Pirates & Revolutionaries: identity

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Summary of

Daniel Smith

[Smith’s academia.edu page]

Essays on Deleuze

Ch.5

Pre- and Post-Kantianism

Logic and Existence: Deleuze on the Conditions of the Real

5.0

“[Introductory material]”

Brief summary (collecting those below):

(5.0.1) At the core of Deleuze’s interest in the rationalists, Leibniz especially, is the philosophical problem of using thought to distinguish the possible from the real. For, nothing would change among the predicates involved in the conception of something were it to exist instead of not exist (or not-exist instead of exist). (5.0.2) Smith will give a cinematic thematization of the material he presents in the text by characterizing its parts as if they were something like parts of a film. (5.0.3) The basic principles of logic, especially the three classic ones – Identity, Non-Contradiction, and Excluded Middle – tell us what is unthinkable and thus impossible. Under Deleuze’s formulations, the Principle of Identity is expressible as “A is A” or “A thing is what it is”; the Principle of Non-Contradiction is (unconventionally) formulated by Deleuze as “A is not non-A,” or “A thing is not what it is not”; and the Principle of Excluded Middle: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” They therefore offer some guidance regarding what is possible. Deleuze will conduct an investigation into the history of philosophy to see if these three laws especially had been reconceived to allow us to think beyond the possible to the real or existant itself. (5.0.4) Smith will follow how Deleuze tells a story in the history of philosophy that is about how philosophers of different times or sorts have reconceived the three classical principles of logic in order to think the real and existant itself and not merely the possible; the Principle of Identity: Pre-Kantian Rationalists, especially Leibniz; the Principle of Non-Contradiction: German Idealists, especially Hegel; and the Principle of Excluded Middle: “existentialists.” Lastly, Smith will explain Deleuze’s critique of these solutions and discuss the one Deleuze proposes instead.

5.0.1

[Deleuze’s Philosophical Question: How Can Thought Think the Real and Not Just the Possible?]

5.0.2

[Smith’s Cinematic Thematization]

5.0.3

[Deleuze’s Particular Formulations for the Three Classic Principles of Logic]

5.0.4

[Previewing the Text]

Bibliography

Summary

5.0.1

[Deleuze’s Philosophical Question: How Can Thought Think the Real and Not Just the Possible?]

[At the core of Deleuze’s interest in the rationalists, Leibniz especially, is the philosophical problem of using thought to distinguish the possible from the real. For, nothing would change among the predicates involved in the conception of something were it to exist instead of not exist (or not-exist instead of exist).]

[ditto] [Deleuze discusses this issue especially in his course of 1983.05.17, Part 1, Part 2, and Part 3.] As Smith explains so well, if we only have our thinking to rely upon, we cannot make a distinction, conceptually speaking, between the concept of a thing were it to exist and the concept of that same thing were it not to exist. [Deleuze says something similar to Smith’s 100 dollars and unicorn illustrations and Kant comment in Course 1983.05.17, Part 1: “Pourquoi est-ce que la pensée en tant que pensée ne dispose d’aucun moyen pour distinguer le possible et le réel ? C’est évident - ou : le possible et l’existant -, c’est évident si vous y réfléchissez. Considérez un concept quelconque, ou une représentation quelconque : représentation soit d’un bœuf, soit d’une chimère, soit d’un triangle. Cette représentation ou ce concept, c’est ce que la pensée pense. Rien n’est changé, que l’objet de la représentation existe ou n’existe pas. Tout est changé pour nous, rien n’est changé pour la pensée, c’est-à-dire : rien n’est changé dans la représentation. C’est ce que Kant disait déjà dans une page célèbre de la Critique de la raison pure. Vous vous faites la représentation de 100 francs - il disait, lui, pour des raisons de nationalité, 100 thalers. Vous vous représentez 100 francs : que ces 100 francs existent, bien plus, que vous les ayez ou que vous ne les ayez pas, c’est très important pour vous ; du point de vue de la représentation, rien n’est changé. Vous vous faites un concept de chimère, animal fabuleux. Qu’il y ait ou qu’il n’y ait pas des chimères, ça change beaucoup ; ça change rien du point de vue du concept, du point de vue de la représentation.” (00:07:17-00:09.32).] Let us work with Smith’s example of the unicorn. We have the idea of the unicorn. And we note that it does not exist. Suppose now that genetic scientists produce a unicorn (or suppose millions of years from now one evolves.) Would the existence of the unicorn change any of its conceptualizable properties? It would seem not. (It would still seem to be a horse with a horn. The fact that it does not presently exist does not change what defines it. It either could exist or it could not. Either way, a unicorn is still conceived as a horse with a horn pointing straight forward from its head.) Thus existence does not seem to be a predicate at least when it comes to predicates that we assign when conceiving concepts or ideas, especially in terms of their essential or defining features. This is because, as Smith explains, “The position of the real is outside the concept; the existing thing is external to the concept.” [Again, note Deleuze’s Course 1983.05.17, Part 1: “Ce qu’on a toujours traduit, en philosophie, en disant que l’existant, c’était extérieur à la représentation. L’existant, c’est la position de l’objet hors du concept. (00:09:35-00:09.44).]

Here is a philosophical problem that lies at the core of Deleuze’s interest in the rationalists, and particularly Leibniz.¹ By itself, thought has no means of distinguishing between the possible and the real. I can have a concept of 100 dollars in my mind, and while it may be important to me practically whether or not I actually have 100 dollars in my pocket, the existence of 100 dollars in reality changes nothing from the point of view of the concept: that is, from the viewpoint of pure thought. The position of the real is outside the concept; the existing thing is external to the concept. (This was Kant’s argument against the ontological argument: existence is not a predicate; from the viewpoint of the concept, an existing God is no more perfect than a non-existing God.) Even though I know that unicorns do not exist, I can still form a concept or a representation of a unicorn, or define the essence of a unicorn.

(72)

1. This paper was originally presented at the conference “Deleuze and Rationalism,” which took place on 16–17 March 2007 at the Centre for Research in Modern European Philosophy at Middlesex University, London.

(377)

5.0.2

[Smith’s Cinematic Thematization]

[Smith will give a cinematic thematization of the material he presents in the text by characterizing its parts as if they were something like parts of a film.]

In this paragraph, Smith explains different philosophical ways to approach this problem, and he describes and defends his cinematic thematization of the material. He firstly reformulates the philosophical problem Deleuze is working on here, namely: how can thought leave its domain of the possible and instead think the real itself (the existant)? He says “Pre-Kantians like Leibniz approached this problem in terms of the distinction between truths of essence (“A triangle has three sides”) and truths of existence (“Caesar crossed the Rubicon”), while post-Kantians like Maimon approached the problem in terms of the distinction between the conditions of possible experience and the conditions of real experience.” (I do not have textual substantiation for these claims, at the moment.) Smith next says that he will approach the problem from a “semi-cinematic perspective,” and he grounds this in a comment Deleuze makes regarding Godard making a film about philosophical texts. [Overall, what Smith is doing here is explaining why he will thematize the presentation of his text as if it were something like a film script. As we will see, it makes the text more playful and charming. It is not entirely obvious if it adds anything substantial to the philosophical material he presents, but it enlivens the text and makes it even more fun and exciting to read.]

For Deleuze, this is one of the fundamental problems of a theory of thought: How can thought leave this meager sphere of the possible in order to think the real: that is, to think existence itself, to think existing things. Pre-Kantians like Leibniz approached this problem in terms of the distinction between truths of essence (“A triangle has three sides”) and truths of existence (“Caesar crossed the Rubicon”), while post-Kantians like Maimon approached the problem in terms of the distinction between the conditions of possible experience and the conditions of real experience. I would like to approach this logical problem from a semi-cinematic perspective. “Theoretically,” Deleuze once mused, “Jean-Luc Godard would be capable of filming Kant’s Critique or Spinoza’s Ethics” (DI 141). In the 1990s, Godard did a multi-part film entitled Histoire(s) du cinéma; following Deleuze’s suggestion, I am imagining Godard undertaking a similar project entitled Histoire(s) de la philosophie. I have no idea, of course, what Godard might have done in such a film, but none the less I am presenting the first part of this essay as a possible scenario for a single sequence of that multi-part film, which has as its title Logic and Existence, which I am borrowing from a well-known book by Jean Hyppolite.²

(72)

2. Jean Hyppolite, Logic and Existence [1952], trans. Leonard Lawlor and Amit Sen (Albany: State University of New York Press, 1997). This book completes the project Hyppolite began with Genesis and Structure of Hegel’s “Phenomenology of Spirit,” trans. Samuel Cherniak and John Heckman (Evanston, IL: Northwestern University Press, 1979), and examines the relation between the phenomenology and the logic. Deleuze wrote an important review of the book in 1954, “Jean Hyppolite’s Logic and Existence,” which is included as appendix to the English translation (191–5). (377)

[DI: Gilles Deleuze, Desert Islands and Other Texts, ed. Sylvère Lotinger, trans. Michael Taormina (New York: Semiotext(e), 2004).]

5.0.3

[Deleuze’s Particular Formulations for the Three Classic Principles of Logic]

[The basic principles of logic, especially the three classic ones – Identity, Non-Contradiction, and Excluded Middle – tell us what is unthinkable and thus impossible. Under Deleuze’s formulations, the Principle of Identity is expressible as “A is A” or “A thing is what it is”; the Principle of Non-Contradiction is (unconventionally) formulated by Deleuze as “A is not non-A,” or “A thing is not what it is not”; and the Principle of Excluded Middle: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” They therefore offer some guidance regarding what is possible. Deleuze will conduct an investigation into the history of philosophy to see if these three laws especially had been reconceived to allow us to think beyond the possible to the real or existant itself.]

Smith next notes that we might turn to the principles of logic in order pursue this question of how can thought thing the real and existant? He then proceeds through the three classical principles of logic, identity, non-contradiction, and excluded middle. Smith will give verbal formulations for each of them. The Principle of Identity says that “A is A” or “A thing is what it is.” Smith notes that the next two can be seen as specifications of this principle [but in what way they are specifications is not explained yet]. The Principle of Non-Contradiction can be seen as saying that “A is not non-A,” or “A thing is not what it is not” [At this point, it is important that we take some care. To all appearances, Smith is closely following Deleuze’s course lecture of 1983.05.17 that we have and will continue to quote from. In this lecture, Deleuze verbally formulates the three classic principles of logic in a way that Smith also does here. But what I find problematic is Deleuze’s formulation of the Principle of Non-Contradiction. “A is not non-A” is not, as far as I know, the conventional way to formulate this principle, even going back to its early appearance in Aristotle. Normally we understand the Principle of Non-Contradiction as being verbally formulated as “It is not that A and not-A.” (See especially Graham Priest, Doubt Truth to be a Liar, 8-9). Deleuze’s and Smith’s formulation is much closer to the Principle of Double Negation (“A if and only if not not-A”) on account of the verb ‘to be’ normally functioning more like the biconditional operator. I have not discovered the source for Deleuze’s seeming confusion about how to formulate the Principle of Non-Contradiction. Given his later focus on Hegel when discussing non-contradiction, we might think that he is getting this formulation from Hegel. However, that is not how it seems to me at the moment. For instance, in Hegel’s Science of Logic, he gives a positive and negative formulation for the Principle of Identity: “the essential category of identity is enunciated in the proposition: everything is identical with itself, A = A. Or negatively: A cannot at the same time be A and not A” (Hegel, Science of Logic, 409). The negative formulation here is close to the normal formulation of the Principle of Non-Contradiction. Hegel later writes, “The other expression of the law of identity: A cannot at the same time be A and not-A, has a negative form; it is called the law of contradiction” (ibid., 416). Yet, Hegel does not next explain what the negative form of this expression is, which would presumably formulate precisely the Principle of Non-Contradiction. So he does not say here that it would be “A is not not-A,” as far as I can tell. I find this issue highly problematic for making more precise determinations of Deleuze’s logic. In other words, we cannot easily say, on the basis of what Deleuze says here, whether he rejects or accepts the Principle of Non-Contradiction. He will seemingly state that he accepts it, but what he would apparently be accepting rather is the Principle of Double Negation.] And the Principle of Excluded Middle is verbally formulable as: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” These three laws tell us what is unthinkable, which thereby tells us what is impossible [presuming that anything unthinkable is impossible]. So, were something to break the Principle of Identity, then it would not be what it is. Were it to break the Principle of Non-Contradiction, then it would be what it is not. [This seems to me to be more a matter of breaking the Principle of Identity than of Non-Contradiction. But Deleuze is seemingly working with the Principle of Double Negation here instead of the Principle of Non-Contradiction. Hence, perhaps, this odd claim.] And finally, something breaks the Principle of Excluded Middle if it is “both what it is and what it is not.” [This seems to be what is more conventionally understood as breaking the Principle of Non-Contradiction. To break Excluded Middle, I think it would be something more like, “To be neither what it is nor what it is not.”] Deleuze’s question will be: “Is there any way in which these three classical principles can be used to exit the sphere of logic and penetrate existence itself?” [Again, see Course 1983.05.17, Part 1, at audio times (00:11:50-16:46): mais c’est très important de comprendre ça, c’est pour ça qu’il y a un problème de la pensée. Je dirais : le problème éternel de le pensée, ç’a été : moi, pensée, comment est-ce que je vais arriver à penser le réel et l’existant ? comment est-ce que je vais sortir de ma sphère des possibles ? comment penser autre chose que l’essence ? Je dirais presque, c’est à partir de là, bon... D’où... d’où, il me semble, la distinction de deux types de principes. La pensée par elle-même pense le possible. Au nom de quoi ? Au nom de certains principes qu’on appellera des principes logiques. Les principes logiques sont des principes qui fixent ce qui est possible et ce qui ne l’est pas ; qui déterminent ce qui est possible et ce qui n’est pas possible. Et ces principes logiques, je dirais : ce sont les principes des essences ou du possible, puisqu’ils discernent, ils distinguent le possible du non-possible ou de l’impossible, ces principes sont au nombre de trois dans la logique classique. → L’un, c’est le principe d’identité, A est A. Et puis deux petits principes qui semblent être comme des spécifications du grand principe d’identité, A est A, c’est-à-dire la chose est ce qu’elle est. → Second principe, dit de non-contradiction : A n’est pas non-A, la chose n’est pas ce qu’elle n’est pas. → Et puis troisième principe, dit du tiers-exclu : la chose est A ou non-A. Ou si vous préférez : entre A et non-A, il n’y a pas de tiers, d’où l’expression « principe du tiers exclu », A ou non-A. Ça m’intéresse déjà, parce que ces trois principes de pure logique, → l’un est un principe de position ou d’affirmation (A est A), → le second est un principe de négation (A n’est pas non-A), → le troisième est un principe d’alternative ou de disjonction (A ou non-A). Je sais donc ce qui est impossible, c’est-à-dire impensable. Ce qui est impossible ou impensable, c’est quelque chose qui ne serait pas ce qu’elle est (donc elle contredirait à l’identité), qui serait ce qu’elle n’est pas (elle contredirait à la non-contradiction), et qui serait à la fois ce qu’elle est et ce qu’elle n’est pas (elle contredirait au tiers exclu). Tout va bien. Sous ces trois principes, je pense les essences, le monde des essences ou le monde du possible, mais je retombe toujours là-dessus : comment penser quelque chose de réel ?]

Here’s the first shot: a radiant sphere hovering in the middle of nowhere. Nothing is written on it, but we know it is the sphere of logic. The film begins here | for an obvious reason: if thought, on its own, is only capable of thinking the possible, it does so on the basis of what can be called logical principles. Classical logic famously identified three such principles. These are the principle of identity (which says that “A is A,” or “A thing is what it is”), and then two smaller principles which seem to be specifications of the principle of identity: the principle of non-contradiction (which says that “A is not non-A,” or “A thing is not what it is not”) and the principle of the excluded middle (which says “either A or not-A,” that is, between A or not-A, there is no middle term). Taken together, these three principles determine what is impossible—that is to say, what is unthinkable without contradiction: something that would not be what it is (which would contradict the principle of identity); something that would be what it is not (which would contradict the principle of non-contradiction); and something that would be both what it is and what it is not (which would contradict the principle of the excluded middle). This sphere of logic would seem to enclose us within the domain of the possible, or what classical philosophy called the domain of essences. But this opening shot sets up the problem with a visual image: Is there any way in which these three classical principles can be used to exit the sphere of logic and penetrate existence itself?

(72-73)

5.0.4

[Previewing the Text]

[Smith will follow how Deleuze tells a story in the history of philosophy that is about how philosophers of different times or sorts have reconceived the three classical principles of logic in order to think the real and existant itself and not merely the possible; the Principle of Identity: Pre-Kantian Rationalists, especially Leibniz; the Principle of Non-Contradiction: German Idealists, especially Hegel; and the Principle of Excluded Middle: “existentialists.” Lastly, Smith will explain Deleuze’s critique of these solutions and discuss the one Deleuze proposes instead.]

Smith next outlines how the text will proceed. First he looks at how pre-Kantian rationalists, especially Leibniz, reconceive the Principle of Identity and extend it to the whole of existence. Secondly, he examines how German Idealists, especially Hegel, do this with the Principle of Non-Contradiction. And thirdly he looks at how “existentialist” sorts of philosopher do this with the Principle of Excluded Middle. He lastly will explain why Deleuze thinks they all fail and how he offers his own response to the problem.

The response to this question will take us through three scenes, which correspond to three broad sequences in the history of philosophy, three attempts to resolve this problem using one of these logical principles. Scene one focuses on the pre-Kantians, the rationalists; its star is Leibniz, since it was he who attempted to extend the principle of identity to the whole of existence. Scene two focuses on the post-Kantians, primarily the German Idealists; its story culminates in Hegel, since it was he who attempted to extend the principle of non-contradiction to the whole of existence. Scene three, finally, looks at that loosely related group of thinkers that often tend to be called, precisely, “existentialists,” since it is they who attempted to extend the principle of the excluded middle to existence. The screenplay reaches its climax with Deleuze: at the end, it briefly examines the reasons why Deleuze is at once fascinated with all three of these philosophical attempts to “think existence,” but none the less thinks they fail, and why he ultimately charts out his own response to the problem. The ending, alas, is somewhat truncated, since the production went over budget, which meant that entire scenes wound up being consigned to the editing room floor.

(73)

Bibliography:

Smith, Daniel. “Logic and Existence: Deleuze on the Conditions of the Real.” In Essays on Deleuze, 72–85. Edinburgh: Edinburgh University, 2012.

(or simply:)

Smith, Daniel. Essays on Deleuze. Edinburgh: Edinburgh University, 2012.

https://www.academia.edu/20805798/Essays_on_Deleuze

Note that an earlier version of this chapter text (which is nearly but not precisely identical) is found in:

Smith, Daniel. “Logic and Existence: Deleuze on the ‘Conditions of the Real.’” Chiasmi International 13 (2011): 361–77.

Smith’s Academia.edu page

Other sources, if otherwise noted:

Deleuze, Gilles. Course 1983.05.17, Part 1. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=236 ; https://www.webdeleuze.com/textes/204.

Deleuze, Gilles. Course 1983.05.17, Part 2. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=250 ; https://www.webdeleuze.com/textes/204.

Deleuze, Gilles. Course 1983.05.17, Part 3. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=251 ; https://www.webdeleuze.com/textes/204.

Hegel, G. W. F. Science of Logic. Translated by A. V. Miller. Oxford/ New York: Routledge, 2002.

Priest, Graham. Doubt Truth to Be a Liar. Oxford: Oxford University, 2006.

21 Aug 2018

Priest (21.9) An Introduction to Non-Classical Logic, ‘Non-classical Identity,’ summary

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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

Many-valued Logics

21.9

Non-classical Identity

Brief summary:

(21.9.1) We now wonder, is it “plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1”? (21.9.2) We might think that identity statements involving the following circumstances could take non-classical values: non-denoting terms, future contingents, verificationism, vague predicates, and paradoxes of self-reference. (21.9.3) Priest will focus here on circumstances involving vagueness with regard to identity statements. “Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values” (468). (21.9.4) Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indeterminate identity statement a = b. But, since it is determinately true – it is 1 rather than i – that a = a, we can infer that a and b have different properties (for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b,) we may infer that a ≠ b. “Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468). (21.9.4) Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. (We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indeterminate identity statement a = b. But, since it is determinately true that a = a – it is 1 rather than i – we can infer that a and b have different properties; for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b, we may infer that a ≠ b. Let me quote Priest so to have it exactly right:) “Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468). (21.9.5) The inference of this argument against non-classical identity is based on a contraposed form of the substitutivity of identicals. (The best I have in my own words right now, to be revised later, is: We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, we must conclude instead that a ≠ b.) (Now the correct account, all in Priest’s words:) “To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that: v(∇A) ∈ D if v(A) = i ; v(∇A) = 0 otherwise . Then the argument is simply:

Suppose that ∇a = b    (1)

Then since ¬∇a = a     (2)

It follows that a ≠ b    (3)

The inference is a contraposed form of SI; SI itself we know to be valid” (468). (21.9.6) Evans’ argument against indeterminate identity must have something wrong about how it proceeds, because the machinery of 3-valued logics do indeed allow for identity statements to take the value i. (21.9.7) Evans’ argument does not hold for gap 3-valued logics, because when a and b are distinct objects, the premises are true but the conclusion is not true: “Consider the K₃ or Ł₃ evaluation in which: v(=)(d, e) = 1 if v(d) = v(e) ; v(=)(d, e) = i if v(d) ≠ v(e) . Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i” (468). (21.9.8) The inference under the above semantic interpretation for identity is still valid in glut logics, but under an alternate interpretation (namely, that identity statements about things that are the same are both true and false), the argument remains valid, yet it concludes that identity statements take the value i, and thus non-classical identity can hold in glut 3-valued logics: “In LP and RM₃, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM₃ interpretation in which: v(=)(d, e) = i if v(d) = v(e) ; v(=)(d, e) = 0 if v(d) ≠ v(e) . Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i” (469).

21.9.1

[Wondering About Assigning Identity Statements Non-Classical Values]

21.9.2

[Non-Classical Values for Identity Statements in Those Cases Where It Was Fitting for Existence Statements]

21.9.3

[Identity and Vague Predicates: The Motorcycle Recomposition Example]

21.9.4

[Evans’ Argument Against Indeterminate Identity]

21.9.5

[A Formal Analysis of the Argument]

21.9.6

[Evans’ Argument Against Indeterminate Identity as Having Something Wrong to It]

21.9.7

[Evans’ Argument Against Indeterminate Identity Fails for Gap Logics]

21.9.8

[Non-Classical Identity in Glut 3-Valued Logics Where Identity Statements Can Be Both True and False]

Summary

21.9.1

[Wondering About Assigning Identity Statements Non-Classical Values]

[We now wonder, is it “plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1”?]

[In the previous section 21.8 we examined the semantics for free logics that allow for identity to take values other than 0 and 1. We now wonder if this is a plausible notion.]

This raises the question of whether it is plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1.

(468)

21.9.2

[Non-Classical Values for Identity Statements in Those Cases Where It Was Fitting for Existence Statements]

[We might think that identity statements involving the following circumstances could take non-classical values: non-denoting terms, future contingents, verificationism, vague predicates, and paradoxes of self-reference.]

[(ditto). (See section 21.6).]

The considerations of 21.6 about existence statements and nonclassical truth values seem to apply just as much to identity statements. I leave the reader to think about plausible candidates for non-classical identity statements in the sorts of situation discussed there.

(468)

21.9.3

[Identity and Vague Predicates: The Motorcycle Recomposition Example]

[Priest will focus here on circumstances involving vagueness with regard to identity statements. “Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values” (468).]

[(ditto). (See Priest’s Logic: A Very Short Introduction ch.10 for another treatment of the motorcycle example.)]

I will just take up one of them in more detail: vagueness. Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values.

(468)

21.9.4

[Evans’ Argument Against Indeterminate Identity]

[Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. (We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indterminate identity statement a = b. But, since it is determinately true that a = a – it is 1 rather than i – we can infer that a and b have different properties; for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b, we may infer that a ≠ b. Let me quote Priest so to have it exactly right:) “Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468).]

[(ditto). (Note: my reasoning in parentheses above is highly uncertain. Please consult the quotation below.)]

There is a well-known argument (due to Gareth Evans) against this possibility, however. Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities.

(468)

21.9.5

[A Formal Analysis of the Argument]

[The inference of this argument against non-classical identity is based on a contraposed form of the substitutivity of identicals. (The best I have in my own words right now, to be revised later, is: We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, we must conclude instead that a ≠ b.) (Now the correct account, all in Priest’s words:) “To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that: v(∇A) ∈ D if v(A) = i ; v(∇A) = 0 otherwise . Then the argument is simply:

Suppose that ∇a = b (1)

Then since ¬∇a = a (2)

It follows that a ≠ b (3)

The inference is a contraposed form of SI; SI itself we know to be valid” (468).]

[Priest then analyzes the argument. I do not follow it so well, so I will stumble through it a bit. It seems that overall he will show how with Evans’ argument, even if we assume that identity statements can take the value i, then we will still infer that the statements in question will take a classical value. Priest has us write “it is indeterminate that” as ∇, and the truth conditions for this operator is:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

In other words, if a statement’s value is i, then it is indeterminate, and if its value is 1 or 0, then it is not indeterminate. Then we have three lines in the argumentation. We begin with the following supposition:

1. ∇a = b

So we begin by assuming ∇a = b, which means that a = b is indeterminate, in other words, that its value is i. Next, we say:

2. ¬∇a = a

Here maybe we are thinking the following, but I am guessing. We know that a = a is true. That means it is not i. That furthermore means that ∇a = a is false. Then, perhaps by some truth-condition for negation, then we say ¬∇a = a is true. From these two lines we conclude:

3. a ≠ b

This part is quite hard for me to follow. Priest says that the inference here is a contraposed form of the substitutivity of identicals. We saw it I think recently in section 21.8.3:

Similarly, it is not difficult to check that a=b ⊨ b=a and a=b, b=c ⊨ a=c. More generally, a = b, A_x(a) ⊨ A_x(b); for the proof of this, see 21.11.4. Note that this fact in no way depends on identities taking only classical values. Identities may well take the value i in LP or RM₃ (or b in FDE).

(467)

I am not sure what is meant by the contraposed form of the substitutivity of identicals. I cannot really guess well here, but I must try. We might think of the inference of the substitutivity as saying something like, “if you have a = b, then whatever you say of a you can say of b.” Maybe the contraposed version would be (and likely not, given how much of a guess this is): “if you cannot say everything of b that you can say of a, then a ≠ b.” But even if I am on track there, I am not exactly sure yet how the inference works. We have as our premises:

1. ∇a = b

2. ¬∇a = a

and our conclusion is

3. a ≠ b

Now, maybe the inference here is simply a matter of the fact that whenever the premises are true and the conclusion is true. But without knowing the tableau rules, I am not sure how to show that. At any rate, let us try to reason through it as formally as I can make it right now, but in fact this is not really formal at all. It seems that basically we are saying the following. We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical to begin with (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, then we must conclude instead that a ≠ b.]

To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

Then the argument is simply:

Suppose that ∇a = b    (1)

Then since ¬∇a = a     (2)

It follows that a ≠ b    (3)

The inference is a contraposed form of SI; SI itself we know to be valid.

(468)

21.9.6

[Evans’ Argument Against Indeterminate Identity as Having Something Wrong to It]

[Evans’ argument against indeterminate identity must have something wrong about how it proceeds, because the machinery of 3-valued logics do indeed allow for identity statements to take the value i.]

[I may not follow Priest’s next point, but maybe it is the following. The argument we saw above in 21.9.5 tried to show that we cannot have indeterminate identity statements, because, on account of the substitutivity of identicals, whenever we say that one thing is indeterminately identical to another thing, we will also need to conclude that they are determinately non-identical (thus in fact they are not indeterminately identical to begin with but are simply determinately non-identical). Priest says now that this argument must fail, because it is possible for identity statements to take the value i in these logics. But I am not sure yet what the point is there. It seems to be that since we know they can take these values, as that is built into their machinery, the problem is not with the machinery of these systems but rather with the argument used against it.]

Now it is clear that as an argument against the possibility of indeterminate identities, the argument must fail. It is quite possible for identity statements to take the value i in all these logics. What, however, is wrong with it?

(469)

21.9.7

[Evans’ Argument Against Indeterminate Identity Fails for Gap Logics]

[Evans’ argument does not hold for gap 3-valued logics, because when a and b are distinct objects, the premises are true but the conclusion is not true: “Consider the K₃ or Ł₃ evaluation in which: v(=)(d, e) = 1 if v(d) = v(e) ; v(=)(d, e) = i if v(d) ≠ v(e) . Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i” (468). ]

[Let us review the three lines of the inference from section 21.9.5:

Suppose that ∇a = b    (1)

Then since ¬∇a = a     (2)

It follows that a ≠ b    (3)

Priest says now that in logics with truth-value gaps (where i means neither true nor false), the inference from ∇a = b and ¬∇a = a to a ≠ b is invalid. Now recall also how we were evaluating sentences with the indeterminacy operator:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

(p.468, section 21.9.5)

Priest now gives truth-conditions for the identity predicate in K₃ or Ł₃ (where i means neither true nor false):

v(=)(d, e) = 1 if v(d) = v(e)

v(=)(d, e) = i if v(d) ≠ v(e)

Here I am not sure what is the thinking behind this, because under this evaluation, identity statements are never false it seems, even though non-identity expressions like v(d) ≠ v(e) (which I think are saying that the domain members are not the same or not identical) can hold. In all, it seems we need to distinguish something being identical to something else, like two members of a set being the same, and an identity statement being true, false, or i. In these rules, if the members are the same, then their identity predication is true. If the members are not the same, then the identity predication is indeterminate. Still I do not know how to make intuitive sense of that, especially when in order to say that an identity statement about two things is i requires a determinate non-identity between them. At any rate, supposing all this, we find that the inference is not valid, because there is an interpretation that makes the premises true but the conclusion false. The first premise is:

Suppose that ∇a = b (1)

We will say that a and b are distinct objects, thus a ≠ b. Now recall from section 7.3.2 that in the gap logics, the only designated value is 1. And also recall that

v(=)(d, e) = i if v(d) ≠ v(e)

So the value of a = b is i. Now recall that:

v(∇A) ∈ D if v(A) = i

So that means ∇a = b is 1. Thus the first premise is true. The second premise is:

Then since ¬∇a = a (2)

Now, a = a has the value of 1, probably because:

v(=)(d, e) = 1 if v(d) = v(e)

Now recall that:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

(p.468, section 21.9.5)

So if a = a has the value of 1, then ∇a = a has the value of 0 (because it is not i). Then, negation would flip its value, so ¬∇a = a has the value of 1. That means now that all the premise are true. But what about the conclusion? It was:

It follows that a ≠ b (3)

Now, does our interpretation make the conclusion not-true so to show the inference to be invalid? Well, recall that a = b is i. In section 7.3.2 we saw that the negation of i is i. So if a = b is i, then a ≠ b (being its negation), is also i. Therefore, the premises are true but the conclusion is not true, and so the inference is invalid. Indeterminate identity holds in gap 3-valued logics.]

That depends. Suppose, for a start, that we are in a logic with truth value gaps. Then the inference from (1) and (2) to (3) is invalid. Consider the K₃ or Ł₃ evaluation in which:

v(=)(d, e) = 1 if v(d) = v(e)

v(=)(d, e) = i if v(d) ≠ v(e)

Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i.

(469)

21.9.8

[Non-Classical Identity in Glut 3-Valued Logics Where Identity Statements Can Be Both True and False]

[The inference under the above semantic interpretation for identity is still valid in glut logics, but under an alternate interpretation (namely, that identity statements about things that are the same are both true and false), the argument remains valid, yet it concludes that identity statements take the value i, and thus non-classical identity can hold in glut 3-valued logics: “In LP and RM₃, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM₃ interpretation in which: v(=)(d, e) = i if v(d) = v(e) ; v(=)(d, e) = 0 if v(d) ≠ v(e) . Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i” (469).]

[Priest next explains why in the glut logics LP and RM₃, the inference above is still valid. And he says that it is valid even without the second premise (but I do not know what is going on with the idea of excluding that premise). So again recall the argument:

Suppose that ∇a = b    (1)

Then since ¬∇a = a     (2)

It follows that a ≠ b    (3)

Priest has us suppose for (1) that ∇a = b takes a designated value. And recall:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

(p.468, section 21.9.5)

So ∇a = b has a designated value, and thus by the first rule, a = b is i. So the first premise is true. That means its negation is i, and so the conclusion a ≠ b is i. Recall from section 7.4.1 that in these glut logics, the designated values are 1 and i. The premises are designated values but the conclusion is also a designated value. Next it gets intuitively hard to grasp, but Priest then proposes evaluation rules that will make the inference invalid, and we will try to think philosophically about the intuitive content of this interpretation. Priest says that the evaluation rules for equality will be:

v(=)(d, e) = i if v(d) = v(e)

v(=)(d, e) = 0 if v(d) ≠ v(e)

So if two things are identically the same, then their identity predication is i (meaning here both true and false). Now, unlike before, a and b denote the same object (I think that means, a and b are constants, and are thus like names. The v function takes as their denotation the same item in the domain (or at least maybe, two items that are established as identical somehow, but I do not know yet how all that works). Thus according to our evaluation rule, since both a and b denote the same item, that means a = b has the value i. Let us see what that does to Evans’ argument:

Suppose that ∇a = b    (1)

Then since ¬∇a = a     (2)

It follows that a ≠ b    (3)

And recall that:

v(∇A) ∈ D if v(A) = i

v(∇A) = 0 otherwise

(p.468, section 21.9.5)

The first line, ∇a = b, is a designated value (being either 1 or i, but it seems not determined which). For the next line I am not sure. I will assume that a = a is i, because of the rule:

v(=)(d, e) = i if v(d) = v(e)

And since

v(∇A) ∈ D if v(A) = i

That makes ∇a = a a designated value. But here it gets less clear to me. Suppose we say it is true. Then its negation ¬∇a = a, is false, and thus all the premises will not be designated values. It seems we need to stipulate that ∇a = a be i and not 1, but I am not sure if I am on the right track, and if indeed I am on the right track, I am not sure how that works. At any rate, somehow or other, we will see that the second line is a designated value. So our premises are all designated values. What of the conclusion? Since a = b has the value i, then its negation, a ≠ b, also has the value i. Therefore, the premises are all designated values and the conclusion is too. The important point it seems is that this argument stays valid, but under these identity evaluation semantics, its validity only leads us to conclude that a = b has the value i and thus that non-classical identity can hold in a 3-valued glut logic. But let us now examine the philosophical intuitions and implications here. We are saying that a and b in a = b denote the same object, but a = b is both true and false. It would also seem that, because of the evaluation rule

v(=)(d, e) = i if v(d) = v(e)

that a = a is both true and false. That means a ≠ b is both true and false. How might this work? I think for example of the cases of development involving vague predicates in section 11.2, section 11.3, and sections 21.6.7 and 21.6.8. But think generally of different expressions of something that is thought be the “same” on account of it being what is undergoing variation, thus both having identity in one sense and not having it in another. After working more on Priest’s philosophy of non-classical identity and multiple-denotation, I will try to say more. But for now I note that identity statements (even self-identity statements) can be both true and false, and thus that you being identical to yourself is both true and false, under this glut many-valued logic with these identity evaluation rules. This might help us with understanding the properties of identity for things that are changing. In one sense the identity holds, in that it is a flux that is a unity by tight temporal contiguities or overlaps of the parts, but it is not unity in that the changes make the composition heterogeneous over time. More on this later.]

In LP and RM₃, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM₃ interpretation in which:

v(=)(d, e) = i if v(d) = v(e)

v(=)(d, e) = 0 if v(d) ≠ v(e)

Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i.

(469)

From:

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

18 Aug 2018

Priest (21.8) An Introduction to Non-Classical Logic, ‘Identity,’ summary

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Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

Part II:

Quantification and Identity

Many-valued Logics

21.8

Identity

Brief summary:

(21.8.1) We define identity in our many-valued quantified logics as:

(=)(d₁, d₂) ∈ D iff d₁ = d₂

(21.8.2) Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb. (21.8.3) Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i. (21.8.4) “If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467). (21.8.5) But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid. (21.8.6) Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).

21.8.1

[Defining Identity]

21.8.2

[Some Properties of Identity]

21.8.3

[Other Properties of Identity]

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

21.8.5

[Certain Valid and Invalid Statements in These Logics]

21.8.6

[Other Valid Formulas]

Summary

21.8.1

[Defining Identity]

[We define identity in our many-valued quantified logics as: (=)(d₁, d₂) ∈ D iff d₁ = d₂.]

[(ditto)]

If we now suppose that one of the predicates in the language is the identity predicate, then the natural truth conditions for this are:

v(=)(d₁, d₂) ∈ D iff d₁ = d₂

(467)

21.8.2

[Some Properties of Identity]

[Under this definition of identity, the following inferences are valid: ⊨ a=a and a=b, Pa ⊨ Pb.]

[(ditto)]

It is not difficult to check that ⊨ a=a and a=b, Pa ⊨ Pb. Thus, for the second of these, suppose that in an interpretation a = b is designated. Then v(a) = v(b). So v(P)(v(a)) ∈ D iff v(P)(v(b)) ∈ D.

(467)

21.8.3

[Other Properties of Identity]

[Under this definition, the following inferences are also valid: a=b ⊨ b=a and a=b, b=c ⊨ a=c. (In other words, identity is reflexive (see above), symmetric, and transitive.) It is also substitutable: a =b, A_x(a) ⊨ A_x(b). This holds even when identity is valued i.]

[(ditto)]

Similarly, it is not difficult to check that a=b ⊨ b=a and a=b, b=c ⊨ a=c. More generally, a = b, A_x(a) ⊨ A_x(b); for the proof of this, see 21.11.4. Note that this fact in no way depends on identities taking only classical values. Identities may well take the value i in LP or RM₃ (or b in FDE).

(467)

21.8.4

[Truth-Conditions for Neutral Gap Free Logics]

[“If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become: if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b) ; if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i (which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function: if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b) ; if either v(a) or v(b) is not defined then v(=)(a, b) = i ”(467).]

[(ditto)]

If we are in a logic where i is thought of as neither true nor false, and we enforce the neutrality constraint, then the truth conditions for identity become:

if v(a) ∈ E and v(b) ∈ E then v(=)(a, b) ∈ D iff v(a) = v(b)

if v(a) ∉ E or v(b) ∉ E then v(=)(a, b) = i

(which makes sense provided that i ∉ D). Or, if we dispense with the outer domain, and take the denotation function to be a partial function:

if v(a) and v(b) are defined then v(=)(a, b) ∈ D iff v(a) = v(b)

if either v(a) or v(b) is not defined then v(=)(a, b) = i

(467)

21.8.5

[Certain Valid and Invalid Statements in These Logics]

[But, if in our logic i is neither true nor false and we also enforce the neutrality constraint, then ⊨ a=a is no longer valid (for, if a is non-existent, then a=a is i, and thus not a designated value). However, a=b, Pa ⊨ Pb and more generally, a=b, A_x(a) ⊨ A_x(b) are valid.]

[(ditto) (Note: I am assuming here, probably incorrectly, that in this logic with i as gap that i is not a designated value. We saw in section 7.3 that the logics with gaps have 1 as the only designated value, while, as we saw in section 7.4, the glut ones have i and 1 as the designated values.)]

It is clear that it will not now be the case that ⊨ a=a. (Take v(a) to be not in E, or undefined.) However it is still the case that a=b, Pa ⊨ Pb. If the first premise is true, then v(a) and v(b) are both in E (or defined), and the argument then proceeds as in 21.8.2. Indeed, more generally, a=b, A_x(a) ⊨ A_x(b). The proof is to be found in 21.11.4.

(467)

21.8.6

[Other Valid Formulas]

[Lastly, Priest notes that “given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a” (467).]

[(ditto)]

Note that, given the neutrality constraint, a=b ⊨ ℭa ∧ ℭb and ℭa ⊨ a=a, as is easy to check.

(467)

From:

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

15 Aug 2018

Priest (16.4) An Introduction to Non-Classical Logic, ‘Rigid and Non-rigid Designators,’ summary

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Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

Part II:

Quantification and Identity

16.

Necessary Identity in Modal Logic

16.4

Rigid and Non-rigid Designators

Brief summary:

(16.4.1) There is a standard objection to quantified modal logic, namely, that it leads to claims of necessity regarding matters that are really contingent, on account of the workings of necessary identity. For example, “Beethoven wrote nine symphonies. Therefore 9 = β, where β is ‘the number of symphonies that Beethoven wrote’. Given NI, ∀x∀y(x = y ⊃ □x = y), it follows that □9 = β; that is, necessarily the number of Beethoven symphonies is nine – which is false, since Beethoven could have died immediately after writing the eighth” (354). (16.4.2) The negativity constraint will not prevent this problem; for, even under the constraint, a = b ⊃ □(ℭa ⊃ a = b) is still valid, and so we must draw the same conclusion: “Since 9 = β, it still follows that □(ℭ9 ⊃ 9 = β), and so □ ℭ9 ⊃ □9 = β” (354). (16.4.3) Priest diagnoses the problem in the following way: “What has gone wrong with the argument is, in fact, that the noun phrase β, ‘the number of symphonies written by Beethoven’ is a noun phrase that may change its denotation from world to world. In some worlds, Beethoven wrote eight symphonies, in some two, in some 147” (354). (16.4.4) When we consider a constant as having world-invariant denotation (like β, ‘the number of symphonies written by Beethoven’, being understood as being 9 in all worlds), it is a rigid designator, and we write it under the form: v(c). However, constants that do vary with the world are called non-rigid designators (like β, ‘the number of symphonies written by Beethoven’, being understood as potentially taking a different value in different worlds, like 2, 9, or 147), and we write them accordingly under the form v_w(c). (“Compare predicates, where extensions may change from world to world, and we write v_w(P), not v(P))” (354-355). (16.4.5) We then “augment the language with a collection of new constants: α₀, α₁, α₂, . . . and call these descriptor constants, or just descriptors. I will use α, β, γ , . . . for arbitrary descriptors. I will call our old constants rigid constants. The terms of the language now comprise descriptors, rigid constants and variables” (355). (16.4.6) Priest next gives the truth-conditions for our semantics.

v_w(¬A) = 1 if v_w(A) = 0, and 0 otherwise.

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, and 0 otherwise.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, and 0 otherwise.

[…]

v_w(◊A) = 1 if, for some w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

(pp.21-22, sections 2.3.4, 2.3.5)

v_w(∃xA) = 1 iff for some d ∈ D_w, v_w(A_x(k_d)) = 1

v_w(∀xA) = 1 iff for all d ∈ D_w, v_w(A_x(k_d)) = 1

(p.331, section 15.3.2)

v_w(Pt₁ . . . t_n) = 1 iff ⟨v_w(t₁), . . . , v_w(t_n)⟩ ∈ v_w(p)

(355)

(16.4.7) Priest then explains how we modify the tableau rules to accommodate descriptors: {1} “the IIR applies only if both terms are rigid constants”; {2} “the rules of universal and particular instantiation (and the NCR if it is present) apply only to rigid constants”; and {3} there is a new identity rule.

Constant-Descriptor Identity (CDI,D)

↓

c = α,i

(c is a constant new to the branch. This rule is applied to every descriptor, α, on the branch, and every i on the branch, for which there is not already a line of this form.)

(355, with names and additional text at the bottom made by me. The name is my own fabrication and probably needs correction.)

(16.4.8) Priest then gives an example tableau for a valid inference in CK(NI) with designators. (16.4.9) Then Priest gives an example tableau for an invalid formula. (16.4.10) “We read off a counter-model from an open branch of a tableau as before. In addition, if there is a line of the form c=β,i on the tableau, we set v_{w_i} (β) to v(c). (Note that if we have lines of the form c₁=β,i and c₂=β,i, then we have a line of the form c₁=c₂,i, by SI, so v(c₁) = v(c₂).)” (356). (16.4.11) Priest then gives an example counter-model. (16.4.12) “Note that various quantifier inferences that hold for rigid constants may fail for descriptors. Thus, □Pα ⊬_CK ∃x□Px” (357). (16.4.13) These tableaux are sound and complete.

16.4.1

[Objection to Quantified Modal Logic: Necessary Identity Leads to False Claims of Necessity]

16.4.2

[The Failure of the Negativity Constraint to Solve the Problem]

16.4.3

[The Problem as Resulting from the Implementation of a Noun Phrase Whose Denotation May Change Between Worlds]

16.4.4

[Rigid and Non-Rigid Designators]

16.4.5

[Rigid Designators as “Rigid Constants” and Non-Rigid Designators as “Descriptor Constants” or as “Descriptors”]

16.4.6

[The Truth-Conditions]

[Tableau Rules]

[Example Tableau 1]

[Example Tableau 2]

[Counter-Models]

[Example Counter-Model 1]

16.4.12

[Failing Quantifier Inferences Under Descriptors That Would Have Held for Rigid Constants. Example Counter-Model 2]

16.4.13

[The Soundness and Completeness of the Tableaux]

Summary

16.4.1

[Objection to Quantified Modal Logic: Necessary Identity Leads to False Claims of Necessity]

[There is a standard objection to quantified modal logic, namely, that it leads to claims of necessity regarding matters that are really contingent, on account of the workings of necessary identity. For example, “Beethoven wrote nine symphonies. Therefore 9 = β, where β is ‘the number of symphonies that Beethoven wrote’. Given NI, ∀x∀y(x = y ⊃ □x = y), it follows that □9 = β; that is, necessarily the number of Beethoven symphonies is nine – which is false, since Beethoven could have died immediately after writing the eighth” (354).]

[We have been looking at quantified modal logics (see section 14 and section 15 and our current section 16.) Priest will now discuss a standard objection to it. He gives the following illustration. We begin by noting that Beethoven wrote nine symphonies. We write “the number of symphonies that Beethoven wrote” as “β”. So 9=β. Now recall necessary identity from section 16.2.4. It expresses the notion that identity will remain the same across all worlds, and it is formulated in the following way:

we will call the formula ∀x∀y(x = y ⊃ □x = y) NI (Necessary Identity)

(p.351, section 16.2.4)

I may get this wrong, but I am guessing we are dealing with the following reasoning. 9=β is an identity that holds between 9 and β. If identity is necessary and it holds in some case, then it holds in all worlds. That means that it is necessary that the number of symphonies Beethoven wrote is 9. But that is not a logical necessity. Beethoven could have died before writing the ninth one. So this is the objection to quantified modal logic, namely, that it leads to claims of necessity that are really contingent.]

Let us now consider a standard objection to quantified modal logic. Beethoven wrote nine symphonies. Therefore 9 = β, where β is ‘the number of symphonies that Beethoven wrote’. Given NI, ∀x∀y(x = y ⊃ □x = y), it follows that □9 = β; that is, necessarily the number of Beethoven symphonies is nine – which is false, since Beethoven could have died immediately after writing the eighth.

(354)

16.4.2

[The Failure of the Negativity Constraint to Solve the Problem]

[The negativity constraint will not prevent this problem; for, even under the constraint, a = b ⊃ □(ℭa ⊃ a = b) is still valid, and so we must draw the same conclusion: “Since 9 = β, it still follows that □(ℭ9 ⊃ 9 = β), and so □ ℭ9 ⊃ □9 = β” (354).]

[Priest’s next point is that the Negativity Constraint will not solve this problem. Recall this tableau from section 16.3.4:

⊢_VK(NI) a = b ⊃ □(ℭa ⊃ a = b)

10.

11.

12.

13.

¬(a = b ⊃ □(ℭa ⊃ a = b)),0

↓

a = b,0

↓

¬□(ℭa ⊃ a = b),0

↓

ℭa,0

↓

ℭb,0

↓

◊¬(ℭa ⊃ a = b),0

↓

0r1

↓

¬(ℭa ⊃ a = b),1

↓

ℭa,1

↓

¬a = b,1

↓

0r1

↓

¬a = b,1

↓

a = b,1

1¬⊃

2NCR

3¬□

6◊r

8¬⊃

10◊r

2,7IRR

(13×12)

valid

(p.353, section 16.3.4, enumeration and step accounting are my own and are probably mistaken)

Here we are restricted by the negativity constraint, but nonetheless, on account of a = b ⊃ □(ℭa ⊃ a = b) being valid, we will need to draw the same conclusion: “Since 9 = β, it still follows that □(ℭ9 ⊃ 9 = β), and so □ ℭ9 ⊃ □9 = β” (354).]

It might be suggested that the failure of NI in necessary identity systems with the Negativity Constraint provides an answer to the problem, but it does not. As we saw in 16.3.4, even with the Negativity Constraint, a = b ⊃ □(ℭa ⊃ a = b). Since 9 = β, it still follows that □(ℭ9 ⊃ 9 = β), and so □ ℭ9 ⊃ □9 = β. But a Platonist about numbers ought to be able to hold that 9 is a necessary existent, without being driven into this absurd conclusion.

(354)

16.4.3

[The Problem as Resulting from the Implementation of a Noun Phrase Whose Denotation May Change Between Worlds]

[Priest diagnoses the problem in the following way: “What has gone wrong with the argument is, in fact, that the noun phrase β, ‘the number of symphonies written by Beethoven’ is a noun phrase that may change its denotation from world to world. In some worlds, Beethoven wrote eight symphonies, in some two, in some 147” (354).]

[(ditto)]

What has gone wrong with the argument is, in fact, that the noun phrase β, ‘the number of symphonies written by Beethoven’ is a noun phrase that may change its denotation from world to world. In some worlds, Beethoven wrote eight symphonies, in some two, in some 147.

(354)

16.4.4

[Rigid and Non-Rigid Designators]

[When we consider a constant as having world-invariant denotation (like β, ‘the number of symphonies written by Beethoven’, being understood as being 9 in all worlds), it is a rigid designator, and we write it under the form: v(c). However, constants that do vary with the world are called non-rigid designators (like β, ‘the number of symphonies written by Beethoven’, being understood as potentially taking a different value in different worlds, like 2, 9, or 147), and we write them accordingly under the form v_w(c). (“Compare predicates, where extensions may change from world to world, and we write v_w(P), not v(P))” (354-355).]

[As we saw in 16.4.3 above, we had a constant, β, with the sense, ‘the number of symphonies written by Beethoven’, but with a denotation that varies with the world in question, (being 9 in our world, but 2 in another, 147 in another, etc.) However, when we try to fix its designation for all worlds, we draw the false conclusion that it is necessary that nine be the number of symphonies Beethoven wrote. When we consider a constant as having world-invariant denotation (like β, ‘the number of symphonies written by Beethoven’, being understood as being 9 in all worlds), it is a rigid designator, and we write it under the form: v(c). However, constants that do vary with the world are called non-rigid designators (like β, ‘the number of symphonies written by Beethoven’, being understood as potentially taking a different value in different worlds, like 2, 9, or 147), and we write them accordingly under the form v_w(c). (“Compare predicates, where extensions may change from world to world, and we write v_w(P), not v(P))” (354-355).]

The constants we have been using so far all have a world-invariant denotation. (Thus, we write v(c), not v_w(c). Compare predicates, where | extensions may change from world to world, and we write v_w(P), not v(P).) Constants of this kind are called rigid designators. Constants like β are, by contrast, non-rigid designators. How do such constants behave logically?

(354-355)

16.4.5

[Rigid Designators as “Rigid Constants” and Non-Rigid Designators as “Descriptor Constants” or as “Descriptors”]

[We then “augment the language with a collection of new constants: α₀, α₁, α₂, . . . and call these descriptor constants, or just descriptors. I will use α, β, γ , . . . for arbitrary descriptors. I will call our old constants rigid constants. The terms of the language now comprise descriptors, rigid constants and variables” (355).]

[(ditto). (I am not certain yet, but the descriptor constants or descriptors seem to be non-rigid designators, since they seem to be distinguished from rigid constants.)]

Let us augment the language with a collection of new constants: α₀, α₁, α₂, . . . and call these descriptor constants, or just descriptors. I will use α, β, γ , . . . for arbitrary descriptors. I will call our old constants rigid constants. The terms of the language now comprise descriptors, rigid constants and variables.

(355)

16.4.6

[The Truth-Conditions]

[Priest next gives the truth-conditions for our semantics (see below).]

[Priest next gives the truth-conditions for our semantics. We first note that “In an interpretation, v assigns each descriptor a denotation, v_w(α), at each world w” Priest says that the semantics are the same except for how closed atomic sentences are evaluated. I am not certain what the semantics are the same as, but for now I will provide what we listed in section 15.3.2, and I will add the new condition mentioned now:

v_w(¬A) = 1 if v_w(A) = 0, and 0 otherwise.

v_w(A ∧ B) = 1 if v_w(A) = v_w (B) = 1, and 0 otherwise.

v_w(A ∨ B) = 1 if v_w(A) = 1 or v_w (B) = 1, and 0 otherwise.

[…]

v_w(◊A) = 1 if, for some w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

v_w(□A) = 1 if, for all w′ ∈ W such that wRw′, v_w′(A) = 1; and 0 otherwise.

(pp.21-22, sections 2.3.4, 2.3.5)

v_w(∃xA) = 1 iff for some d ∈ D_w, v_w(A_x(k_d)) = 1

v_w(∀xA) = 1 iff for all d ∈ D_w, v_w(A_x(k_d)) = 1

(p.331, section 15.3.2)

v_w(Pt₁ . . . t_n) = 1 iff ⟨v_w(t₁), . . . , v_w(t_n)⟩ ∈ v_w(p)

(355)

]

In an interpretation, v assigns each descriptor a denotation, v_w(α), at each world w. If we define v_w(a) to be v(a) for all rigid constants, a, we can write the truth conditions of closed atomic sentences uniformly as:

v_w(Pt₁ . . . t_n) = 1 iff ⟨v_w(t₁), . . . , v_w(t_n)⟩ ∈ v_w(p)

In all other ways, the semantics remain the same. In particular, the truth conditions of the quantifiers are still given in terms of the canonical constants, k_d, which are rigid.

(355)

16.4.7

[Tableau Rules]

[Priest then explains how we modify the tableau rules to accommodate descriptors: {1} “the IIR applies only if both terms are rigid constants”; {2} “the rules of universal and particular instantiation (and the NCR if it is present) apply only to rigid constants”; and {3} there is a new identity rule (see below).]

[The tableau rules will vary according to the system and constraints we are employing. But we note three important things about the rules, whatever they be. {1} “the IIR applies only if both terms are rigid constants”; {2} “the rules of universal and particular instantiation (and the NCR if it is present) apply only to rigid constants”; and {3} there is a new identity rule:

Constant-Descriptor Identity (CDI,D)

↓

c = α,i

(c is a constant new to the branch. This rule is applied to every descriptor, α, on the branch, and every i on the branch, for which there is not already a line of this form.)

(355, with names and additional text at the bottom made by me. The name is my own fabrication and probably needs correction.)

]

To obtain tableaux for the extended language, the identity rules (whatever they are) are extended to include all closed terms, descriptors or rigid constants, except that the IIR applies only if both terms are rigid constants. All of the other rules remain the same. In particular, the rules of universal and particular instantiation (and the NCR if it is present) apply only to rigid constants. There is, in addition, one further rule:

Constant-Descriptor Identity (CDI,D)

.

↓

c = α,i

(c is a constant new to the branch. This rule is applied to every descriptor, α, on the branch, and every i on the branch, for which there is not already a line of this form.)

(355, with names and additional text at the bottom made by me. The name is my own fabrication and probably needs correction.)

c is a constant new to the branch. The rule is applied to every descriptor, α, on the branch, and every i on the branch, for which there is not already a line of this form.³

(355)

3. The effect of applying the other rules to descriptors, where this is legitimate, is obtained by applying this rule. Thus, consider UI, for example. Given ∀xPx, i, we have a line of the form c = α, i, so we can infer Pc, i by UI, and Pα, i by SI.

(355)

16.4.8

[Example Tableau 1]

[Priest then gives an example tableau for a valid inference in CK(NI) with designators.]

[Priest will now give an example tableau in CK(NI). I am going to guess that the rules will be the following, but likely I have this wrong:

Double Negation

Development (¬¬D)

¬¬A,i

↓

A,i

Conjunction

Development (∧D)

A ∧ B,i

↓

A,i

B,i

Negated Conjunction

Development (¬∧D)

¬(A ∧ B),i

↓

¬A ∨ ¬B,i

Disjunction

Development (∨D)

A ∨ B,i

↙ ↘

A,i B,i

Negated Disjunction

Development (¬∨D)

¬(A ∨ B),i

↓

¬A,i

↓

¬B,i

Conditional

Development (⊃D)

A ⊃ B,i

↙ ↘

¬A,i B,i

Negated Conditional

Development (¬⊃D)

¬(A ⊃ B),i

↓

A,i

↓

¬B,i

Negated Necessity

Development (¬□D)

¬□A,i

↓

◊¬A,i

Negated Possibility

Development (¬◊D)

¬◊A,i

↓

□¬A,i

Relative Necessity

Development (□rD)

□A,i

irj

↓

A,j

(both □A,i and irj must occur somewhere on the same branch, but in any order or location)

Relative Possibility

Development (◊rD)

◊A,i

↓

irj

A,j

(j must be new: it cannot occur anywhere above on the branch)

(p.24, section 2.4.4)

Negated Existential

Development (¬∃D)

¬∃xA

↓

∀x¬A

Negated Universal

Development (¬∀D)

¬∀xA

↓

∃x¬A

(p.266, section 12.4.1)

Universal Instantiation

Development (UI,D)

∀xA,i

↓

A_x(a),i

where a is any rigid constant on the branch, (if there are not any, we select one at will).

Particular Instantiation

Development (PI,D)

∃xA,i

↓

A_x(c),i

where c is any rigid constant that does not occur so far on the branch.

(p.331, section 15.4.1)

Principle of Identity

Development (=D)

↓

a = a,i

(You can always add a line of the form a = a,i)

Substitutivity of Identicals (SI,D)

a = b,i

↓

A_x(a),i

↓

A_x(b),i

(where A is any atomic sentence distinct from a = b.)

(Note: the world index on every line is the same, so substitution is licensed only within a world.)

Identity Invariance Rule (IIR,D)

a = b,i

↓

a = b,j

(where j is any world parameter on the branch distinct from i, and both a and b are rigid constants)

(350, with names and additional text at the bottom made by me. See section 16.2.5.)

Constant-Descriptor Identity (CDI,D)

↓

c = α,i

(c is a constant new to the branch. This rule is applied to every descriptor, α, on the branch, and every i on the branch, for which there is not already a line of this form.)

(355, with names and additional text at the bottom made by me. The name is my own fabrication and probably needs correction.)

]

Here is a tableau to show that ∀x□Px ⊢ □Pα in CK(NI).

∀x□Px ⊢_CK(NI) □Pα

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

.

8.

.

9.

.

10.

.

∀x□Px,0

↓

¬□Pα,0

↓

◊¬Pα,0
↓
0r1

↓

¬Pα,1

↓

a = α,o

↓

b = α,1

↓

□Pb,0

↓

Pb,1

↓

Pα,1

×

P

.

P

.

2¬□

.

3◊r

.

3◊r

.

4,5CDI

.

4,5CDI

.

1,7UI

.

8,4□r

.

7,9SI

(10×5)

valid

(enumeration and step accounting are my own and are probably mistaken)

Lines six and seven apply the new rule, and the last line is obtained by SI from line seven.

(356)

16.4.9

[Example Tableau 2]

[Then Priest gives an example tableau for an invalid formula.]

[(ditto) (See section 16.4.8 above for the rules we will apply below.)]

Here is a tableau to show that ⊬ a = α ⊃ □a = α in the same system.

⊬_CK(NI) a = α ⊃ □a = α

1.

.

2.

.

3.

.

4.

.

5.

.

6.

.

7.

¬(a = α ⊃ □a = α),0

↓

a = α,0

↓

¬□a = α,0
↓
◊¬a = α,0

↓

0r1

↓

¬a = α,0

↓

b = α,1

P

.

1¬⊃

.

1¬⊃

.

3¬□

.

4◊r

.

4◊r

.

5,6CDI

(open)

invalid

(enumeration and step accounting are my own and are probably mistaken)

The last line is provided by the new rule, but its addition has no further consequences.

(356)

16.4.10

[Counter-Models]

[“We read off a counter-model from an open branch of a tableau as before. In addition, if there is a line of the form c=β,i on the tableau, we set v_{w_i} (β) to v(c). (Note that if we have lines of the form c₁=β,i and c₂=β,i, then we have a line of the form c₁=c₂,i, by SI, so v(c₁) = v(c₂).)” (356).]

[(ditto)]

We read off a counter-model from an open branch of a tableau as before. In addition, if there is a line of the form c=β,i on the tableau, we set v_{w_i} (β) to v(c). (Note that if we have lines of the form c₁=β,i and c₂=β,i, then we have a line of the form c₁=c₂,i, by SI, so v(c₁) = v(c₂).)

(356)

16.4.11

[Example Counter-Model 1]

[Priest then gives an example counter-model.]

[(ditto)]

Thus, in the counter-model given by the tableau of 16.4.9, W = {w₀,w₁}, D = {∂_a, ∂_b}, w₀R w₁, v(a) = ∂_a, v(b) = ∂_b, v_w₀ (α) = ∂_a, v_w₁(α) = ∂_b. | That is:

+--------+xxw0x→xw1xx+--------+
|xαxxxxxx|xxw0x→xw1xx|xxxxxxαx|
|x∂axx∂bx|xxw0x→xw1xx|x∂axx∂bx|
+--------+xxw0x→xw1xx+--------+

The descriptor is written above the object that it denotes at each world. v_w₀(a) = v(a) = ∂_a = v_w₀(α). Hence, a = α is true at w₀. But v_w₁(a) = v(a) = ∂_a = ∂_b = v(b) = v_w₁(α). Hence, a = α is false at w₁, so □a = α is false at w₀.

(356-357)

16.4.12

[Failing Quantifier Inferences Under Descriptors That Would Have Held for Rigid Constants. Example Counter-Model 2.]

[“Note that various quantifier inferences that hold for rigid constants may fail for descriptors. Thus, □Pα ⊬_CK ∃x□Px” (357).]

[(ditto)]

Note that various quantifier inferences that hold for rigid constants may fail for descriptors. Thus, □Pα ⊬_CK ∃x□Px. The tableau for this is infinite. Here is a finite counter-model:
+---------+xxw0x→xw1xx+---------+
|xxαxxxxxx|xx↷xx→xw1xx|xxxxxxαxx|
|xx∂axx∂bx|xxw0x→xw1xx|xx∂axx∂bx|
|Px√xxx×xx|xxw0x→xw1xx|Px×xxx√xx|
+---------+xxw0x→xw1xx+---------+

I leave it as an exercise to check that this works.

(357)

16.4.13

[The Soundness and Completeness of the Tableaux]

[These tableaux are sound and complete.]

[(ditto)]

All the tableau systems described in this chapter are sound and complete with respect to the appropriate semantics. This is proved in 16.6 and 16.7.

(357)