) and no items (where neither rule is satisfied) should be at ceiling and floor, respectively. In this ideal world, the responses to target items give us a direct measure of preference between the two rules. In actual fact, various biases, errors, and additional rules can play into the task, and hence yes and no items are not at ceiling/bottom. Still, the positioning of target items within the range between the two extremes provided by yes and no items provides a measure of preference: the closer target is to yes, the more the weaker quantifier is preferred to the stronger. This rescaling of the range for target items to the span between yes and no eliminates noise created by general yes/no biases and rules that are weaker or stronger than both quantifiers in the pair.

One source of simplification here is that we ignore the possibility that participants might infer a rule that is ‘intermediate’ between the two members of the pair; for example, ‘at least 3’ and ‘more than half’ are intermediate between ‘some’ and ‘all’. Depending on the specific rule participants infer and the particular test box showing, this could alter the rate of ‘yes’ responses, but not because participants opt for quantifiers from our pair. We thus work on the assumption that the two rules we compare are the most salient in their range. However, we also note that our main interpretations are based not only on pairs, but on the comparison between two pairs of quantifiers. We submit that the intermediate rules relevant for these two pairs are similar across the pairs we use, so that their role should be cancelled out in our analysis. For instance, when we compare ‘all’ and ‘SBNA’ through the pairs (some, all) and (some, SBNA), we assume that the relevant intermediate rules, which are compatible with the positive and negative examples being what they are, and which could make the test box true, are of the form ‘at least x’ in both cases.

1.2 Models and analyses

The logit models we fit thus had a slightly different structure from the generalized linear models commonly used in the analysis of binary response data. Their basic form was as follows:

$$\begin{aligned}&\displaystyle Y_{pi} \sim \mathsf {bernoulli}(\mathsf {logit}^{-1}(\pi _{pi}))\\&\displaystyle \text {with: }\pi _{pi} = \alpha _p \cdot \textsc {no}_i + \beta _p \cdot \textsc {yes}_i + (\alpha _p + \gamma _p \cdot (\beta _p - \alpha _p)) \cdot \textsc {target}_i. \end{aligned}$$

Given a pair of quantifiers $p=(x,y)$, the parameter $\gamma _{(x,\,y)}$ thus represents the preference for the weaker member over the stronger member. Given two pairs of quantifiers $(x,\,y_1)$ and $(x,\,y_2)$ that share a common member x, we can then compare $y_1$ and $y_2$ by fitting a model to the data from both pairs and seeing whether there is evidence for a difference between $\gamma _{(x,\,y_1)}$ and $\gamma _{(x,\,y_2)}$. If there is one, then one of the y’s is preferred to the other, in that it is more distinct from x than the other is.

We partitioned our data into three sets based on the quantifier pairs that shared a member: (i) a set comprising data from (some, all) and (some, SBNA); (ii) a set comprising data from (not all, no) and (not all, SBNA); and (iii) a set comprising data from (at least n, exactly n) and (at most n, exactly n). On each of these three data sets, we fitted models that included either one or two $\gamma$ parameters (allowing, or not allowing, for a relative preference between the two non-shared quantifiers in the two pairs, as discussed in the main text),⁹ and different varying-effects structures: there was (or was not) a subject intercept $u_{0s}$, and either no modification of the parameters, parameter modifiers differing by subject, or parameter modifiers differing by both subject and pair. All models were fitted with MCMC methods using STAN through the rstan package in R, with STAN’s default uniform (improper) prior over $\alpha$, $\beta$, and $\gamma$ (in the latter case restricted to [0, 1]) as well as the standard deviation hyperparameters for the varying subject effects (which themselves were assumed to be normally distributed around 0). Models were evaluated by leave-one-out cross-validation, approximated by Pareto-smoothed importance sampling with the loo package (Vehtari et al. 2016) on the basis of 5000 samples of the likelihood for each data point, drawn after 5000 burn-in iterations.

1.3 Results

For all three data sets, the best model turned out to be the two-$\gamma$ model with a subject intercept and parameter modifiers varying by subject, but not by pair:¹⁰

$$\begin{aligned} Y_{spi} \sim \mathsf {bernoulli}(\mathsf {logit}^{-1}(\pi _{spi})) \end{aligned}$$

$$\begin{aligned} \text {with: } \pi _{spi}= & {} (\alpha _p + u_{\alpha s}) \cdot \textsc {no}_i + (\beta _q + u_{\beta s}) \cdot \textsc {yes}_i\\&+ (\alpha _p + u_{\alpha s} + (\gamma _p + u_{\gamma s})((\beta _p + u_{\beta s}) - (\alpha _p + u_{\alpha s}))) \cdot \textsc {target}_i + u_{0s}. \end{aligned}$$

In order to see whether the preference for two $\gamma$-parameters was meaningful, we compared the optimal models to the variants with identical varying effects structure, but only one (pair-independent) $\gamma$-parameter. Differences in estimated log pointwise predictive likelihood and their standard errors for models with one and two $\gamma$-parameters are given in Table 2.

In addition, posterior means and 95% credible intervals for the difference between $\gamma$-parameters are shown in Table 3.

Table 2 Differences in estimated log pointwise predictive likelihood and their standard errors for models with one or two $\gamma$ parameters for sets of quantifier pairs sharing a weaker member

Table 3 Posterior means and 95% credible intervals for differences between $\gamma$ parameters, reflecting relative preference between quantifiers not shared between the pairs

With a $\Delta _{elpd}$ of $-32.0$, there is strong evidence for a preference for ‘all’ over ‘SBNA’.¹¹ The direction of this preference is given by the negative difference in $\gamma$ parameters. If x is stronger than y in a pair (x, y), then a low $\gamma _{(x,y)}$ indicates a stronger preference for x, and since $\gamma _{(\text {some,all})}$ is smaller than $\gamma _{(\text {some,SBNA})}$, we know that ‘all’ is preferred to ‘some’ more strongly than ‘SBNA’ is preferred to ‘some’, hence that ‘all’ is preferred to ‘SBNA’.

The same reasoning applies to the comparison of ‘no’ with ‘SBNA’ via comparison of each with ‘not all’. In this instance, however, the effect is smaller and the strength of the evidence, with a $\Delta _{elpd}$ of $-3.8$, much more modest.

For ‘at least’ and ‘at most’, a greater $\gamma$ value indicates a stronger preference over ‘exactly’, because now the quantifiers of interest are the weaker members of the pair. Since $\gamma _{(\text {at least,exactly})}$ is greater than $\gamma _{(\text {at most,exactly})}$, we can infer a preference for ‘at least’ over ‘at most’. The effect size here is in the same ballpark as with ‘all’ versus ‘SBNA’, and the evidence quite strong.