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by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
11.
Fuzzy Logics
11.2
Sorites Paradoxes
Brief summary:
(11.2.1) Priest first illustrates the sorites paradox. A person begins at age five and is thus a child. One second after that the person is still a child. Thus also one second after that new second the person is still a child. No additional second will cause the child to definitively cease being a child and start being an adult. However, after 30 years, we know that the person is now an adult. (11.2.2) The sorites paradox results from vague predicates like “is a child,” where , “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221). (11.2.3) Many other vague predicates, like “is tall,” “is drunk,” “is red,” “is a heap,” and even “is dead,” can all be used to construct sorites paradoxes. (11.2.4) We can structure the sorites paradox as a chain of modus ponens inferences where we say that something begins at a certain state at a certain time, and next that if something is so at that time it is so in the next second, and we repeat that indefinitely, never arriving upon the state we know it will change into.
[Sorites’ Paradox Age Illustration]
[Vague Predicates Causing the Sorites Paradox]
[Other Vague Predicates]
[The Inferential Structure of Sorites Paradoxes]
Summary
[Sorites’ Paradox Age Illustration]
[Priest first illustrates the sorites paradox. A person begins at age five and is thus a child. One second after that the person is still a child. Thus also one second after that new second the person is still a child. No additional second will cause the child to definitively cease being a child and start being an adult. However, after 30 years, we know that the person is now an adult.]
[Priest first illustrates the sorites paradox. (He gives a similar example in his book Logic: A Very Short Introduction, chap.10.) Here we have a graduated transition between two distinct and exclusive states, where there seems to be no way to know when along the transition it can be said that the change of state happens; or rather, we cannot on the basis of the addition of each small alteration induce that the alternate state will ever be attained.]
Suppose that Mary is aged five, and hence is a child. If someone is a child, they are a child one second later: there is no second at which a person turns from a child to an adult. (We are talking about biological childhood here, not legal childhood. The latter does terminate at the instant someone turns eighteen, in many jurisdictions.) So in one second’s time, Mary will still be a child. Hence, one second after that, she will still be a child; and one second after that; and one second after that ... Hence, Mary will be a child after any number of seconds have elapsed. But this is, of course, absurd. After an appropriate number of seconds have elapsed, so have thirty years, by which time Mary is thirty-five, and so certainly not a child.
(221)
[Vague Predicates Causing the Sorites Paradox]
[The sorites paradox results from vague predicates like “is a child,” where , “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221).]
Priest now explains the problem involved in the sorites paradox. Mary has the predicate “is a child.” But this predicate is vague; for, “very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate” (221). [This notion of vagueness as understood here is still a bit unclear to me, so I will not try to further comment until learning more in forthcoming sections.]
The argument of 11.2.1 is known as a sorites paradox. It arises because the predicate ‘is a child’ is vague in a certain sense. Specifically, very small changes to an object (in this case, a person) seem to have no effect on the applicability of the predicate.
(221)
[Other Vague Predicates]
[Many other vague predicates, like “is tall,” “is drunk,” “is red,” “is a heap,” and even “is dead,” can all be used to construct sorites paradoxes.]
Priest then gives other examples of vague predicates that can result in sorites paradoxes:
In fact, most of the predicates we commonly use are vague in this sense: ‘is tall’, ‘is drunk’, ‘is red’, ‘appears red’, ‘is a heap of sand’ (‘sorites’ comes from the Greek soros meaning ‘heap’) – even ‘is dead’ (dying takes time: one nanosecond makes no difference). One can construct sorites arguments for all such predicates.
(222)
[The Inferential Structure of Sorites Paradoxes]
[We can structure the sorites paradox as a chain of modus ponens inferences where we say that something begins at a certain state at a certain time, and next that if something is so at that time it is so in the next second, and we repeat that indefinitely, never arriving upon the state we know it will change into.]
Priest lastly provides an inferential structuration for sorites paradoxes. We chain together a series of modus ponens inferences where we say that “if the thing is of a certain state at a certain time, then it is still in that state one second later.” We also affirm the antecedent and infer the consequent. We repeat the same procedure on the predication for that next second, and do so indefinitely.
Sorites arguments can often be put in the form of a sequence of modus ponens inferences. Thus, if Mi is the sentence ‘Mary is a child after i seconds’, then the sorites of 11.2.1 is just:
M0 xxx M0 → M1
____________
xxxxxM1 M1 → M2
xxxxx____________
xxxxxxxxxM2
xxxxxxxxx.
xxxxxxxxxx.
xxxxxxxxxxx.
xxxxxxxxxxxxxxMk-1 Mk-1 → Mk
xxxxxxxxxxxxxx____________
xxxxxxxxxxxxxxxxxxxxMk
where k is some very large number.
(222)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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