Humans have a wide choice of dimensions they can quantify. Objects and the stuff they are made of have multiple quantitative properties—a berry has dimensions like weight, volume, and temperature, and a group of berries dimensions like numerosity, total weight, total volume, and average temperature. Likewise, water and other nonsolids have distinct quantifiable properties—volume, density, viscosity, temperature, and many others. Scientists, probing further into the nature of things, reveal novel properties that also warrant quantifying. Operationalists (e.g., Bridgman, 1938) believed that the measurement of these properties exhausts the content of scientific concepts like temperature or mass. But even realists about the referents of such concepts believe that measurement is a means of understanding these concepts. Quantification depends on principles that allow scientists to map degrees of properties onto portions of corresponding number systems. The numerosity of a set of objects gets mapped to the natural numbers; continuous dimensions like weight or volume to the reals. Measurement theory provides criteria for the correctness of measurement of this kind (Krantz et al., 1971, Roberts, 1979).

Proper quantification of things depends on the choice of the things, the choice of a dimension for them, and an appropriate measure function from the dimension to numbers. In line with this approach, the Principles and Standards of the National Council of Teachers of Mathematics (2024) advises teachers of pre-K through second-grade to make sure that “each and every student should recognize the attributes of length, volume, weight, area, and time; compare and order objects according to these attributes; understand how to measure using nonstandard and standard units; [and] select an appropriate unit and tool for the attribute being measured.”.

Semantic analysis of natural language uses a similar tactic. According to some theories (e.g., Kennedy and Brown, 2006, Wellwood, 2019), gradable adjectives, such as “big” or “hot,” are associated with “degrees,” which are fundamental quantitative units belonging to scales for the dimension in question. For example, there is a scale of degrees of size and a scale of degrees of temperature. The answers to questions like “Which is bigger?” applied to two animals, for instance, depends on whether the degree associated with the size of the first animal exceeds that of the second (for other linguistic approaches to degrees and comparatives, see Schwarzschild & Wilkinson, 2002, and Scontras, 2014).

This paper examines possible difficulties people have when trying to answer questions about relative amounts of physical objects and substances. We consider cases in which the dimension of comparison is not always clear and look at how people resolve these difficulties by considering linguistic cues and cues from the objects themselves. Our focus is on the ambiguities of counting and measuring: Whether to decide which is more of two groups of objects by counting them or by calculating their total size. We will argue that this decision depends on a combination of linguistic and referential cues, and we will offer a mathematical model for this combination process.

Cognitive theories of how people mentally quantify physical entities bear similarities to the methods of scientists and semanticists. People start by: (a) encoding or recalling the entities to be assessed, then (b) selecting the quantitative property of interest, and finally (c) estimating or comparing the degree of the entities on a corresponding mental scale. Figuring out which is bigger of two objects means singling out the objects, isolating their size, and comparing their degrees on the mental scale for size. We will call steps (a)–(c) “the standard picture” of quantifying.

Experiments in this vein have presented participants with pairs of names or pictures of well-known objects (e.g., “dog” and “bear”) and have asked the participants to choose which is larger or smaller (Moyer, 1973, Paivio, 1975). To predict the response time and accuracy for these decisions, cognitive theories assume that people retrieve from memory information about the items’ size, compute an estimate of size from that information, and then compare the estimates to determine the answer (see Banks, 1977, and Moyer & Dumais, 1978, for reviews of earlier experiments of this type, and Chen et al., 2014, and Leth-Steensen & Marley, 2000, for more recent distributed-memory models in this tradition). A consistent finding, which we make use of here, is that response times are faster the larger the difference between the items: a “distance effect.” For example, participants take less time to compare the sizes of “dog” and “bear” than to compare the sizes of “dog” and “wolf.”.

However, the standard picture runs into complications when we apply it to everyday instances of counting and measuring. This is especially true when it comes to step (b)—selecting the right quantitative dimension for comparison—and this will be the focus of the experiments we report here. A first complication is that the standard picture fails to recognize directly the distinction between counting and measuring as means of assessing amounts. Although both types of comparison produce distance effects and related phenomena, cardinality seems special among quantifiable properties. You could try to reduce counting to measuring or measuring to counting. You could regard the cardinality of a group of items as a kind of measurement of those items: as a measure of the items’ natural units—for example, a measure of berry units in a set of berries (Krifka, 1989). Conversely, you could regard measurement as a way of counting continuous amounts by imposing (abstract) countable units on them: A measure like grams provides gram-sized units by which to count amounts of water, sand, and other quantities (Doetjes, 2021).1

But despite some important analogies between them (e.g., Sutherland, 2022), the distinction between counting and measuring persists in the nature of the scales we use in quantifying: We measure the number of items along a scale of natural numbers, but most other dimensions along a scale of reals. Evidence also exists that people use separate mental processes in counting and measuring. According to some researchers, the distinction is present even in infants. When viewing contrasting image streams of pictures, infants are more likely to attend to the stream that changes in cardinality rather than in total area, even when the area change is of greater magnitude than the number change (Libertus et al., 2014). Especially with large sets of items, infants can detect changes in cardinality over and above changes in continuous dimensions, and these sensitivities require smaller ratios of change to be detectable (e.g., Lipton and Spelke, 2004, Starr and Brannon, 2015). Moreover, children’s abilities to detect discrete and continuous quantities improve on different trajectories (Odic, 2018). Neuropsychological studies have also found differences in the way adults compare displays (Castelli et al., 2006) and solve arithmetic problems (Bagnoud et al., 2018) when the tasks depend on counted versus measured amounts. Although debate exists about whether infants can distinguish cardinality from continuous quantity—we will return to this debate in the General Discussion—there is no doubt that adults can do so, since adults can count. In the rest of the paper, we will observe this distinction by using “measuring” for assessing continuous quantities, “counting” for assessing the cardinality of discrete ones, and “quantifying” as a general expression for either measuring or counting.

A second complication for the standard picture derives from the first. The nature of an item not only limits the dimensions of the object that we can quantify, it also biases the choice between counting and measuring. Both children and adults decide “Which is more?” for solid, individuated objects by counting them, but for nonsolid items by measuring their area or volume (Barner and Snedeker, 2005, Scontras et al., 2017). For example, given a choice between three small shoes and one big shoe whose volume exceeds the sum of the small ones, participants chose the three small shoes as “more shoes.” But given a choice between three small dabs of toothpaste and one large dab, participants chose the large dab as “more toothpaste.” This was true even though both shoes and dabs can be counted, and both have measurable total volume. Nonsolids invite continuous measurement whereas solids invite counting. Likewise, an object’s apparent non-arbitrariness—for example, the regularity or the repeatability of its shape—biases toward treating it as a countable thing, even for novel objects (Prasada et al., 2002, Wellwood et al., 2018; see Rips and Hespos, 2015, Rips and Hespos, 2019, for reviews).

A third issue for the standard picture also stems from the first. Because the standard picture does not differentiate counting and measuring, it fails to recognize the way natural language syntax and semantics distinguish expressions of cardinality from similar expressions of measure and thus determine the result of a comparison (e.g., Landman, 2016, Partee and Borschev, 2012a, Partee and Borschev, 2012b, Rothstein, 2017, Snyder, 2021, Sutton et al., 2021, Chierchia, 1998, Scontras, 2014). For example, distributive expressions like “each” or “apiece” can appear with counted quantities (“Five boxes of sugar cost $5 each”) but seem odd with measured ones (?“Five pounds of sugar cost $5 each”). “Five pounds of sugar” seems to denote a single entity; so “each” has nothing to distribute over. Similarly, the result of a comparison could differ if phrased in terms of the amount of sugar in boxes versus the amount of sugar in pounds (or so we will try to show).

The language of comparison can be ambiguous or underdetermined. “Which is more?” leaves open both the items to be quantified and the appropriate dimension. “Which is more N?” where N is a plural count noun (e.g., “dogs”) or mass noun (e.g., “sand”) can leave open the dimension of comparison (e.g., weight or volume). Even “Which is more N in Q?” (or “Which is more Q of N?”) where Q is a “quantizing noun,” or “classifier” can be ambiguous between different dimensions.2 For example, “Who has more sugar in boxes?” might refer to the total volume of sugar in the boxes or to the number of portions of sugar that the boxes individuate.

The perspective on quantifying that we take here derives from the observation that a range exists for the meanings of quantitative expressions from those that require counting to those that require measuring. Partee and Borschev, 2012a, Partee and Borschev, 2012b point out that phrases like “a box of sand” have several readings that include those focusing on the container (the box), those focusing on the combination of the container and its contents (the box plus the sand), those focusing on the contents (a portion of sand in the box), and those focusing on a measure of the contents (amount of sand in box-sized units). Fig. 1 sketches this range of meanings.

When numbers appear with such phrases (“two boxes of sand”), the interpretation of the phrase varies with that of the container, according to the linguistic theories. At one end of this range (left end of Fig. 1), the numbers simply count the containers (the number of boxes), but they shift to counting the containers plus their contents (the number of boxes + sand) to counting the individuated contents (the number of portions of sand) to measuring those contents. Measuring (right end of Fig. 1) can use either ad hoc units like boxes or standardized units like liters. The size of the ad hoc units may be inexact or unknown, increasing ambiguity and flexibility (Dodge & Wright, 2002).

Near one end of this range, (1a) suggests that Jennifer and Kelly each stacked two boxes filled with sand into the trucks; but near the other end, (1b) suggests that Martha and Nora each poured amounts of sand, measuring two boxfuls, into the mixers.“Stacked” in (1a) selects a counted reading, and “poured” in (1b) a measured reading. In line with this difference, (1a) seems odd with fractional boxes, since it deals with counted (integer) quantities. But (1b) seems sensible with fractional boxes, since it deals with real-number measurement of volume (Partee and Borschev, 2012a, Partee and Borschev, 2012b).

In this article, we develop the Fig. 1 picture in two ways. First, a similar range of meanings occurs even with expressions that focus on the contents of containers rather than on the container itself. Phrases like “sand in boxes” are unambiguously about sand, and as mass noun phrases, numbers can’t directly quantify them (*“two sand(s) in boxes”). But if numbers apply to the containers within these phrases, the phrases can display a similar array of meanings as those in Fig. 1, from counted to measured interpretations. For example, (2a), like (1a), seems to require a boxes + sand reading, but (2b), like (1b), requires a measure of sand:If Jennifer and Kelly had stacked different quantities (say, sand in two vs. three boxes), the number of boxes would seem to determine who stacked more sand in boxes. But if Martha and Nora poured different quantities, then the volume of sand would seem to decide who poured more sand in boxes.

We can also identify factors that determine where on the counted-to-measured scale a given “more” comparison will fall. One of these is the type of quantizer Q, as the semanticists cited earlier have noted. At one extreme, Q can convey an individual object composed of N (as in (3b)), and at the other extreme, an explicit measure of N (as in (3e)). In between these readings, are cases in which Q is a container (or container + ful), as in (3c) and (3d), and where the reading can shift from a count of the containers to a measure of their contents.However, our second proposed extension to the counted-to-measured theory is that the noun itself or its referent can play a role in the interpretation. “Who has more N in Q?” where N is a mass noun, can strengthen the content reading, leading to more measure interpretations than “Who has more in Q?” Even the interpretation of “Who has more?” can vary with the intended referents of the comparison, with counted answers more prominent for individuated objects, like boxes or beakers, and measured answers for substances, like water or sand (see Rips and Hespos, 2015, Rips and Hespos, 2019 for reviews of earlier research on the object/substance distinction). Experiment 1 examines the first of these factors, varying the type of quantizer. Experiments 2 and 3 look at the effect of adding a mass noun and varying the intended referent. The General Discussion proposes a model of how these factors combine.

Comparative questions sometimes bias toward a particular interpretation—either counted or measured—as in the case of “Who has more apples?” and “Who has more pounds of apples?” For these questions, choice of interpretation might depend on a single feature (e.g., ±count) that the noun or the quantizer supplies. But as we have noted, the interpretation is not always clear cut (see Landman, 2016, for a critique of feature-based approaches). In the case of quantizers, consider the questions in (3), as applied to the situation in Fig. 2, where the square-shaped items in each of the two open “containers” consist of or are completely full of sugar. As we’ve noted earlier, the constructions in (3) make it clear that all the questions are about “more sugar” rather than directly about grams, boxfuls, boxes, or cubes, as might be the case for questions like “Which has more cubes of sugar?” That is, the questions attempt to hold constant the entity (the sugar) that is the subject of the comparison. We will therefore focus on constructions of type (3) in these experiments.

Both “cubes” (as in “sugar cubes”) and “boxes” in (3b) and (3c) are ordinary count nouns. The counted interpretation is prominent for “cubes” in (3b), so the likely answer is the left-hand set in Fig. 2. However, “boxes” can also be understood as a temporary unit of measure for its contents, according to the semantic theories mentioned earlier. If the measure interpretation is available for “boxes” in (3c), the “Which has more?” question should result in more choices of the right-hand set than for “cubes” in (3b). The “-ful” suffix for “boxfuls” in (3d) lends more emphasis to the measure interpretation, as we saw earlier, and should produce further choices of the right-hand set in the case of Fig. 2. Finally, “grams” in (3e) is an explicit measure term and should make the choice of the right-hand set the more reasonable one. (Although “grams” is a measure of mass, it correlates with volume under the conditions just mentioned.) So questions (3b)-(3e) vary from a more clearly counted interpretation to a more clearly measured one. Experiments 1 and 2 attempt to document the effects of a range of quantizers using examples like these. Experiments 2 and 3 examine the relation between these quantizers and the nouns and referents they quantify.

To pin down the participants’ understanding of these questions, we vary orthogonally the number of squares in the two containers and their total area across trials. Participants choose, under speeded conditions, which container correctly answers questions like those in (3). We take as the signature of a count interpretation faster response times when the two sets differ by a larger number of items (e.g., faster times when the sets contain six vs. four squares than five vs. four). This follows from the many studies of comparison, mentioned earlier, that have found faster choice times when disparities are bigger. Likewise, we take the signature of a measure interpretation to be faster response times when the two sets differ by a larger total area.

We note that a similar technique appears in earlier studies that have looked at whether people privilege measuring or counting in quantifying visual displays and at how changes in area or number affect their estimates of the other. In experiments with adults, some researchers have found that area biases numerosity estimates to a greater extent than the reverse (Hurewitz et al., 2006, Yousif and Keil, 2020); that biases exist in both directions (Nys & Content, 2012); and that little bias exists in either direction when proper controls are in place (Barth, 2008, Tomlinson et al., 2020). These studies are the topic of current methodological debate about the proper way to control continuous variables, such as area, density, and perimeter (e.g., Aulet and Lourenco, 2021, Barth, 2008, Park, 2022, Yousif and Keil, 2020; see also related debate about infants’ quantitative abilities by, e.g., Clearfield & Mix, 1999, and Feigenson et al., 2002). As we have noticed, however, when adults decide which is more (in settings like that of Fig. 2), their answers differ when they are judging number than when they are judging area. The main goal of the present experiments is not to examine people’s method of estimating absolute number or area, but to study the way quantizers and other variables change the extent to which people rely on these two dimensions. The stimulus items here are also constant for each of the quantizers. For this reason, resolving these controversies about estimates of area or number is not crucial to the experiments’ aims.

A related point is that area is just one of the many continuous measures that apply to groups of physical objects, such as the displays in Fig. 2. The two groups of squares also differ in height, total contour length, density, total projected volume, area of their convex hulls, and an indefinite number of other variables. The aim of the experiments is not to show that area is the measure that people use when answering questions like (3e) in preference to other continuous measures. Instead, the experiments attempt to determine the relative importance they attach to measuring versus counting as a function of quantizers and other factors. Although we will refer to the continuous measurable variable in the stimulus displays as “area,” since that is the one we manipulate explicitly, we acknowledge that area may be a proxy for other measurable dimensions. However, as long as participants’ choices differ for counting versus measuring as a function of the quantizers, their selection of a specific measured dimension is not critical.

We note too that although we are examining the effects of linguistic factors, such as the quantizers in (3), our aim is not to describe the word-by-word processing of such questions (see Grant et al., 2019, for a processing analysis of other types of degree constructions). Instead, the more modest goal is to see how people decide on a counting or measuring interpretation when they have full access to the question at issue and the materials to be quantified.