Editing Sorites paradox (section)

==General conditional sorites==
A formal generalization of the paradoxical sorites argument is as follows:<ref>{{Cite book |last=Ronzitti |first=Giuseppina |url=https://www.google.com/books/edition/Vagueness_A_Guide/WhutdYQz6mMC?hl=en&gbpv=1&dq=vagueness+a+guide&printsec=frontcover |title=Vagueness: A Guide |date=2011-03-03 |publisher=Springer Science & Business Media |isbn=978-94-007-0375-9 |pages=4 |language=en}}</ref>

:''<math>Fa_{1}</math>''.
:''If <math>Fa_{1}</math>, then <math>Fa_{2}</math>.''
:''If <math>Fa_{2}</math>, then <math>Fa_{3}</math>.''
:''<math>\vdots</math>''
:''If <math>Fa_{n-1}</math>, then <math>Fa_{n}</math>.''
:''<math>\overline{\therefore Fa_{n}}</math>'' (where <math>n</math> can be arbitrarily large)
This formalization is in [[first-order logic]], where <math>F</math> is a predicate and <math>a_{1}, a_{2}, a_{3}, \ldots, a_{n}</math> are different subjects to which it may be applied; for each subject <math>a_{i}</math>, the notation <math>Fa_{i}</math> signifies the application of the predicate <math>F</math> to <math>a_{i}</math>, i.e., the proposition that "<math>a_{i}</math> is <math>F</math>". ([[Jonathan Barnes]] originally represented each "if <math>Fa_{n}</math>, then <math>Fa_{n+1}</math>" proposition using the symbol <math>\supset</math> for the [[Material conditional|material implication]] [[Logical connective|connective]], so his argument originally ended with <math>Fa_{n-1} \supset Fa_{n}</math>.)<ref name=":1" />

[[Jonathan Barnes]] has discovered the conditions for an argument of this general form to be soritical.<ref name=":1">{{Cite book |last=Barnes |first=Jonathan |url=https://www.google.com.br/books/edition/Science_and_Speculation/yWF8s27NpmUC |title=Science and Speculation |last2=Brunschwig |first2=J. |date=2005-11-10 |publisher=Cambridge University Press |isbn=978-0-521-02218-7 |pages=30-31 |language=en}}</ref> First, the '''series''' <math>\langle a_{1}, . . . , a_{n} \rangle</math> must be ordered; for example, heaps may be ordered according to number of grains of sand in them, or, in the ''falakros'' version (see {{section link||Variations}}), heads may be ordered according to the number of hairs on them. Second, the '''predicate''' <math>F</math> must be ''soritical relative to the series'' <math>\langle a_{1}, . . . , a_{n} \rangle</math>, which means: first, that it is, to all appearances, true of <math>a_{1}</math>, the first item in the series; second, that it is, to all appearances, false of <math>a_{n}</math>, the last item in the series; and third, that all adjacent pairs of subjects in the series, <math>a_{i}</math> and <math>a_{i+1}</math>, are, to all appearances, so similar as to be indiscriminable in respect of <math>F</math> – that is, it must seem that either both of <math>a_{i}</math> and <math>a_{i+1}</math> satisfy <math>F</math>, or neither do.

This last condition on the predicate is what [[Crispin Wright]] called the predicate's ''tolerance'' of small degrees of change, and which he considered a condition of a predicate's being [[Vagueness|vague]].<ref name=":0">{{Cite book |last=Keefe |first=Rosanna |url=https://www.google.com.br/books/edition/Vagueness/mywTDgAAQBAJ |title=Vagueness: A Reader |last2=Smith |first2=Peter |date=1996 |publisher=MIT Press |isbn=978-0-262-11225-3 |pages=156-157 |language=en}}</ref> As Wright said, supposing that <math>\phi</math> is a concept related to a predicate <math>F</math> such that "any object which <math>F</math> characterizes may be changed into one which it does not simply by sufficient change in respect of <math>\phi</math>", then "<math>F</math> is ''tolerant'' with respect to <math>\phi</math> if there is also some positive degree of change in respect of <math>\phi</math> insufficient ever to affect the justice with which <math>F</math> applies to a particular case."<ref name=":0" />