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 (7.7) An Introduction to Non-Classical Logic, ‘Truth-value Gluts: Paradoxes of Self-reference,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

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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

 

7.

Many-Valued Logics

 

7.7

Truth-value Gluts: Paradoxes of Self-reference

 

 

 

 

Brief summary:

(7.7.1) We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts. (7.7.2) One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129). (7.7.3) Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “ (7.7.4) There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts. (7.7.5) We will now examine briefly a couple claims that these paradoxical arguments are not sound. (7.7.6) Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’. (7.7.7) Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither. (7.7.8) Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.

 

 

 

 

 

 

 

Contents

 

7.7.1

[Paradoxes of Self-Reference as Motivation for Gluts]

 

7.7.2

[The Liar’s Paradox]

 

7.7.3

[Russell’s Paradox]

 

7.7.4

[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]

 

7.7.5

[Turning to Claims that the Paradoxes are not Sound]

 

7.7.6

[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]

 

7.7.7

[Objection 2: The Liar Sentence Is Neither True nor False]

 

7.7.8

[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]

 

 

 

 

Summary

 

 

7.7.1

[Paradoxes of Self-Reference as Motivation for Gluts]

 

[We will now consider paradoxes of self-reference as motivation for advocating for truth-value gluts.]

 

[In the previous section 7.6, we examined a motivation for arguing for truth-value gluts, namely, inconsistent laws. We consider now another motivation: paradoxes of self-reference. There are both old and modern ones.]

A second argument for the existence of truth-value gluts concerns the paradoxes of self-reference. There are many of these; some very old; some very modern. Here are a couple of well-known ones.

(129)

[contents]

 

 

 

 

 

 

7.7.2

[The Liar’s Paradox]

 

[One paradox of self-reference is the liar’s paradox. For example, ‘this sentence is false’. “Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false” (129).]

 

[(ditto)]

THE LIAR PARADOX: Consider the sentence ‘this sentence is false’. Suppose that it is true. Then what it says is the case. Hence it is false. Suppose, on the other hand, that it is false. That is just what it says, so it is true. In either case – one of which must obtain by the law of excluded middle – it is both true and false.

(129)

[contents]

 

 

 

 

 

 

7.7.3

[Russell’s Paradox]

 

[Another paradox of self-reference is Russell’s Paradox: “Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false. “]

 

[(ditto) (See  section P.6 of One and ch.5 of Logic: A Very Short Introduction.)]

RUSSELL’S PARADOX: Consider the set of all those sets which are not members of themselves, {x; x ∉ x}. Call this r. If r is a member of itself, then it is one of the sets that is not a member of itself, so r is not a member of itself. On the other hand, if r is not a member of itself, then it is one of the sets in r, and hence it is a member of itself. In either case – one of which must obtain by the law of excluded middle – it is both true and false.

(129)

 

[contents]

 

 

 

 

 

 

7.7.4

[These Paradoxes of Self-Reference as Showing Truth-Value Gluts]

 

[There are many such arguments that come to a conclusion of the form A∧¬A, and supposing they are sound, that makes the conclusions true and thus means there really are truth-value gluts.]

 

[(ditto)]

These (and many others like them) are both prima facie sound arguments, and have conclusions of the form A∧¬A. If the arguments are sound, the conclusions are true, and hence there are truth-value gluts.

(129)

[contents]

 

 

 

 

 

 

7.7.5

[Turning to Claims that the Paradoxes are not Sound]

 

[We will now examine briefly a couple claims that these paradoxical arguments are not sound.]

 

[(ditto)]

Many people have claimed that the arguments are not, despite appearances, sound. The reasons given are many and complex; let us consider, briefly, just a couple.

(129)

[contents]

 

 

 

 

 

 

7.7.6

[Objection 1: Self-Referential Sentences Are Meaningless. Reply 1: Not So in Many Cases]

 

[Objection 1: All self-referential sentences are meaningless. Reply 1: But, there are many such meaningful sentences, like, ‘this sentence has five words’.]

 

[(ditto)]

Some have argued that any sentence which is self-referential, like the liar sentence, is meaningless. (Hence, such sentences can play no role in logical arguments at all.) This, however, is clearly false. Consider: ‘this sentence has five words’, ‘this sentence is written on page 129 of Part I of An Introduction to Non-Classical Logic’, ‘this sentence refers to itself’.

(129)

[contents]

 

 

 

 

 

 

7.7.7

[Objection 2: The Liar Sentence Is Neither True nor False]

 

[Objection 2: The liar sentence is neither true nor false. Thus our logical assumptions remove excluded middle, and we cannot develop the argument as, “either it is true or false; if false, then thus; if true then false; thus ...”. For, now we have a third situation, that it is neither.]

 

[The second objection is the most popular one. It says that the liar sentence is neither true nor false. This means that we cannot appeal to the law of excluded middle (because it is no longer the case that the sentence is either true nor false. How did we use it previously? I am not really sure. Maybe it goes like this, but I am guessing. We have the sentence, “this sentence is false.” Then we say, “Either it is true or it if false. If it were true, then it is false, and if it is false, then it is true. Either way, it is both true and false.” So maybe, the argument works by having the original proposal that it is either true or false. And maybe the idea now is that were it neither value, then the law of excluded middle does not hold, and so we cannot start off with the assumption “Either it is true or it is false; if true ...”. Or maybe we can start it that way, but it cannot end that way, because we have the third possibility to assess, that it is neither, meaning that we cannot further derive another value from it in addition to it being neither. I am not sure.) “Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too” (129).]

The most popular objection to the argument is that the liar sentence is neither true nor false. In this case, we can no longer appeal to the law of excluded middle, and so the arguments to contradiction are broken. (Thus, the paradoxes of self-reference are sometimes used as an argument for the existence of truth-value gaps, too.)

(129)

[contents]

 

 

 

 

 

 

7.7.8

[Reply 2: Extended Paradoxes Still Produce Contradiction. Reply 3: Not All Paradoxes of Self-Reference Invoke the Law of Excluded Middle]

 

[Reply 2: “Extended Paradoxes” still present a contradiction. For example: “This sentence is either false or neither true nor false”. If true, it is either false or neither value. Either way, it is not true, which contradicts our assumption that it is true. If it is either false or neither valued (meaning that it is not true), then its value is what it claims to be, and thus it is true, which contradicts what we assumed. Reply 3: Some paradoxes of self-reference, like Berry’s paradox, do not invoke the law of excluded middle.]

 

[(ditto)]

This suggestion does not avoid contradiction, however, because of ‘extended paradoxes’.³ Consider the sentence ‘This sentence is either false or neither true nor false.’ If it is true, it is either false or neither. In both cases it is not true. If, on the other hand, it is either false or neither (and so not true), then that is exactly what it claims, and so it is true. In either case, therefore, it is both true and not true.

(130)

3. Moreover, and in any case, not all of the paradoxical arguments invoke the law of excluded middle. Berry’s paradox, for example, does not.

(130)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

 

 

 

 

 

.

 

Priest (4.8) An Introduction to Non-Classical Logic, ‘The Explosion of Contradictions,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

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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

 

4.

Non-Normal Modal Logics; Strict Conditionals

 

4.8

The Explosion of Contradictions

 

 

 

 

Brief summary:

(4.8.1) One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider. (4.8.2) The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles. (4.8.3) The second counter-example: we can have inconsistent laws without their contradiction entailing everything. (4.8.4) The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.

 

 

 

 

 

 

 

Contents

 

4.8.1

[The Strict Conditional Involves the Explosion of Contradictions]

 

4.8.2

[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]

 

4.8.3

[Counter-Example 2: Inconsistent Legislation]

 

4.8.4

[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]

 

 

 

 

 

 

 

 

Summary

 

4.8.1

[The Strict Conditional Involves the Explosion of Contradictions]

 

[One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider.]

 

[Let us first recall some notions regarding the strict conditional. In section 4.5.2 and section 4.5.3 we learned that the strict conditional is defined as “□(A ⊃ B),” and it is symbolized as A⥽B. In section 4.6.2 and section 4.6.3, we learned that modal systems that can handle conditionality should be systems where modus ponens holds: A, A⥽B ⊨ B. (I did not know why exactly this is necessary, but I guessed it was for the following reason. Suppose modus ponens does not hold. That would mean by affirming the antecedent, we could not obtain the consequent. But were that the case, then we have lost a basic intuition we have about conditionality, namely, that the consequent will follow necessarily from the antecedent.) We learned in section 4.6.2 that for modus ponens to hold in a modal system, it needs the ρ-constraint (reflexivity). (Recall it from section 3.2.3: “ρ (rho), reflexivity: for all w, wRw” p.36.) But we then learned in section 4.6.3 that no matter how many other constraints we add to ρ, we will always obtain the paradoxes of strict implication, with one being: ‘⊨ (A ∧ ¬A) ⥽ B’. Now in our current section, Priest says that by modus ponens, from ⊨ (A ∧ ¬A) ⥽ B we can derive (A∧¬A)⊨B. (I do not know exactly how that works, however. I guess the idea is that if we establish the conditional, and if we have modus ponens, then that means simply from the antecedent being affirmed we can infer the consequent as a semantic consequence. The important philosophical point here is that) the strict conditional in any modal system that can handle conditionality leads us to being able to derive any arbitrary formula whatsoever from a contradiction. As Priest puts it: “Contradictions would entail everything.” But this is counter-intuitive. Priest will now give three counter-examples of situations or theories that are inconsistent but also where we should not be able thereby to infer that everything whatsoever holds.]

The toughest objections to a strict conditional, at least as an account of the indicative conditional, come from the fact that ⊨(A∧¬A)⥽B. If this were the case, then, by modus ponens, we would have (A∧¬A)⊨B. Contradictions would entail everything. Not only is this highly counterintuitive, | there would seem to be definite counter-examples to it. There appear to be a number of situations or theories which are inconsistent, yet in which it is manifestly incorrect to infer that everything holds. Here are three very different examples.

(74-75)

[contents]

 

 

 

 

 

4.8.2

[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]

 

[The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles.]

 

[I do not know much about the first example, so please see the quotation below. The basic idea is that Bohr knowingly combined two inconsistent assumptions in his model of the atom, namely, he assumes “the standard Maxwell electromagnetic equations” but also “that energy could come only in discrete packets (quanta).” Yet, despite its obvious inconsistency, both assumptions were needed for the model to work and “many of its observable predictions were spectacularly verified.” Priest’s philosophical point here is that on the basis of this contradiction, we cannot infer everything else. “Bohr did not infer, for example, that electronic orbits are rectangles” (75).]

The first is a theory in the history of science: Bohr’s theory of the atom (the ‘solar system’ model). This was internally inconsistent. To determine the behaviour of the atom, Bohr assumed the standard Maxwell electromagnetic equations. But he also assumed that energy could come only in discrete packets (quanta). These two things are inconsistent (as Bohr knew); yet both were integrally required for the account to work. The account was therefore essentially inconsistent. Yet many of its observable predictions were spectacularly verified. It is clear though that not everything was taken to follow from the account. Bohr did not infer, for example, that electronic orbits are rectangles.

(75)

[contents]

 

 

 

 

4.8.3

[Counter-Example 2: Inconsistent Legislation]

 

[The second counter-example: we can have inconsistent laws without their contradiction entailing everything.]

 

[In Priest’s second counter-example, we have two laws that together function together non-problematically in most cases, but in a particular situation they come into contradiction. Priest then says that on the basis of this contradiction, “it would be stupid to infer from this that, for example, the traffic laws are consistent” (75). (I did not quite get how that works. Are we saying that we can consider our two inconsistent laws as presenting a structure like A∧¬A, and “the traffic laws are consistent” is some arbitrary B that we try to derive from it? At any rate, surely at least we might say that from this contradiction we cannot derive any other traffic law we want.)]

Another example: pieces of legislation are often inconsistent. To avoid irrelevant historical details, here is an hypothetical example. Suppose that an (absent-minded) state legislator passes the following traffic laws. At an unmarked junction, the priority regulations are:

(1) Any woman has priority over any man.

(2) Any older person has priority over any younger person.

(We may suppose that clause 2 was meant to resolve the case where two men or two women arrive together, but the legislator forgot to make it subordinate to clause 1.) The legislation will work perfectly happily in three out of four combinations of sex and age. But suppose that Ms X, of age 30, approaches the junction at the same time as Mr Y, of age 40. Ms X has priority (by 1), but has not got priority (by 2 and the meaning of ‘priority’). Hence, the situation is inconsistent. But, again, it would be stupid to infer from this that, for example, the traffic laws are consistent.

(75)

[contents]

 

 

 

 

4.8.4

[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]

 

[The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.]

 

[The third example is that there are perceptual illusions that can give us inconsistent impressions. For example, the waterfall illusion causes us to see something both in motion and not in motion. But thereby we do not perceive everything else, like for example that everything is red all over. The fourth example is that in fictional situations where there are contradictions, that does not entail that everything holds in that fictional situation. (For some reason the fourth one is placed in a  footnote, despite being an excellent and convincing counter-example.)]

Third example: it is possible to have visual illusions where things appear contradictory. For example, in the ‘waterfall effect’, one’s visual system is conditioned by constant motion of a certain kind, say a rotating spiral. If one then looks at a stationary situation, say a white wall, it appears to move in the opposite direction. But, a point in the visual field, | say at the top, does not appear to move, for example, to revolve around to the bottom. Thus, things appear to move without changing place: the perceived situation is inconsistent. But not everything perceivable holds in this situation. For example, it is not the case that the situation is red all over.⁵

(75-76)

5. A fourth kind of example is provided by certain fictional situations, in which contradictory states of affairs hold. This may well be the case without everything holding in the fictional situation.

(76)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

.

 

Priest (4.6) An Introduction to Non-Classical Logic, ‘The Paradoxes of Strict Implication,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

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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

 

4.

Non-Normal Modal Logics; Strict Conditionals

 

4.6

The Paradoxes of Strict Implication

 

 

 

 

Brief summary:

(4.6.1) We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) –  is adequate. But first we need to address the matter of its variance under different systems of modal logic. (4.6.2) To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle  that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint . (4.6.3) We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication:  ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.

 

 

 

 

 

Contents

 

4.6.1

[The Question of the Adequacy of the Strict Conditional ⥽]

 

4.6.2

[The Need for the ρ-Constraint]

 

4.6.3

[The Paradoxes of Strict Implication]

 

 

 

 

 

Summary

 

4.6.1

[The Question of the Adequacy of the Strict Conditional ⥽]

 

[We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) –  is adequate. But first we need to address the matter of its variance under different systems of modal logic.]

 

[Recall from section 4.5 the notion of the strict conditional. In section 4.5.3 we learn that the strict conditional, symbolized as ⥽, is defined in the following way: A⥽B is defined as □(A⊃B). (p.72, section 4.5.3). We now ask if this definition of the conditional is adequate. But as the properties of the strict conditional will vary according to the modal logic system at hand, we need to say more on this matter.]

Does it provide an adequate account of the conditional? Each system of modal logic gives ⥽ different properties. Hence, before we can answer that question, we need to address the question of which system of modal logic it is that is at issue. Let me make two comments on this.

(72)

[contents]

 

 

 

 

4.6.2

[The Need for the ρ-Constraint]

 

[To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle  that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint .]

 

[Priest first notes that modus ponens fails in systems without the ρ-constraint (the reflexivity constraint; see section 3.2.3). It seems that for conditionality we would want modus ponens to hold: A, A⥽B ⊨ B. I do not know the exact reason why, but it would seem that conditionality should allow us to infer the consequent from an affirmation of the antecedent. For otherwise, what is the sense of the conditional without that also holding? But, Priest says, in systems without the reflexivity constraint, modus ponens will not hold. Thus we at least need the reflexivity constraint.]

First, it is natural to suppose that any notion of necessity that is to be employed in defining a notion of conditionality must be at least as strong as Kρ (or Lρ if one is countenancing non-normal systems). This is because, without ρ, modus ponens fails: A, A⥽B ⊭ B. With it, it holds, as simple tableau tests verify.

(73)

[contents]

 

 

 

 

4.6.3

[The Paradoxes of Strict Implication]

 

[We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication:  ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.]

 

[Priest’s next point I might not summarize properly, but I am guessing it is the following. So far we specified that for the strict conditional we need systems with the reflexivity constraint. But we learn now that we need not narrow our systems down any further, because no matter how constrained we make them, all systems will lead to certain paradoxes. And it is these paradoxes that lead us to question the notion that the strict conditional models the English conditional.]

Second, a further determination of this question is not very important for what follows. This is because the major objections to the claim that English conditionals are strict hinge on a feature that the strict conditional possesses in all systems of modal logic. In all systems of modal logic the following hold:

□B ⊨ A ⥽ B

¬◊A ⊨ A ⥽ B

These facts are sometimes called the ‘paradoxes of strict implication’. A tableau test verifies that these hold in L, and so in all the normal and non-normal systems that we have looked at. Since, in all systems, we also have ⊨□(B∨¬B) and ⊨¬◊(A∧¬A), this gives us as special cases:

⊨ A ⥽ (B ∨ ¬B)

⊨ (A ∧ ¬A) ⥽ B

(73)

[contents]

 

 

 

 

 

 

 

From:

 

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

 

 

.

 

Priest (1.4) One. ‘The Bradley Regress,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

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[The following is summary. You will find typos and other distracting mistakes, because I have not finished proofreading. Bracketed commentary is my own. Please consult the original text, as my summaries could be wrong.]

 

 

 

Summary of

 

Graham Priest

 

One:

Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness

 

Ch.1

Gluons and Their Wicked Ways

 

1.4

The Bradley Regress

 

 

 

Brief summary:

(1.4.1) We will now discuss why the gluon cannot be an object on account of a vicious regress. (1.4.2) In the Bradley regress, a binding factor is posited as being a member of the unity it binds. But that only leaves us to find yet another binding factor that would bind the first into the whole. There can be no end so long as the binding factors are consider as object/parts. In terms of gluons, if we make the gluon be an object/part, then we will always need yet another gluon to explain how the prior gluon is bound into the whole. (1.4.3) We cannot simply account for unity by saying it is gluons all the way down. For, no such gluon is sufficient to explain the unity. All of them require something in addition. So, simply saying the unity is found in yet another part never tells us in what the unity consists. (1.4.4) In conclusion, on account of the Bradley Regress, we cannot explain the unity of objects as being another object.

 

 

 

 

Contents:

 

1.4.1

[Preview: The Gluon as a Non-Object on Account of a Vicious Regress]

 

1.4.2

[The Bradley Regress and Composition]

 

1.4.3

[Unity is Not Gluons All the Way Down]

 

1.4.4

[Unity is Not Another Object]

 

Bibliography

 

 

 

 

Summary

 

1.4.1

[Preview: The Gluon as a Non-Object on Account of a Vicious Regress]

 

[We will now discuss why the gluon cannot be an object on account of a vicious regress.]

 

[Recall from section 1.3.4 the notion of the gluon. From the brief summary of that section: “The unifying factor in a thing is called its gluon. It both is and is not an object/part. It is an object insofar as we name it and conceive it. But it is not an object insofar as it is what constitutes the organizing and unifying factor of the thing, because as such it needs to be over and beyond any of the parts rather than simply being another part.” And here is the paragraph in full, where the gluon is given a slightly more formal account:

Here, then, is our problem of unity. Let me lay it out in abstract terms. Take any thing, object, entity, with parts, p₁, .. , p_n. (Suppose that there is a finite number of these; nothing hangs on this.) A thing is not merely a plurality of parts: it is a unity. There must, therefore, be something⁹ which constitutes them as a single thing, a unity. Let us call it, neutrally (and with a nod in the direction of particle physics), the gluon of the object, g.¹⁰ Now what of this gluon? Ask whether it itself is a thing, object, entity? It both is and is not. It is, since we have just talked about it, referred to it, thought about it. But it is not, since, if it is, p₁, .. , p_n, g, would appear to form a congeries, a plurality, just as much as the original one. If its behaviour is to provide an explanation of unity, it cannot simply be an object.

(p.9, section 1.3.4)

9. Or some things; but it will turn out that there is only one.

10. The name was coined, with essentially this meaning, in the Conclusion to Priest (1995a).

(p.9, section 1.3.4)

Although it may seem like the gluon could be an object, as it in some sense is a nameable something with regard to a thing’s unity, Priest now will explain why it cannot be an object.]

It will pay to become clearer about why a gluon cannot be an object. A vicious regress stands behind this.¹²

12. This kind of regress argument is very old. In the form of the “third man argument” it is used in Plato’s Parmenides as an argument against the theory of forms. Plato is there concerned with what makes all, for example, red things one (namely, red). Invoking a form of redness produces the regress. Being one by being red is not the same thing as being one by being parts of something, and Plato’s form is not (obviously) a gluon. However, structurally, the situations are similar. We will come to the third man argument itself in Chapter 8.

(9)

[contents]

 

 

 

1.4.2

[The Bradley Regress and Composition]

 

[In the Bradley regress, a binding factor is posited as being a member of the unity it binds. But that only leaves us to find yet another binding factor that would bind the first into the whole. There can be no end so long as the binding factors are consider as object/parts. In terms of gluons, if we make the gluon be an object/part, then we will always need yet another gluon to explain how the prior gluon is bound into the whole.]

 

[In section 1.3.3 we discussed Frege’s problem of accounting for the unity of a proposition. The unity is to be found in the relation between a function-part (like a predicate) and an argument part (like a subject to the predicate). That unity comes undone when trying to make statements about concepts themselves, because then something which is not an object is also bestowed that status by means of the propositional structure.] Priest returns now to the problem of unity in a proposition, but this time turning to Russell rather than Frege. [The idea seems to be the Following. Russell wants to account for the unity of the proposition, and he locates it in the copula ‘is’. For, it is what unites the subject and predicate. He furthermore claims that the copula cannot be a constituent of the proposition, and it can only be a “way in which the constituents are put together.” For, suppose that it were a constituent. We would still need to find something else that puts those constituents together (the subject, with the ‘is’, with the predicate). And supposing that binding element to be a component too, we would need yet another such binding factor. Under such a structuring pattern, we would reach no ultimate binding factor, despite that being our very aim.]

Return to the matter of the unity of the proposition again. At one stage in his career, Russell was much concerned with this, and one possibility he considered was that it was the copula, ‘is’, that binds the constituents together. (So, in Fregean terms, there is just one concept, which is the copula.1³) He then explains why the copula cannot be on a footing with the other constituents:¹⁴

It might be thought that ‘is’, here, is a constant constituent. But this would be a mistake: ‘x is a’ is obtained from ‘Socrates is human’, which is to be regarded as a subject-predicate proposition, and such propositions, we said, have only two constituents [Socrates and humanity]. Thus ‘is’ represents merely the way in which the constituents are put together. This cannot be a new constituent, for if it were there would have to be a new way in which it and the two other constituents are put together, and if we take this way as again a constituent, we find ourselves embarked on an infinite regress.

(10)

13. A discussion of this view, in the context of its regress, is given in Gaskin (1995).

14. Eames and Blackwell (1973), p. 98.

(10)

Priest says that Russell here is using an argument by F.H. Bradley that was also related to the issue of the unity of the proposition and to unity in general.

Russell is using an argument used earlier to great effect by Bradley.¹⁵ Again, addressing the problem of the unity of the proposition, Bradley starts by supposing that a proposition has components A and B. What constitutes them into a unity? A natural thought is that it is some relation between them, C. But, he continues:¹⁶

[we] have made no progress. The relation C has been admitted different from A and B ... Something, however, seems to be said of this relation C, and said, again, of A and B ... [This] would appear to be another relation, D, in which C, on one side, and, on the other side, A and B, stand. But such a makeshift leads at once to the infinite process ... [W]e must have recourse to a fresh relation, E, which comes between D and whatever we had before. But this must lead to another, F; and so on indefinitely ... [The situation] either demands a new relation, and so on without end, or it leaves us where we were, entangled in difficulties.

And Bradley is, in fact, aware that this is not just a problem concerning the unity of the proposition. It is much more general. Thus, in discussing the unity of the mind, Bradley writes:¹⁷

When we ask ‘What is the composition of Mind,’ we break up that state, which comes to us as a whole, into units of feeling. But since it is clear that these units, by themselves, are not all the ‘composition’, we are forced to recognize the existence of the relations ... If units have to exist together, they must stand in relation to one another; and, if these relations are also units, it would seem that the second class must also stand in relation to the first. If A and B are feelings, and if C their relation is another feeling, you must either suppose | that component parts can exist without standing in relation to one another, or else that there is a fresh relation between C and AB. Let this be D, and once more we are launched off on the infinite process of finding a relation between D and C–AB; and so on forever. If relations are facts that exist between facts, then what comes between the relations and the other facts? (10-11)

15. In fact, it had been used some 600 years earlier by Jean Buridan in his Questiones in Metaphysicam Aristotelis (Bk V, q. 8). (See Normore (1985), p. 197f.) It should therefore be called the Buridan/Bradley regress.

16. Allard and Stock (1994), p. 120. 

17. Allard and Stock (1994), pp. 78–9. (10)

Priest then reformulates this in terms of gluons. Suppose that the gluon is a member among the other parts. We would then need another gluon to bind it with them. And that gluon would need yet another, and so on without end.

We can state the regress problem generally in terms of gluons. Suppose that we have a unity comprising the parts, a, b, c, d, for example. There must be something which, metaphysically speaking, binds them together.This is the object’s gluon, g. But then there must be something which binds g and a, b, c, d together, a hyper-gluon, g′. There must, then, be something which binds g′, g, and a, b, c, d together, a hyper-hyper-gluon, g′′. Obviously we are off on an infinite regress. Moreover, it is a vicious one.

(11)

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1.4.3

[Unity is Not Gluons All the Way Down]

 

[We cannot simply account for unity by saying it is gluons all the way down. For, no such gluon is sufficient to explain the unity. All of them require something in addition. So, simply saying the unity is found in yet another part never tells us in what the unity consists.]

 

Priest next explains why we cannot simply say that it is gluons all the way down, or in other words, that there is an infinity of gluons. [I may not capture his insight here. My best attempt for now is the following. What we want is an explanation for unity. Suppose we say it is gluons all the way down. This fails, because at no point in the going down is there a structuring part that unifies the whole. For, given any gluon in the infinite chain, none is sufficient to account for the unity. And to say that it is always to be found in yet another part only makes this problem unsolvable, because it makes it impossible to ever identify the ultimate unifying component. Let me quote, as I am probably not putting that in the best way.]

Perhaps it is not immediately obvious that this is so. Could there not just be a whole lot’a gluin’ goin’ on? To understand why this is not a valid response, we must come back to what is at issue here. Our original problem was how a unity of parts is possible. We need an explanation. Given a bunch of parts, simply invoking another object does not do this. We still have the original problem of how a unity of parts is possible. Thus is a new step triggered, and so on indefinitely. Even invoking an infinite regress of objects does not solve the problem. We still have no explanation of how a unity is constituted. If one is asked how to join two links of a chain together, it helps not one iota to say that one inserts an intervening link. (And adding that one might need an infinite number of such links merely makes the matter worse.) In vicious regresses of this kind (I do not think it is the only kind) the infinity has, in fact, precious little to do with matters. The point is that something has already gone wrong at the first step: a failure of explanation.¹⁸

(11)

18. ‘[I]t is the first step in the regress that counts, for we at once, in taking it, draw attention to the fact that the alleged explanation or justification has failed to advance matters; that if there was any difficulty in the original situation, it breaks out in exactly the same form in the alleged explanation. If this is so, the regress at once develops . . . ’ Passmore (1961), p. 31.

(11)

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1.4.4

[Unity is Not Another Object]

 

[In conclusion, on account of the Bradley Regress, we cannot explain the unity of objects as being another object.]

 

Thus: “As Frege realized, if something is to perform the role of explaining how it is that a unity of objects is achieved, it cannot just be another object” (11).

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Bibliography:

 

Priest, Graham. 2014. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University.

 

 

Or if otherwise cited:

 

Allard, J. W., and Stock, G. (eds.) (1994), F. H. Bradley: Writings on Logic and Metaphysics, Oxford: Oxford University Press.

 

Eames, E., and Blackwell, K. (eds.) (1973), Collected Papers of Bertrand Russell, vol. 7: Theory of Knowledge, London: Allen and Unwin.

 

Gaskin R. (1995), ‘Bradley’s Regress, the Copula and the Unity of the Proposition’, Philosophical Quarterly 45: 161–80. E

 

Normore, C. (1985), ‘Buridan’s Ontology’, pp. 189–203 of J. Bogen and E. McGuire (eds.), How Things Are, Dordrecht: Reidel Publishing Company.

 

Passmore, J. (1961), ‘The Infinite Regress’, ch. 2 of Philosophical Reasoning, London: Duckworth.

 

 

 

 

 

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