Dimensional Feature Analysis of English Prepositions

CJ Quines

The missing preposition. Consider the three English prepositions in, on, and at. When we use these as spatial prepositions, we use them for referring to being located in a three-dimensional container, on a two-dimensional surface, or at a zero-dimensional point, respectively:

The horse sleeps in the stable.

The apple is on the table.

Brian stood at the origin of the Cartesian plane.

Missing from this paradigm is a preposition corresponding to one-dimensional locations. At first glance, we might believe that this function is also served by the preposition on, as in the sentences:

The boat is on the Charles River.

Point P lies on line segment AB.

While the Charles River or line segment AB seem like one-dimensional locations, we argue that, based on the semantics of how they are used in these sentences, they actually have two dimensions. For indeed, the Charles River is not actually a one-dimensional object: at several points along it, the river is wide enough to have bridges spanning it.

As a mathematical abstraction, line segment AB does not seem to have the same property at first. But we can apply another diagnostic to determine its semantic dimension in this sentence, based on the judgments¹ of:

Point P lies on line segment AB, which is [two-dimensional / *one-dimensional].

Another candidate for the one-dimensional spatial preposition is along, as we see in:

Varying the endpoints of PQ along their corresponding sides of triangle ABC traces out a catenary.

At several points along it, the Charles River is wide enough to have bridges spanning its width.

Indeed, along is a genuinely one-dimensional preposition. Note that we cannot say *along the stable, *along the table, or *along the origin of the Cartesian plane. Does that mean, then, the dimensions from zero to three correspond to the prepositions at, along, on, in?

Feature analysis. We now argue that the sequence at, along, on, in is a manifestation of an inflectional paradigm, where the one-dimensional along is an irregular form. In particular, we derive the sequence via a new theory we call dimensional feature analysis.

In science, the term dimensional analysis refers to manipulating abstract physical dimensions when performing calculations. As a simple example, a measurement of volume has dimension L³, where L represents length, and a measurement of velocity has dimension L/T, where T represents time. One can conclude, then, that multiplying a velocity (L/T) with a time interval (T) and an area (L²) produces a measurement of volume (L³).

Under dimensional analysis, to go from a two-dimensional area (L²) to a three-dimensional volume (L³), we multiply by some one-dimensional length factor (L). In our theory of dimensional feature analysis, we analogously propose that we can transform the two-dimensional spatial preposition on to the three-dimensional spatial preposition in by “multiplying” some one-dimensional factor.

The natural candidate for such a factor is a phonological feature. Indeed, note that three of the four inflections have the same structure of a vowel followed by a consonant: at is [æt], on is [ɔn], in is [ɪn]. Taking the feature [high] for the vowels in the sequence, where we have an unknown position for the first-dimension, we get the binary values [−], [?], [−], [+]. If we then take the numerical representatives 1 for [+] and −1 for [−], we get the sequence −1, ?, −1, 1. But we can produce such a sequence by starting with −1, then multiplying by a factor of −1 each time:

• at: −1 [−high]
• ?: −1 × −1 = 1 [+high]
• on: 1 × −1 = −1 [−high]
• in: −1 × −1 = 1 [+high]

We call −1 the dimensional factor for the [high] feature. Now what about the [back] feature of the vowel? We have [−], [?], [+], [−]. It is not possible to produce this sequence if our representatives were −1 for [−], and 1 for [+]. Instead, we must use complex numbers for our dimensional factor. If we take the set of representatives 1 for [+] and i, −1, −i for [−], we can get the following sequence:

• at: −1 [−back]
• ?: −1 × i = −i [−back]
• on: −i × i = 1 [+back]
• in: 1 × i = i [−back]

In general dimensional feature analysis, we take 1 as the representative for [+], and every other number as the representative for [−]. With complex numbers, there exists a second possibility for the [high] feature. We present this in the table² below, along with our analysis for the remaining vowel features in the sequence. Here, ω represents a primitive cube root of unity, a complex number which satisfies ω ≠ 1 but ω³ = 1.

As English does not have a vowel that is [+high] and [−] in every other feature, we must then choose the possibility of [−], [−], [−], [+]. Our dimensional feature analysis then predicts that the vowel for the one-dimensional spatial preposition is [ə].

A similar analysis⁴ for the consonant of this sequence does not narrow it down to a single possibility, but it does predict that the consonant must be an alveolar that is [+consonantal] and [+sonorant]. Among English consonants, this leaves the candidates [ən] and [əl].

Of these, we find [əl] most likely, both because it is the beginning of along, and because it is cognate to prepositions in other Indo-European languages. As to the origin of the irregular inflection along, we hypothesize its formation via suppletion, perhaps by analogy with the word long. We leave this as an open question for further research.

Conclusion. We presented the theory of dimensional feature analysis, and applied it to determine a likely regular form for the one-dimensional English spatial preposition. This use of dimensional analysis in linguistics suggests some further topics of research:

Temporal topology. Note that we use the zero-dimensional at for times of day, like at 9 PM or at night, but we use the two-dimensional on for days of the week, like on Tuesday. This suggests a semantic temporal topology of a day, as a two-dimensional object built from the path of a zero-dimensional point. This matches the description of a space-filling curve. How does this square with the phrase in an hour, which suggests that hours are three-dimensional?

Fractional dimension. Every space-filling curve is a fractal with Hausdorff dimension 2. But fractals are notable for having dimensions that can be non-integral; examples include the Koch snowflake with dimension ≈1.26, the Sierpiński triangle with dimension ≈1.59, and the coastline of Great Britain with dimension ≈1.25. What English prepositions are used to refer to these locations? Do they match the predictions from dimensional feature analysis?

The quantity of mass. Of the remaining physical quantities in dimensional analysis, the next most frequently used is certainly mass. Consider phrases like in trouble, on fire, or at ease. We see a potential relationship with these and the metaphor of emotions as weights: one can carry a heavy burden or feel an unbearable lightness of being. What relates these constructions? Are there other cases of adpositions of mass?

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