#### Chisholm’s paradox (co-authored with Juhani Yli-Vakkuri)

The following form of modal sorites puzzle has come to be known as Chisholm’s paradox: (i) For all n: if x could have been F(n), then x could have been F(n+1). (ii) x is F(1). (iii) x could not have been F(n*). Or, equivalently (in S5): (i) For all n: necessarily, if x is F(n), then x could have been F(n+1). (ii) x is F(1). (iii) x could not have been F(n*). F(n) might be, for example, the property of being at least n seconds long, while x is a particular performance of a symphony, and n* is some very large number such that it is implausible that that very performance of the symphony could have been at least n* seconds long. Like a number of earlier authors, we do not think that Chisholm-paradoxical arguments call for a treatment different from more familiar forms of sorites puzzle: what must be rejected (at least in a wide variety of cases) is the ‘tolerance’ principle (ii). Yet, in many cases, rejecting the tolerance principle of a Chisholm-paradoxical argument leads to a new puzzle: if we reject it, we will have to accept that there is a possible situation that is qualitatively exactly like (or very similar to) the actual situation but in which we are in the presence of an object that could not have been even slightly different with respect to a certain parameter: for example, a performance of a symphony that could not have been even one second longer, a table that could not have been made without a particular microfibfril of cellulose, or a painting that could not have been created without a particular cadmium atom. In such a situation, it appears, we speak falsely when we assert: ‘Every table could have been made without any given microfibril of cellulose.’ Such mysterious possibilities of error threaten to undermine our apparent knowledge that every table could have been made without any given microfibril of cellulose. We call this the new Chisholm paradox. We discuss and evaluate a range of solutions to it that can be extrapolated from the literature on the old Chisholm paradox, in order of increasing plausibility: S4-denial (Chandler, Salmon), contextualism about metaphysical modality (Murray and Wilson), accepting that in many cases we do not know that an object could have been slightly different with respect to a certain parameter (Williamson), and a theory of semantic drift that assumes a plenitudinous metaphysics (Hawthorne and Yli-Vakkuri, inspired by a suggestion Williamson attributes to Eli Hirsch).