**Speaker:** Elizabeth Coppock (BU)

**Date and time:** Wednesday February 23, 1-2pm

**Location:** 32-D461

**Title: **Triangle Equivalences

**Abstract: **Denominator phrases like “per ton” can combine either with expressions denoting particular degrees, as in “$100 per ton”, or with expressions denoting measure functions, as in “cost per ton”. As we see regularly when we look at translations of “per” into other languages, there is a systematic pattern of equivalences involving these two options:

(1) a. The cost of wheat is $100 per ton. / b. The cost of wheat per ton is $100.

(2) a. the EUR 100 billion per annum shortfall / b. the per annum shortfall of EUR 100 billion

(based on English-Bulgarian aligned sentences in Europarl)

In the (a) examples, an expression denoting a specific quantity combines with a denominator to yield a fractional degree-denoting term (“$100 per ton”, “EUR 100 billion per annum”) that is somehow equated with a measure function (“cost”, “shortfall”), or its value when applied to some implicit object. In the (b) examples, an expression denoting a measure function combines with a denominator (“cost … per ton”, “per annum shortfall”) to yield an expression that is somehow equated with a specific quantity. The goal of the talk is to explain these equivalences in a way that is mathematically coherent, respecting the principle that quantities cannot be equated unless they are of the same (possibly complex) dimension (building on foundations for degree multiplication drawn from the field of quantity calculus), and in a way that captures my own intuition that in the (b) variants, division is somehow taking place at the level of the measure function. To do so, I propose that the (a) sentences are derived via a “quotient operator” analysis of “per” (one I will advocate for in my upcoming SALT contribution), and the (b) sentences are derived using two tricks: (i) a binary-Geached version of a “quotient term” analysis of “per” (one I advocated for in my contribution to last year’s SALT); and (ii) a type-shifting operation that converts units like “ton” (type d) to unit functions (type <e,d>) like “lambda x . the weight of x in tons”.