# The Ockham Society (Friday - Week 6, MT19)

On first sight, it seems as if it is quite easy to give the semantics for counting sentences such as “Exactly two oranges are on the table”. On the simple account, this sentence is true if there are two things, which are both on the table, both oranges and not identical, and everything else is either not on the table or not an orange. However, this simple account becomes problematic when partial objects are considered.

Imagine there are three oranges on a table. Roman eats half of one of the three oranges and puts the remaining half back on the table. If one asked somebody, how many oranges are on the table now, this person would probably confidently reply “Two and a half oranges are on the table”. This is problematic for the simple account. By law of the excluded middle, the half orange either is an orange or it is not an orange. If on the one hand it is an orange, the simple account predicts that “There are exactly three oranges on the table” is true, for there are three different things, which are oranges and on the table (the first orange, the second orange and the orange half) and everything else is either not an orange or not on the table. If on the other hand the orange half is not an orange, the simple account predicts that “There are exactly two oranges on the table” is true, for there are two different things, which are oranges and on the table (the first orange, the second orange) and everything else is either not an orange or not on the table. Both predictions are not compatible with the intuitively correct reply “Two and a half oranges are on the table”.

What makes this even more problematic is that the difficulty of getting the right semantics for counting sentences involving fractions carries over to counting sentences that only involve natural numbers.

Salmon provides a tentative solution for the problem and Liebesman provides a more detailed solution (Liebesman 2015 & 2016, Salmon 1997). However, both Liebesman and Salmon assume that half oranges are not oranges, i.e. that half oranges are not in the extension of “orange”. While this does not make a difference to whether the problem arises, it makes a difference for what kinds of solutions can be given. I will argue that half oranges are in the extension of “orange”. This is because of observations regarding how we use “orange” and “half orange”. I will then give a different account of counting which deals with the problem for counting sentences and is compatible with the assumption that half oranges are in the extension of “orange”.

Ockham Society Convenor: Sean Costello | Ockham Society Webpage