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).
[Outer and Inner Domain Quantifiers in 3-Valued Logics]
[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]
[Existence Statements as Gaps]
[Existence Gaps and Non-Denotation]
[Existence Gaps and Future Contingents]
[Existence Gaps and Verificationism]
[Dying as Involving a Vague, Valueless Existential Predication]
[Borderline Existence Statements as Both True and False]
[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]
Summary
[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, Ax(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(ℭkd) ∈ 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(ℭkd ∧ Pkd) = 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(ℭke ∧ Pke) = 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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
[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)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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