~(Fn & ~Fn+1)
Fn -> Fn+1.
In order to fix it, we could weaken it (or its consequences) by (i) weakening the force of the negation on the outside by using many-valued logic, metalinguistic negation, etc.; (ii) adding operators like "Knowably", "Determinately", etc. in various places. Both of these things have been done; one could try [with "D" for box and "C" for diamond]:
(0) DFn -> Fn+1 [Williamson's Margin for Error Principle--taken as constitutive of vagueness?...Not quite; taken as constitutive of Inexact Knowledge, of which vagueness is a species.]
(1) DFn -> CFn+1
(2) C(DFn -> DFn+1)
...I don't know whether any entailment relations hold between these. (iii) We could question the truth-preservingness of the many, many applications of MP (an infinite number?) required to get us from the plausible premises to the absurd conclusion. (iv) We could say that the premise is true or valid in some way, but not in the same way as the other premises that get us to a contradiction: for example, that it is true in some metalinguistic sense, or that it expresses a truth about use, rather than a truth about meaning. [A model for this sort of suggestion: some of the premises are logical truths in the logic of indexicals, while others are logical truths in the standard sense.]
But here is another way we could try to fix it: what is wrong with all of these principles is that they appeal to a "next" item in the sorites series---a next tile, a next grain of sand, etc. Clearly, for heaps and rows of tiles, there will *be* a next grain, and a next tile. However, the suggestion is that what we have in mind when we have tolerance in mind is something like:
Each shade of red is next to another shade of red.
or
Each point on the spectrum, if it is red, is next to another red point.
We naturally wish to say that this other red shade, or red point, is very close to the original red one, and so we seek to express this thought with the original tolerance principle, (An)Fn -> Fn+1. The thought seems right; it is its expression in terms of "next" which is wrong, for there is no well-defined way of getting to the next point or the next shade. We go further wrong when we substitute "next tile" for "next point"; since there is only an infinite number of tiles, repeated applications of the principle will get us all the way to the end of the spectrum.
Can we better understand the sorites---originally, a paradox of heaps, where heaps are made of discrete grains of sand---by this route---by arguing that we mistakenly transform a principle which is valid for nondiscrete quantities into one which could apply to discrete quantities, and then so apply it? Furthermore, how should we fix our reasoning?---can some modified version of Tolerance, or the Margin of Error principle, be found, and if so, what use would it be? (Note, of course, that it wouldn't be of use in getting us to the contradiction again.)
Tolerance was originally offered as a criterion for vagueness; one which replaced the older description in terms of indecision in light of all the nonsemantic facts. If a newer version could be found, we could avoid the despairing conclusion (which has been aired) that what makes a term vague that we use it incoherently as a matter of semantic competence.
What, finally, would success in this endeavor suggest---should we conclude that there is a continuum of operators, truth-predicates, or colors themselves? Since it appears that the most plausible starting assumption is about red, and not truth, or the "determinately" operator, can we say that vagueness originates here?
****Wright quotes
"In these examples we encounter the feature of a certain tolerance in the concepts respectively involved, a notion of a degree of change too small to make any difference, as it were. There are degrees of change in point of size, maturity and colour which are insufficient to alter the justice with which some specific predicate of size, maturity or color is applied. This is quite palpably an incoherent feature since, granted that any case to which such a predicate applies may be linked by a series of 'sufficiently small' changes with a case where it is not, it is inconsistent with the exclusivity of the predicate." (333-334)
****Related quote by Rayo 2008 ["Vague Representation"]
In addressing the Sortes Paradox, it is not enough to tell a story whereby [the conditional premise] fails to be true. One must also explain our inclination to accept [it]. It seems to me that the localist is unusually well-placed to supply the necessary explanation. We are tempted to think that [the conditional premise] is true because we make a certain kind of mistake. We think of tolerance---which is a feature of our ability to use linguistic representations---as a semantic principle governing the correctness of our assertions. This leads us from the unobjectionable obser4vation that we are unable to use 'bald' to discriminate between man n and man n+1 to the mistaken conclusion that 'bald' can only be correctly applied to man n if it is also correctly applied to man n+1. It is easy to make this mistake if one is under the grip of a certain conception of language: the idea that there are semantic rules corresponding to sentences, and that language-mastery i sa matter of gaining cognitive access to the relevant rules and learning to apply them in the right sorts of ways. For---as emphasized in Wright 1976 ["Language-Mastery and the Sortes Paradox"]---this picture makes it natural to suppose that one ca nuncover the semantic rules governing our language by straightforward reflection on our usage. But5 the localist thinks of matters very differently. Language-mastery is not a matter of applying semantic rules; it is a matter of making sensible semi-principled decisions about how to partition the context set in light of past linguistic usage and the location of the gap.
(1) DFn -> CFn+1
(2) C(DFn -> DFn+1)
...I don't know whether any entailment relations hold between these. (iii) We could question the truth-preservingness of the many, many applications of MP (an infinite number?) required to get us from the plausible premises to the absurd conclusion. (iv) We could say that the premise is true or valid in some way, but not in the same way as the other premises that get us to a contradiction: for example, that it is true in some metalinguistic sense, or that it expresses a truth about use, rather than a truth about meaning. [A model for this sort of suggestion: some of the premises are logical truths in the logic of indexicals, while others are logical truths in the standard sense.]
But here is another way we could try to fix it: what is wrong with all of these principles is that they appeal to a "next" item in the sorites series---a next tile, a next grain of sand, etc. Clearly, for heaps and rows of tiles, there will *be* a next grain, and a next tile. However, the suggestion is that what we have in mind when we have tolerance in mind is something like:
Each shade of red is next to another shade of red.
or
Each point on the spectrum, if it is red, is next to another red point.
We naturally wish to say that this other red shade, or red point, is very close to the original red one, and so we seek to express this thought with the original tolerance principle, (An)Fn -> Fn+1. The thought seems right; it is its expression in terms of "next" which is wrong, for there is no well-defined way of getting to the next point or the next shade. We go further wrong when we substitute "next tile" for "next point"; since there is only an infinite number of tiles, repeated applications of the principle will get us all the way to the end of the spectrum.
Can we better understand the sorites---originally, a paradox of heaps, where heaps are made of discrete grains of sand---by this route---by arguing that we mistakenly transform a principle which is valid for nondiscrete quantities into one which could apply to discrete quantities, and then so apply it? Furthermore, how should we fix our reasoning?---can some modified version of Tolerance, or the Margin of Error principle, be found, and if so, what use would it be? (Note, of course, that it wouldn't be of use in getting us to the contradiction again.)
Tolerance was originally offered as a criterion for vagueness; one which replaced the older description in terms of indecision in light of all the nonsemantic facts. If a newer version could be found, we could avoid the despairing conclusion (which has been aired) that what makes a term vague that we use it incoherently as a matter of semantic competence.
What, finally, would success in this endeavor suggest---should we conclude that there is a continuum of operators, truth-predicates, or colors themselves? Since it appears that the most plausible starting assumption is about red, and not truth, or the "determinately" operator, can we say that vagueness originates here?
****Wright quotes
"In these examples we encounter the feature of a certain tolerance in the concepts respectively involved, a notion of a degree of change too small to make any difference, as it were. There are degrees of change in point of size, maturity and colour which are insufficient to alter the justice with which some specific predicate of size, maturity or color is applied. This is quite palpably an incoherent feature since, granted that any case to which such a predicate applies may be linked by a series of 'sufficiently small' changes with a case where it is not, it is inconsistent with the exclusivity of the predicate." (333-334)
****Related quote by Rayo 2008 ["Vague Representation"]
In addressing the Sortes Paradox, it is not enough to tell a story whereby [the conditional premise] fails to be true. One must also explain our inclination to accept [it]. It seems to me that the localist is unusually well-placed to supply the necessary explanation. We are tempted to think that [the conditional premise] is true because we make a certain kind of mistake. We think of tolerance---which is a feature of our ability to use linguistic representations---as a semantic principle governing the correctness of our assertions. This leads us from the unobjectionable obser4vation that we are unable to use 'bald' to discriminate between man n and man n+1 to the mistaken conclusion that 'bald' can only be correctly applied to man n if it is also correctly applied to man n+1. It is easy to make this mistake if one is under the grip of a certain conception of language: the idea that there are semantic rules corresponding to sentences, and that language-mastery i sa matter of gaining cognitive access to the relevant rules and learning to apply them in the right sorts of ways. For---as emphasized in Wright 1976 ["Language-Mastery and the Sortes Paradox"]---this picture makes it natural to suppose that one ca nuncover the semantic rules governing our language by straightforward reflection on our usage. But5 the localist thinks of matters very differently. Language-mastery is not a matter of applying semantic rules; it is a matter of making sensible semi-principled decisions about how to partition the context set in light of past linguistic usage and the location of the gap.
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