Priest (21.6) An Introduction to Non-Classical Logic, ‘Existence and Quantification,’ summary

 

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 II:

Quantification and Identity

 

21

Many-valued Logics

 

21.6

Existence and Quantification

 

 

 

 

Brief summary:

(21.6.1) We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive. (21.6.2) We now wonder if it makes sense for the existential predicate to have  non-classical values. (21.6.3) According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false. (21.6.4) One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464). (21.6.5) Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464). (21.6.6) There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464). (21.6.7) Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”. (21.6.8) We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464). (21.6.9) There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).

 

 

 

 

 

Contents

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

21.6.3

[Existence Statements as Gaps]

 

21.6.4

[Existence Gaps and Non-Denotation]

 

21.6.5

[Existence Gaps and Future Contingents]

 

21.6.6

[Existence Gaps and Verificationism]

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

 

 

 

 

Summary

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

[We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive.]

 

[We will now recall some matters from section 13.5  regarding the useful distinction and addition of inner and outer quantifiers:

(13.5.1) We might want a free logic where quantifiers range over all objects and not just existent ones. (13.5.2) Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃^E and ∀^E. (13.5.3) We read ∀xA as ‘Every x is such that A’; ∀^ExA as ‘Every existent x is such that A’; ∃xA as ‘Some x is such that A’ or as ‘Something is A’; and ∃^ExA as ‘there exists an x such that A’ or as ‘there is an x such that A’. (13.5.4) We should not think that the existential quantifier of natural language necessarily implies existence. (13.5.5) There is an argument for reading the existential quantifier as “there exists”. The argument wants to avoid problems like the ontological argument, so it does not allow existence to be a predicate. Instead, it sees as the only other viable option for expressing existence as being the existential quantifier. Part of the thinking is that only things that are there can be predicated. But this is not a convincing argument, because there are many examples of predication of non-existing objects, like Zeus being worshipped. (13.5.6) If we wish, we can define inner quantifiers in terms of outer ones, which means that “in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether,” namely, in the following way:

∃^ExA     ∃x(ℭx ∧ A)

∀^ExA     ∀x(ℭx ⊃ A) 

(p297). However, “There is no way of defining outer quantifiers in terms of inner quantifiers” (297). (13.5.7) These new semantics make one problematic inference no longer problematic, namely, A_x(a) ⊨ ∃xA, now meaning that if something can be predicated, it is either an existent or non-existent object (previously it implied that any predicable thing must be existent). But it may not make the logical truth ∃x(A ∨ ¬A) unproblematic (it implies now that there must be at least a non-existent object, while before it implied there must be at least an existent object.)

(From the brief summary of section 13.5)

Priest notes now that for our 3-valued logics, we can have outer-quantifiers ranging over the whole of the domain D. But, for certain technical reasons (see below, as I have not yet tried to grasp and summarize them yet), we cannot define the inner quantifiers by means of outer-ones in 3-valued logics where existence predicates take non-classical values (like i), and so “inner quantifiers will have to be taken as primitive.” (I am not sure yet how that works, but it seems in the next section we should figure that out as an exercise.) ]

As with the two-valued case, in the free 3-valued logics we have been talking about, one can have outer quantifiers, ranging over the whole of D. The definability of the inner (existentially loaded) quantifiers in terms of the outer quantifiers and the existence predicate is, however, more problematic. If, as in 13.5.3, we write the outer quantifiers as ∀ and ∃, and use a superscript E to indicate the existentially loaded quantifiers, what we require is:

1. v(∃^ExA) = v(∃x(ℭx ∧ A))

2. v(∀^ExA) = v(∀x(ℭx ⊃ A))

We know that v(ℭk_d) ∈ D iff d ∈ E. If ℭ is a classical predicate, in the sense that for all d ∈ D, v(ℭkd) = 1 or v(ℭkd) = 0, these equations hold. The details are straightforward, and left as an exercise. (Check that if the lefthand side is 1, so is the righthand side. Then check the opposite direction. Do the same thing for 0. The case for i then follows.) If, however, existential statements may take the value i, things may go wrong. Consider an interpretation with | two members, d and e, as follows:

 

       D

+---------------+
|               |
|     E         |
|  +-----+      |
|  |     |      |
|  |  e  |   d  |
|  |     |      |
|  +-----+      |
+---------------+

If v is as follows:

 

	v(ℭ)	v(P)
d	i	1
e	1	0

 

this is a K₃ and Ł₃ interpretation. It is not difficult to check that v(∃^ExPx) = 0, but v(ℭk_d ∧ Pk_d) = i = v(∃x(ℭx ∧ Px)).

 

If v is as follows:

 

	v(ℭ)	v(P)
d	0	0
e	i	1

 

this is an LP and RM₃ interpretation. It is not difficult to check that v(∃^ExPx) = 1, but v(ℭk_e ∧ Pk_e) = i = v(∃x(ℭx ∧ Px)). Hence, if the existence predicate is allowed to take non-classical values, inner quantifiers will have to be taken as primitive.

(463)

[contents]

 

 

 

 

 

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

[We now wonder if it makes sense for the existential predicate to have  non-classical values.]

 

[(ditto)]

Arranging for this is a simple matter, and left as an exercise. However, it does raise the question of whether it makes sense for the existence predicate to have a non-classical value, the answer to which is not so obvious.

(463)

[contents]

 

 

 

 

 

 

21.6.3

[Existence Statements as Gaps]

 

[According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false.]

 

[(ditto)]

Suppose that we are in a logic where i is interpreted as neither true nor false. Could a sentence of the form ℭa take this value? The answer depends on what sorts of thing one takes to be neither true nor false; but on certain views about this, the answer could be ‘yes’.

(463)

[contents]

 

 

 

 

 

 

21.6.4

[Existence Gaps and Non-Denotation]

 

[One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464).]

 

[(ditto)]

Some have argued that a sentence containing a non-denoting name has no truth value (see 7.8). If this is the case, and a does not denote anything, | ℭa has no truth value. But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless.

(463-464)

[contents]

 

 

 

 

 

 

21.6.5

[Existence Gaps and Future Contingents]

 

[Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464).]

 

[(ditto)]

Aristotle argued that statements about a future state of affairs that is not, as yet, determined are neither true nor false (see 7.9). If this is correct then, arguably, ‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true.

(464)

[contents]

 

 

 

 

 

 

21.6.6

[Existence Gaps and Verificationism]

 

[There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464).]

 

[(ditto)]

Better arguments can be found if one subscribes to verificationism of some kind. This might be a philosophy of mathematics which identifies mathematical truth with provability; or it might be a philosophy of science which identifies truth with empirical verifiability. If one subscribes to such a view, and one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind.

(464)

[contents]

 

 

 

 

 

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

[Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”.]

 

[(ditto)]

As another example: some have argued that statements about the borderline range of some vague predicate are neither true nor false (see 11.3.6, 11.3.7). Thus, ‘Dana is an adult’, said of Dana around puberty, might be thought to be neither true nor false. But can existence be a vague predicate? Certainly: when people die they go out of existence (let us suppose). But dying can be a gradual process. Bodily functions do not normally all cease at once; there can therefore be a grey area where it is vague as to whether or not someone exists.

(464)

[contents]

 

 

 

 

 

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

[We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464).]

 

[(ditto) (Note: the argument of gaps for dying from section 21.6.7 above seems odd to me, and the gluts version here seems much more reasonable. If it is neither true nor false that one is alive (exists), and it is neither true nor false that one is not alive, than what can we say about the person’s state of being? Can we say that it is true it is some third state? To me it seems more reasonable to say it is both true and false that one exists when in the transitional process of dying, as there is not really another predicate I can think of that applies here that would be just true (and surely some predicate or other regarding its state of being should hold, because we cannot simply say they are dead, but the dying person is still there in some state of being that is also not life, under these gap assumptions). The gap thinking seems to be the following. Can we say the dying person is dead (does not exist)? No, because they are not dead enough to be such. Can we say they are alive? No, because they are not alive enough to be such. In other words, there is a window during which neither the predicate “exists” nor its negation holds. But, as I pointed out, there is still a person there in some state of being, and presumably that state can be given a name and serve as a predication to the dying person. Now, under this gap reasoning, that predicate cannot be “exists”. But I have two problems with that. If it is not “exists,” then you are saying it is false that they exist. That to me seems like you are saying that “they exist” is false and not valueless (otherwise you might be saying they exist only partly, but then you are using a fuzzy value or maybe even a glut, which is not what we are assuming here for gaps). My other problem is that if you insist that “they exist” is neither true nor false, but that moments later after they fully die, “they exist” is false, then, as I noted, something true can be predicated of their existential state which is between existence and non-existence. So my final point here is that to be in a state between existence and non-existence would not be like jumping into a third state that is completely different from existence and non-existence but would rather seem to have certain properties of existence and certain properties of non-existence. For, it is a continuous variation from the one state to the other. So for example, you may have consciousness but not cell life-sustenance on account of stopped blood flow, or maybe you do not have consciousness but you have blood flow. I am not sure what really is involved in death processes. At any rate, it seems to me that existence in the dying transition phase would seem to be a glut, in that it is both true that you exist (in that you have enough functions at this very moment to say that you have not completely passed out of existence and thus that you still are existing, even if barely so) but it is also true to say that you do not exist (as you have a lack of certain functions that will sustain your existence for much longer and thus you are practically dead. To put it phenomenologically, when you are passing away, you will be consciousness of your own fading consciousness, and thus in one instant you will experience both your state of existence and non-existence simultaneously).)]

What of a logic where i is interpreted as both true and false. Could a sentence of the form ℭa be both true and false? Some have suggested that the statements about the borderline range of some vague predicate are both true and false. What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false.

(464)

[contents]

 

 

 

 

 

 

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

[There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).]

 

[(ditto)]

One final example. Some have argued that paradoxical sentences generated by the paradoxes of self-reference are both true and false (see 7.7). Some of these can be existence statements, as in Berry’s paradox, which is as follows. Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist. So paradoxes of self-reference may deliver existence statements that are both true and false.

(465)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

Priest, Ch10 of Logic: A Very Short Introduction, “Vagueness: How Do You Stop Sliding Down a Slippery Slope?”, summary

by Corry Shores

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Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.10
Vagueness: How Do You Stop Sliding Down a Slippery Slope? 

 

 

Brief Summary: 
A thing can change gradually over time. A true statement about that thing’s status at the beginning can later be false at the end of the development. But in many cases, it is not clear when exactly during that development the status changes without ambiguity. “Jack is a child” is true when Jack is very young and not true when Jack is old; but, when precisely in his young adult years does it cease being entirely true and instead “Jack is an adult” becomes entirely true? This issue is related to sorites paradoxes. Consider that “Jack is a child” is true at the beginning, and “If Jack is a child at the beginning, then he is still a child one second later” also is probably also true. That means by modus ponens, “Jack is a child one second later” is true. Using this same sort of reasoning, we can then conclude that Jack is a child two seconds later, and so on, meaning that he never ceases being a child. (We reiterate the structure, taking the affirmed prior conclusion that Jack is still a child in the  succeeding second, and use it as a premise in an argument of the same structure, allowing us to conclude he is a child in yet the next succeeding second, and so on infinitely).  One solution to these issues is to use fuzzy truth values. We can say for example that when he is 3 years old, the statement “Jack is a child” has a full truth value of 1. At 9 years “Jack is  child” has a truth value of 0.75. At 14 years, 0.5. At 19 years, 0.25. And at 24 years, 0. And when we apply truth functional operators to statements with  values between 1 and 0, we can determine the different resulting fuzzy values. Also, we can say that an inference is valid when both the conclusion and the premises meet a certain minimum level of truth value, which is determined by the actual context to which the statements apply. What we find then is that the sorites paradox does not hold when we use this fuzzy system. [For, in order for the modus ponens inference to work in all steps, we will need the minimum value to be 0 (in order to accommodate the final transitional step), which is too low to be meaningful.] Also, fuzzy values do not clear up the situation entirely, because we have the same problem when we need to determine precisely at what point the values change from 1 to something less than 1.

 

Summary

[We previously discussed identity, and we noted that something’s properties can change, which then raises the difficulty of determining whether or not we can say that the thing’s identity has changed.] Priest will discuss another problem with identity; in this case it is the problem of vagueness. He poses a very interesting thought experiment to illustrate the issue. What if we replace something’s parts one by one until it has all new parts. For certain things like machines, we would seem to still have the same identical machine even after all its parts were gradually replaced. [Or at least, it would be very hard to determine at which point its identity changed. If we say it changed with the first replacement, what if that replacement were negligible? And on the atomic level, are there  not such changes happening always anyway? Or, consider if we say it happened with the most final replacement. But also, could not that final replacement be negligible as well? Yet, if we choose some replacement happening between, what would be our criteria for saying it was one middle replacement and not the one coming right before or after it? Priest’s motorcycle example is especially effective, and it makes the issue even more complicated. What if, along with the gradual replacements, we kept the parts and reconstructed the original machine? We would on the  one hand have the replacement machine, which we said could be the same as the original, and we then have the machine made of all the replaced parts, which would also seem to be the original. But how can both be identical to the original machine?]

While we are on the subject of identity, here is another problem about it. Everything wears out in time. Sometimes, parts get replaced. Motor bikes and cars get new clutches; houses get new roofs; and even the individual cells in people’s bodies are replaced over time. Changes like this do not affect the identity of the object in question. When I replace the clutch on my bike, it remains the same bike. Now suppose that over a period of a few years, I replace every part of the bike, Black Thunder. Being a careful fellow, I keep all the old parts. When everything has been replaced, I put all the old parts back together to recreate the original bike. But I started off with Black Thunder; and changing one part on a bike does not affect its identity: it is still the same bike. So at each replacement, the machine is still Black Thunder; until, at the end, it is – Black Thunder. But we know that that can't be right. Black Thunder now stands next to it in the garage.
(70)

 

Priest gives another example. If a minute incremental change does  not change a basic state of something, then how do we know when there are enough such tiny changes to constitute the change in state? His example of seconds added to childhood illustrates this very well.

Here is another example of the same problem. A person who is 5 years old is a (biological) child. If someone is a child, they are still a child one second later. In which case, they are still a child one second after that, and one second after that, and one second after that, ... So after 630,72o,ooo seconds, they are still a child. But then they are 25 years old!
(70)

 

[Note: page 71 is a full page figure of the motor cycle situation.]

 

[Recall from chapter 5 that Eubulides is credited for inventing the liar paradox.] Such arguments as these are now called sorites paradoxes, and they are thought to have been invented by Eubulides. Priest explains where the name comes from: “A standard form of the argument is to the effect that by adding one grain of sand at a time, one can never form a heap; ‘sorites’ comes from ‘soros’, the Greek for heap” (72). He explains that we encounter sorites paradoxes “when the predicate employed (‘is Black Thunder’, ‘is a child’) is vague, in a certain sense; that is, when its applicability is tolerant with respect to very small changes: if it applies to an object, then a very small change in the object will not alter this fact” (72). He notes that in fact, very many of the predicates in our everyday usage are vague. For example, ‘is red’ is vague [because something’s redness can vary slightly to the point of no longer being red], as is ‘is awake’ [perhaps because we “drift-off” to sleep and at times awaken gradually]. He even says that ‘is dead’ is a vague predicate, because it takes time to die [I am not entirely sure I understand this one. It seems the idea is that because we taper-off into death, there is no way to know exactly at what point in that passing-away is the point of death. So imagine the instant before all life is gone. There would seem to be too little life left to really say the person is alive. But I am not sure.] Thus such “slippery slope” sorties arguments “are potentially endemic in our reasoning” (72).

 

Priest will now present a symbolic formulation of the paradox with the childhood example illustrating it. We have a five year old child whom we will call ‘Jack’, and we will consider Jack after certain  numbers of seconds have passed. We will write ‘Jack is a child after 0 seconds’ as a₀. And thus, a₁ will mean ‘Jack is a child after 1 second,” and so on for each subscript numeral. We can then think more generally of the formulation using the variable n, and say that a_n means “Jack is a child after n seconds,” when n stands for some number value. We will pick an unspecified very large number that is “at least as great as 630,720,000,” and we will just call it k. [So in the diagram below, a_k-1 would mean, the second right before this very large number.] We will then  make a series of inferences using modus ponens. [He discusses modus ponens in chapter 6 and chapter 7.] We first note that Jack is still a child at five years old, that is, after zero seconds have passed. So we first assert a₀, which states this fact. We then formulate a conditional, “If Jack is a child after 0 seconds, then he is a child after 1 second”, or: a₀ → a₁. Now, since we already asserted a₀ , then by modus ponens, we can infer a₁. We then take as a premise in the next inference, where we, following the same reasoning, say that if Jack is a child after 1 seconds, then he is a child after 2 seconds.

 

As we noted, a_k means 630,720,000 seconds later Jack is still a child, which we know to be false. “Something has gone wrong, and there doesn’t seem much scope to manoeuvre” (72).

 

Priest then notes one solution using fuzzy logic. The idea seems to be that since in reality one gradually changes from childhood to adult through a series of degrees of variation, then there are degrees of truthfulness to the claim that “Jack is a child” at different point along his growth.

So what are we to say? Here is one answer, which is sometimes called fuzzy logic. Being a child seems to fade out, gradually, just as being a (biological) adult seems to fade in gradually. It seems natural to suppose that the truth value of ‘jack is a child’ also fades from true to false. Truth, then, comes by degrees. Suppose we measure these degrees by numbers between 1 and o, 1 being complete truth, o complete falsity. Every situation, then, assigns each basic sentence such a number.
(73)

 

[The next idea seems to be that we will want to incorporate operators to sentences with fuzzy truth values, and the way to do that for negation is to give the negated term the value that the non-negated term lacks in comparison to the full 1 value.]

What about sentences containing operators like negation and conjunction? As Jack gets older, the truth value of  ‘Jack is a child’ goes down. The truth value of ‘Jack is not a child’ would seem to go up correspondingly. This suggests that the truth value of ¬a is 1 minus the truth value of a. Suppose we write the truth value of a as |a|; then we have:

|¬a | = 1 – |a|

Here is a table of some sample values:

(73)

[Priest then will show how we calculate the truth value for a conjunction of sentences with fuzzy values. The reasoning here is less obvious I think. One element of the thinking seems to be that by conjoining such sentences, we are not adding all their truth values together nor are we finding the average. So let us suppose that in addition to “Jack is a child” we also assume that Jack is training to become a cook, and so we also have “Jack is a cook”. Suppose also that Jack is halfway to becoming an adult and halfway to becoming a cook. When we conjoin those statements, “Jack is a child and Jack is a cook”, the value should still be 0.5, because that represents his situation on both accounts. However, suppose that Jack is halfway to becoming an adult and only a quarter of the way to becoming a cook. For some reason, in this case, the conjunction “Jack is a child and Jack is a cook” takes the value of 0.25. I am not sure why. For instance, why not the average of the two values? One possible thing to consider is that the normal truth evaluation for conjunction seems to have a similar sort of pattern. If even one conjunct is false, then the whole thing is false. So the value of the whole can be no “greater” than the value of the “lowest” term. But I am not sure if that helps us understand why in the fuzzy value assessment of conjunction, we always take the lowest value as being the value for the whole conjunction.]

What about the truth value of conjunctions? A conjunction can only be as good as its worst bit. So it’s natural to suppose that the truth value of a & b is the minimum (lesser) of |a| and |b|:

|a & b| = Min(|a|, |b|)

|

Here is a table of some values. Values of a are down the left hand column; values of b are along the top row. The corresponding values of a & b are where the appropriate row and column met. For example, if we want to find |a & b|, where |a| = 0.25 and |b| = 0.5, we see where the italicized row and column meet. The result is in boldface.

(73-74)

[For disjunction, we go with the value of the greater disjunct. I am not sure why, again. But using the reasoning we had before, it could be because normally a disjunction is true so long as at least one disjunct is true. And it is false only if both terms are false. So perhaps the idea is that we always go with the highest value in the two-value system, so we likewise go with the highest value in the fuzzy value system.]

Similarly, the value of a disjunction is the maximum (greater) of the values of the disjuncts:

|a ∨ b| = Max(|a|, |b|)

I leave it to you to construct a table of some sample values.

(p.74)

[I am not sure I have it right, but below is possibly an option.]

Priest  notes that ¬, &, ∨ are truth functional in fuzzy logic, because the truth value when these operators are applied is determined on the basis of the truth values of the original terms. [We discussed truth functions in chapter 2. What makes operators truth functional is if the output values are computable on the basis of the input values. The only difference now is that the values that we input and receive as output are between 0 and 1 rather than being T or F.] This is apparent when we compare the truth tables for the two systems. (74). [Below we see that clearly with negation (just ignore the fuzzy values in the middle).

With conjunction, look just at the places where 1 and 0 intersect in the fuzzy table. You see that it is only where 1 and 1 intersect that the output value is 1, just like how the time the output value is T is where both conjuncts are T.

And look again at the fuzzy disjunction for where 1 and 0 intersect, and you will see it corresponds with the T and F of the other table.

]

Priest now addresses the question of the conditional, →, which we will consider here as truth functional. It is not obvious how we compute fuzzy values for the conditional. But he gives a standard solution, “which at least seems to give the right sorts of results”.

If |a| ≤ |b|: |a→b| = 1
If |b| < |a|: |a→b| = 1 – (|a| – |b|)

(< means ‘is less than’; ≤ means ‘is less than or equal to’.) Thus, if the antecedent is less true than the consequent, the conditional is completely true. If the antecedent is more true than the consequent, then the conditional is less than the maximal truth by the difference between their values. Here is a table of some sample values. (Recall that the values of a are down the left-hand column and those of b are along the top row.)
(75)

[I do not understand why the values are the way they are.  Let us take an example of a conditional from chapter 6: “If she works out regularly then she is fit.” If the truth value of a is 1 (she indeed works out regularly) and b is 0.25 (she is only a little fit) then the whole conditional’s value is 0.25 (it is only a little true that if she works out regularly she will be fit). This makes sense, because she worked out so she should be very fit, but in fact it is only a little so. Now, if a is 0.25 (she only works out a little), and b  is 1 (she is fit), then the whole conjunction is completely true, 1. I am not sure why. Perhaps the idea is that this situation demonstrates that even a little bit of exercise is already enough to result in fitness. This holds even if she is only 0.25 fit. But if it is 0.25 true that she exercised but she is not fit at all, then the whole conditional is 0.75. So she exercised a little, but she is not fit, and for some reason the whole value of “if she works out she will be fit” is 0.75. This perhaps makes sense because she did not really fulfill the conditions enough to qualify entirely. But it is hard to make a general statement about how it all works. It seems one thing we can say is that the conditional on a whole cannot be less true than the consequent, but it can be less true than the antecedent. And any time that the antecedent is at least as true as the consequent, the whole conditional is completely true (this is perhaps the result of the condition: If |a| ≤ |b|: |a→b| = 1). Yet, I cannot really grasp why all this is like this. If we compare it with the two-valued table for the conditional, we see also here that it favors truth, which we see as well in the fuzzy table.

Perhaps the idea with the distribution of values in the fuzzy table is that the degree to which we have a being true (1) with respect to b being false (0) determines the degree to which the whole conditional is false (0). Probably not, but I am not sure how else to generalize it.]

Priest next discusses what validity would be in this fuzzy context. Normally we say that “An inference is valid if the conclusion holds in every situation where the premisses hold” (75). But now the difference is that we have premises and conclusions which “hold” to different degrees. Priest says now in a fuzzy system a proposition holds when it is “true enough,” and that is determined by the context. He gives an example for a vague predicate: “is a new bike”. If a bike dealer tells us this, it is reasonable for us to expect that the bike has never been used before. In other words, we expect the proposition, “This is a new bike” to have the value of 1. But, “Suppose, on the other hand, that you go to a bike rally, and are asked to pick out the new bikes. You will pick out the bikes that are less than a year or so old. In other words, your criterion for what is acceptable as a new bike is more lax. ‘This is a new bike’ need have value, say, 0.9 or greater” (75).

Priest then formulates the notion validity more precisely for a fuzzy system. We determine some value which is what is acceptable in this context for a proposition to hold. Then an inference is valid if the conclusion is at least as much as that chosen value, whenever the premises also have at least that value.

So we suppose that there is some level of acceptability, fixed by the context. This will be a number somewhere between 0 and 1 – maybe 1 itself in extreme cases. Let us write this number as ε. Then an inference is valid for that context just if the conclusion has a value at least as great as ε in every situation where the premisses all have values at least as great as ε.
(76)

 

We return now to the problem of the sorites paradoxes that we began with. First we suppose that we have a sorites sequence a₀ to a₄ for a_n, meaning, “Jack is a child after n seconds” (76). Before we noted that it takes many many seconds for Jack to grow from child to adult. But to make things easier here, we just assume it takes only four seconds. In that case, the truth values could be:

Now recall the chart for evaluating conditionals.

Priest will note that to have a conditional where the antecedent is a_n and the consequent is a_n+1, the overall value of the conditional will always be 0.75. We can see this in chart [see the diagonal where all the values are 0.75]. [The idea might be that to say, ‘Jack is a child from any moment to the next’, where there are just 4 moments, is 0.75 true no matter which transition we consider. I am not sure, but I suppose the idea is that with each change some more of that truth value in comparison with the first state is lost, until the last status, when all of it is gone.]

a₀  → a₁  has value 0.75 (=(1(1-0.75)); so does a₁  → a₂ ; in fact, every conditional of the form a_n → a_n+1 has the value 0.75.
(76)

 

[Priest then will show that no matter what level of acceptability we assign such situations leading to sorites paradoxes, the final inference in the sequence will be invalid. I do not grasp the reasoning here, so I will quote it below, and you can skip to it now. The idea seems to be the following. Recall the schema for how the sorites paradoxes work.

We see that they use modus ponens throughout, since for each line we affirm the antecedent, then in the next line we begin with the inferred consequent, which now becomes the affirmed antecedent in the next conditional. Recall also our example, that since Jack is a child after just one second, he is therefore a child after no matter how many seconds. To see how Priest’s explanation works, it seems we need to note a couple things first. First recall the idea of validity. For an inference to be valid when values are fuzzy, we first assign an acceptable minimum value for a proposition to hold; then, for the inference to be valid, the conclusion must be at least that value, when the premises also meet at least that value. Now, suppose that we set the minimum value at 0.75. In our examples, we have for instance a₀  → a₁. Here the values of the parts and the whole inference would be, 1 → 0.75, 0.75. Here everything is fine so far. Everything meets the minimum value. However, at some point, we need to get to a₄, whose value is 0. Perhaps the idea here is that its conditional a₃  → a₄ cannot be valid, because here one of the terms has the value 0, which means the minimum value would need to be 0, but that is too low for its validity to have any meaning. So on the one hand, in order for the modus ponens operation to work throughout the series of inferences, we need to set the minimum too low. If however on the other hand we set it at 1, none of the inferences will fulfill that minimum. This does not seem to be Priest’s reasoning exactly, so I probably have it wrong. Just go with what he says:]

What this tells us about the sorites paradox depends on the level of acceptability, ε, that is in force here. Suppose the context is one that imposes the highest level of acceptability; ε is 1. In this case, modus ponens is valid. For suppose that |a| = 1 and |a → b| = 1. Since |a → b| = 1, we must have |a| ≤|b|. It follows that |b| = 1. Thus the sorites argument is valid. In this case, though, each conditional premiss, having value 0.75, is unacceptable.

If, on the other hand, we set the level of acceptability lower than 1, then modus ponens turns out to be invalid. Suppose, for the sake of illustration, that ε is 0.75. As we have already seen, a, and a₀  → a₁ both have value 0.75, but a₂ has value 0.5, which is less than 0.75.

Either way you look at it, the argument fails. Either some of the | premisses aren’t acceptable; or, if they are, the conclusions don’t follow validly. Why are we taken in by sorites arguments so easily? Maybe because we confuse complete truth with near-complete truth. A failure to draw the distinction doesn’t make much difference normally. But if you do it again, and again, and again, ... it does.
(76-77)

 

Priest’s next point seems to be that even with fuzzy values, we still have not clarified the situation. [It seems the problem is the following. Consider the example of Jack growing up. In our fuzzy system, we know that at some point near the beginning he is not entirely a child (that is, the truth value of “Jack is a child” is not 1). But probably one millisecond after the first designation he is still fully a child. So we will need still to make an arbitrary division to know how long it takes before he becomes slightly less than a child. Now, the original motivation for using fuzzy values was that there was no clear-cut point along the development where Jack goes from simply being a child to simply not being one. But now, in order to go from a value of 1 to less than 1, we have the same problem. There does not seem to be an obvious place where this transition happens. Thus we have not solved the problem; we have merely relocated it to a place very near the beginning.]

That's one diagnosis of the problem. But with vagueness, nothing is straightforward. What was the problem about saying that ‘Jack is a child’ is simply true, until a particular point in time, when it becomes simply false? Just that there seems to be no such point. Any place one chooses to draw the line is completely arbitrary; it can be, at best, a matter of convention. But now, at what point in Jack’s growing up does he cease to be 100% a child; that is, at what point does ‘Jack is a child’ change from having the value of exactly 1, to a value below 1? Any place one chooses to draw this line would seem to be just as arbitrary as before. (This is sometimes called the problem of higher-order vagueness. ) If that is right, we haven’t really solved the most fundamental problem about vagueness: we have just relocated it.
(77)

 

 

[The following is quotation.]

Main Ideas of the Chapter

● Truth values are numbers between o and 1 (inclusive).
● |¬a | = 1 – |a|
● |a ∨ b| = Max(|a|, |b|)
● |a & b| = Min(|a|, |b|)
● If |a| ≤ |b|: |a→b| = 1
    If |b| < |a|: |a→b| = 1 – (|a| – |b|) otherwise.
● A sentence is true in a situation just if its truth value is at least as great as the (contextually determined) level of acceptability.

(quoted from Priest, 77)

From:

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.