by Corry Shores

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[Priest,* Introduction to Non-Classical Logic*, entry directory]

[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

11.3

. . . and Responses to Them

Brief summary:

(11.3.1) We will now consider responses to the sorites paradox, where for

M_{0},M_{1}, . . . ,M_{k}

*M*_{0} is definitely true but *M _{k}* is definitely false. We will try to understand what is going on, logically speaking, between

*M*

_{0}and

*M*. (11.3.2) If we simply think that every sentence is either simply true or simply false, then we can break the paradoxical chain at some point where we say for example that the person is a child at this moment and in the next one they are an adult (“there must be a unique

_{k}*i*such that

*M*is true, and

_{i}*M*

_{i}_{+1}is false. In this case, the conditional

*M*→

_{i}*M*

_{i}_{+1}is false, and the sorites argument is broken” (222).) But, while that solves the paradox, it goes against our intuition that the change is continuous and thus there can be no such discrete leap happening from one instant to the next (and thus no discrete jump from truth to falsity). (11.3.3) Some still think that there are these discrete leaps of truth value in continuous changes, and the reason it strikes us as counterintuitive is simply because we lack the means to know where exactly the change takes place. (11.3.4) Those arguing the above claim – that there is a discrete truth-value break but we cannot know it – use the following reasoning. We can only know true things, and we can only make judgements from evidence. The discrete truth-value shift in the actual change will make

*M*true but

_{i}*M*

_{i}_{+1}false. However, the evidential basis will be the same for both. This means that when we make the judgment

*M*

_{i}_{+1}(on the basis of the misleading evidence that is the same from the prior moment), we are making a false judgment, and so we can never know when the shift happens. (11.3.5) The main problem with this argument is that the real problem with the paradox is not that we cannot know where the change happens but that there could even be such a sharp cut-off point in a continuous change. (11.3.6) Another proposal is that cases of vagueness require that we reject a bivalent dichotomy between simple truth and falsity, and so for sorites changes, there would be a middle part where the sentences are {1} neither true nor false, or {2} both true or false. (11.3.7) One three-valued solution is using

*K*

_{3 }(and perhaps in addition supervaluation). “In this case, there is some

*i*, such that

*M*is true and

_{i}*M*

_{i}_{+1}is neither true nor false. Again,

*M*→

_{i}*M*

_{i}_{+1}is not true, and so the sorites argument fails” (223). (11.3.8) But three-valued solutions suffer from the same counter-intuitiveness: it is hard to accept that there is a discrete boundary between truth and the middle value. (11.3.9) So since the changes are continuous, we might want to use a fuzzy logic where the truth-values come in continuous degrees too. (11.3.10) But even fuzzy logic has this same problem, because somewhere there must be a change from

*completely true*to

*less than completely true*.

[Turning to Responses to the Sorites Paradoxes]

[The Failure of a Bivalent Classical Solution]

[Claiming There Is Such a Break, but It Is Unknowable]

[The Reasoning Behind the Not-Knowing Theory]

[The Failure of the Not-Knowing Theory]

[Using a Third Value Solution]

[One Particular Three-Valued Solution]

[A Shortcoming of Three-Valued Solutions]

[Using Fuzzy Logic for Cases of Continuous Change]

[The Same Shortcoming in Fuzzy Logic, Too]

Summary

[Turning to Responses to the Sorites Paradoxes]

[We will now consider responses to the sorites paradox, where for

*M*_{0}, *M*_{1}, . . . , *M _{k}*

*M*_{0} is definitely true but *M _{k}* is definitely false. We will try to understand what is going on, logically speaking, between

*M*

_{0}and

*M*.]

_{k}

[Recall the sorites paradox from section 11.2. Here is our 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 ponensinferences 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.(our brief summary of section 11.2)

Priest formulated it in section 11.2.4 as:

Sorites arguments can often be put in the form of a sequence of

modus ponensinferences. Thus, ifM_{i}is the sentence ‘Mary is a child afteriseconds’, then the sorites of 11.2.1 is just:

M_{0 xxx }M_{0 }→M_{1 }____________

xxxxx

M_{1 }M_{1 }→M_{2}xxxxx____________

xxxxxxxxx

M_{2 }xxxxxxxxx.

xxxxxxxxxx.

xxxxxxxxxxx.

xxxxxxxxxxxxxx

M_{k-1}_{ }M_{k-1 }→M_{k}xxxxxxxxxxxxxx____________

xxxxxxxxxxxxxxxxxxxx

M_{k}where

kis some very large number.

(p.222, section 11.2.4)

Priest notes how in the sequence, the first item is certainly true, while the last is certainly false. We now wonder, what logically speaking happens between the two? We will consider now different responses to the sorites paradox.]

Various, very different, responses to the sorites paradox have been given. To see what some of these are, consider the sequence:

M_{0},M_{1}, . . . ,M._{k}M_{0}is definitely true;Mis definitely false. What is one to say about what goes on in between?_{k}(222)

[The Failure of a Bivalent Classical Solution]

[If we simply think that every sentence is either simply true or simply false, then we can break the paradoxical chain at some point where we say for example that the person is a child at this moment and in the next one they are an adult (“there must be a unique* i *such that *M _{i}* is true, and

*M*

_{i}_{+1}is false. In this case, the conditional

*M*→

_{i}*M*

_{i}_{+1}is false, and the sorites argument is broken” (222).) But, while that solves the paradox, it goes against our intuition that the change is continuous and thus there can be no such discrete leap happening from one instant to the next (and thus no discrete jump from truth to falsity).]

[(*ditto*)]

If we suppose that every sentence is either simply true or simply false, and given that the change from child to adult is not reversible, then there must be a unique

isuch thatMis true, and_{i}M_{i}_{+1}is false. In this case, the conditionalM→_{i}M_{i}_{+1}is false, and the sorites argument is broken. The problem with this supposition is obvious, however: the discrete nature of the change (that is, the jump from truth to falsity) would seem to be incompatible with the relatively continuous nature of the change from being a child to being an adult.(222)

[Claiming There Is Such a Break, but It Is Unknowable]

[Some still think that there are these discrete leaps of truth value in continuous changes, and the reason it strikes us as counterintuitive is simply because we lack the means to know where exactly the change takes place.]

[(*ditto*)]

Some have bitten the bullet, and accepted that there is, indeed, such a point. The most notable defence of this line (given by

epistemicists) attempts to argue that we find the existence of the point counterintuitive because, | as a matter of principle, we cannot know where it is; and we cannot know this for the following reason.(222-223)

[The Reasoning Behind the Not-Knowing Theory]

[Those arguing the above claim – that there is a discrete truth-value break but we cannot know it – use the following reasoning. We can only know true things, and we can only make judgements from evidence. The discrete truth-value shift in the actual change will make *M _{i}* true but

*M*

_{i}_{+1}false. However, the evidential basis will be the same for both. This means that when we make the judgment

*M*

_{i}_{+1}(on the basis of the misleading evidence that is the same from the prior moment), we are making a false judgment, and so we can never know when the shift happens.]

[Those arguing the above claim from section 11.3.3 (namely, that there is a discrete truth-value break but we cannot know it) use the following reasoning. (I do not follow it so well. I will be guessing it is the following. We are speaking of a moment *i*, which is the last moment where the sentence holds. So in our example it is the last moment when the person is a child, and in the next moment, they will be an adult. So we can say that *M _{i}* is true (“the person is a child” at the last moment they are a child) and

*M*

_{i}_{+1}is false (“the person is a child” at the first moment they are not a child). The next idea here is that you cannot know a falsehood. And finally, we have the idea of evidential bases for our knowledge. Whatever we do know we do so on account of some evidential basis that tells us it is so. When we see the person at that last moment of being a child, we have some kind of evidence to tell us that it is a child. Now, in the next moment, in reality, the person has become an adult. But for some reason, the evidential basis has remained the same. Maybe the child has all appearances the same, and any attempt to make some kind of a measurement of features of the child (that are indicative of their status) could never be precise enough to catch that sudden, imperceptible change. So for all practical purposes, the evidential basis remains the same, even though what it gives us evidence of is now false. Since we cannot know a falsity, that means we cannot know that the person is no longer a child. That thus means that for any such continuous change, we can never know when the discrete truth-value shift happens, because we can only know true things but the evidence will only give us falsehoods. Please see the quote below, as I did not put that together well.)]

If you know something, this has to be on some evidential basis. Thus, if you know something about a situation, you must know the same thing about any situation that is evidentially the same. Now suppose that you know that

M. Since,_{i}M_{i}_{+1}is evidentially the same (you could not tell the difference), you would have to knowM_{i}_{+1}too. But you cannot, sinceM_{i}_{+1}is false.(223)

[The Failure of the Not-Knowing Theory]

[The main problem with this argument is that the real problem with the paradox is not that we cannot know where the change happens but that there could even be such a sharp cut-off point in a continuous change.]

[(*ditto*)]

Whatever one makes of this argument itself, it cannot really serve to explain why we find the existence of a semantic discontinuity counterintuitive. For it is not just the fact that we do not

knowwhere the cut-off point is that is odd; it is the very possibility of a cut-off point at all: the changes involved in one second of a person’s life just do not seem to be of the kind that could ground a difference between childhood and adulthood.(223)

[Using a Third Value Solution]

[Another proposal is that cases of vagueness require that we reject a bivalent dichotomy between simple truth and falsity, and so for sorites changes, there would be a middle part where the sentences are {1} neither true nor false, or {2} both true or false.]

[(*ditto*)]

Some philosophers have suggested that vagueness requires us to reject a simple dichotomy between truth and falsity. In a sorites transition, there is a middle ground: some sentences in the middle of the transition are neither true nor false – or, perhaps, both true and false – something symmetric between truth and falsity, anyway.

(223)

[One Particular Three-Valued Solution]

[One three-valued solution is using *K*_{3 }(and perhaps in addition supervaluation). “In this case, there is some *i*, such that *M _{i}* is true and

*M*

_{i}_{+1}is neither true nor false. Again,

*M*→

_{i}*M*

_{i}_{+1}is not true, and so the sorites argument fails” (223).

[So one way to solve the sorites paradox is using the three-valued logic *K*_{3 }(see section 7.3), which might be combined with a supervaluation technique. (See sections 7.10.3–7.10.5a. We have not summarized them yet, so see supervaluation in Nolt’s *Logics* section 15.3.1). “In this case, there is some *i*, such that *M _{i}* is true and

*M*

_{i}_{+1}is neither true nor false. Again,

*M*→

_{i}*M*

_{i}_{+1}is not true, and so the sorites argument fails” (223). ]

Thus, a popular suggestion is that

K_{3 }(7.3), possibly in conjunction with some supervaluation technique (7.10.3–7.10.5a), is an appropriate logic for vagueness. In this case, there is somei, such thatMis true and_{i}M_{i}_{+1}is neither true nor false. Again,M→_{i}M_{i}_{+1}is not true, and so the sorites argument fails.(223)

[A Shortcoming of Three-Valued Solutions]

[But three-valued solutions suffer from the same counter-intuitiveness: it is hard to accept that there is a discrete boundary between truth and the middle value.]

[(*ditto*) (**But I wonder if one could say that any “becoming” is one involving both truth an falsity. So at M_{1 }the person is a child, and at M_{k }the person is not a child, but in the interval of change from 1 to k, the person both is and is not a child. I suppose this would be like the fuzzy logic we later consider, but here from the very beginning of the change, it is equally true that the person is both a child and not a child (rather than a fraction of being true). My reasoning for taking this view is that if we make a snapshot analysis, we can say that at any instant the person is more a child than an adult or more an adult than a child. However, insofar as we are dealing with intervals of change, that is, with movements, we no longer have quantifiable statuses. So again, we can take an interval close to the end of the change and say, at certain snapshot moments of that interval, the person is much more adult than child. However, if we only take into account the interval itself as a block of transformational movement, the only thing we could quantify is how much change is happening. So in one second near the end, there was little change. But with regard to statements about the person undergoing that movement of change, we can only say that they are both a child and are not a child. For, to quantify how much they are a child is to deal with statuses at instants and not with actual movements; for, statuses at instants can have quantities that are not quantities of motion, movement, change etc. (but rather of certain extensive determinations, that is to say, they are so far away from some point or other; while quantities of motion and change are intensive, that is to say, being faster or slower, or the like**).]

The problem with any 3-valued approach is obvious, however. The existence, in a sorites progression, of a discrete boundary between truth and the middle value is just as counterintuitive as that of one between truth and falsity.

(223)

[Using Fuzzy Logic for Cases of Continuous Change]

[So since the changes are continuous, we might want to use a fuzzy logic where the truth-values come in continuous degrees too.]

[(*ditto*)]

Moreover, the existence of relatively continuous change along a sorites progression would seem to be incompatible with any discrete boundaries. It is natural to suppose, therefore, that truth values must themselves change continuously. Thus, we must consider a logic in which truth comes | in continuous degrees. This is fuzzy logic, and will concern us for the rest of this chapter.

^{1}(223-224)

1. There are, in fact, sorites progressions where each step is clearly discrete: for example, the addition of a single grain of sand. So, in principle, one could use a finitely-many valued logic for these. But the continuum-valued semantics is more general, and can be applied to all sorites paradoxes, giving, what is clearly desirable, a uniform account.

(224)

[The Same Shortcoming in Fuzzy Logic, Too]

[But even fuzzy logic has this same problem, because somewhere there must be a change from *completely true* to* less than completely true*. ]

[(*ditto*)]

It should be noted, though, that even fuzzy logic is not entirely unproblematic. For if truth comes by degrees, there must be some point in a sorites transition where the truth value changes from

completely truetoless than completely true. The existence of such a point would itself seem to be intuitively problematic.(224)

From:

Priest, Graham. 2008 [2001]. *An Introduction to Non-Classical Logic: From If to Is*, 2nd edn. Cambridge: Cambridge University.

.

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