On a Broader Notion of Rigidity
Marián Zouhar2012
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According to S. Kripke, an expression is rigid provided it refers to the same object in all possible worlds in which the object exists. On the other hand, H. Putnam claims that an expression is rigid provided it refers to the same object in all possible worlds in which it refers to anything at all. The paper shows that the two notions of rigidity are not equivalent because (i) Putnam's rigidity is much broader than Kripke's; (ii) unlike Putnam's rigidity, Kripke's is interwoven with essentialism; and (iii) identity statements between rigid designators in Putnam's sense need not be necessarily true (if true at all).
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At least since Kripke (1980) it has been generally accepted that true identity statements involving proper names are necessarily true. This view is allegedly supported by our most ordinary, pretheoretic intuitions according to which ordinary proper names are rigid designators. This paper challenges the established status of this view. Section 1 develops the context of the debate by presenting the intuitions of rigidity and of contingency of identity found among competent speakers. Section 2 shows how the latter constitute a serious problem for the received view, one that cannot be easily ignored. Section 3 considers three available proposals intended to solve the problem and shows why they fail. Section 4 briefly describes a way to make compatible the intuitions of rigidity and of contingency, a consequence of which is the acceptance of the possibility of contingently true identity statements. Finally, section 5 considers some philosophical consequences of accepting such a view.
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1 shall raise here a number of difficulties with the very possibility of any philosophically interesting rigid designators. Rigid designation is a theory central to Saul Kripke's views developed in "Naming and Necessity"and "Identity and Necessity." Although rooted in the theory of reference, the notion of rigid designation governs his views on necessity and contingency as well as the a priori and a posteriori. Consequently, it has important metaphysical and epistemological ramifications. However, I shall argue that no referring expressions can satisfy the standard Kripke has set for the most important class of rigid designat ors.
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The notion of rigidity looms large in philosophy of language, but is beset by difficulties. This paper proposes a simple theory of rigidity, according to which an expression has a world-relative semantic property rigidly when it has that property at, or with respect to, all worlds. Just as names, and certain descriptions like 'The square root of 4', rigidly designate their referents, so too are necessary truths rigidly true, and so too does 'cat' rigidly have only animals in its extension. After spelling out the theory, I argue that it enables us to avoid the headaches that attend the misbegotten desire to have a simple rigid/non-rigid distinction that applies to expressions, giving us a simple solution to the problem of generalizing the notion of rigidity beyond singular terms.
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Although it is commonly denied that the semantic principles of rigid designation and direct reference have essentialist implications it is still common to assume that the principles imply the necessity of identity. Previous attempts to combine rigidity and contingent identity have, accordingly, involved several controversial semantic or metaphysical claims. In this article, however, it is argued that rigidity and the necessity of identity are logically independent principles. Both are needed to derive the data usually claimed to be evidence for the idea that names are rigid designators; thus, there may be little motivation for combining rigidity with contingent identity. The fact that rigidity can consistently be combined with contingent identity shows, importantly, that no non-trivial metaphysical implications can be derived from the semantic principles of rigidity and direct reference alone.
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Rigid designation' is Kripke's name for a concept that has been in the air at least since the development of quantified modal logics: (a token of) a designator is rigid if and only if it designates the same individual in every possible world in which the individual exists. Two seminal conclusions for which Kripke (, ) argues are that proper names are rigid designators, and that there are some deep semantic anities between proper names and various sorts of general terms. However, even though he does, at places, explicitly attribute rigidity to certain general terms, Kripke nowhere gives a definition of rigidity that applies to general terms. This presents a challenge: Precisely which general terms ought to be classified as rigid designators? More fundamentally: What should we take the criterion for rigidity to be for general terms? There is a considerable sub-literature, stretching back
1 Rigid Designation and Semantic Structure
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Forthcoming in " Philosophers' Imprint " Comments to: [email protected] 1. the issue 'Rigid designation' is Kripke's name for a concept that has been in the air at least since the development of quantified modal logics: (a token of 1) a designator is rigid if and only if it designates the same individual in every possible world in which the individual exists. Two seminal conclusions for which Kripke (1971, 1972) argues are that proper names are rigid designators, and that there are some deep semantic affinities between proper names and various sorts of general terms. However, even though he does, at places, explicitly attribute rigidity to certain general terms, 2 Kripke nowhere gives a definition of rigidity that applies to general terms. This presents a challenge: Precisely which general terms ought to be classified as rigid designators? More fundamentally: What should we take the criterion for rigidity to be, for general terms? There exists a considerable s...
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General Terms and Rigidity: Another Solution to the Trivialization Problem
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In this paper I am concerned with the problem of applying the notion of rigidity to general terms. In Naming and Necessity, Kripke has clearly suggested that we should include some general terms among the rigid ones, namely, those common nouns semantically correlated with natural substances, species and phenomena, in general, natural kinds - ‘water’, ‘tiger’, ‘heat’- and some adjectives - ‘red’, ‘hot’, ‘loud’. However, the notion of rigidity has been defined for singular terms; after all, the notion that Kripke has provided us with is the notion of a rigid designator. But general terms do not designate single individuals: rather, they apply to many of them. In sum, the original concept of rigidity cannot be straightforwardly applied to general terms: it has to be somehow redefined in order to make it cover them. As is known, two main positions have been put forward to accomplish that task: the identity of designation conception, according to which a rigid general term is one that designates the same property or kind in all possible worlds, and the essentialist conception, which conceives of a rigid general term as an essentialist one, namely, a term that expresses an essential property of an object. My purpose in the present paper is to defend a particular version of the identity of designation conception: on the proposed approach, a rigid general term will be one that expresses the same property in all possible worlds and names the property it expresses. In my opinion, the position can be established on the basis of an inference to the best explanation of our intuitive interpretation and evaluation, relative to counterfactual circumstances, of statements containing different kinds of general terms, which is strictly analogous to our intuitive interpretation and evaluation, relative to such circumstances, of statements containing different kinds of singular ones. I will argue that it is possible to offer a new solution to the trivialization problem that is thought to threaten all versions of the identity of designation conception of rigidity. Finally, I will also sketch a solution to the so-called ‘over-generalization and under- generalization problems’, both closely related to the above-mentioned one.
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In this article the following criticisms of the essentialist conception of general term rigidity presented in the previous papers are considered and responded: (i) the identity of designation conception of rigidity can provide us with a better alternative account for general term rigidity (Orlando), and (ii) the essentialist conception fails to meet the condition of extensional adequacy, both because it (allegedly) over-and undergeneralizes (Zerbudis). Against (i), it is claimed that the proposed definition of general term rigidity cannot feature in lost rigidity arguments against description theories because it is circular, and then fails to do the primary work that rigidity is supposed to do, namely, distinguishing terms that are covered by a description theory from those that are not. As regards (ii), after insisting that the essentialist view need not be commited to the condition of extensional adequacy, both charges of over-and undergeneralization are addressed: while the argumentation aimed at showing that some examples (such as 'paperweight') are cases of overgeneralization is rejected, the cases of undergeneralization (of the likes of 'frog') are admitted to be still in need of a better explanation than the one given in Devitt (2005).
General Terms as Rigid Designators
Philosophical Studies, 2006
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❑r✐♣❦❡✬s ✐s ✐♥t❡r✇♦✈❡♥ ✇✐t❤ ❡ss❡♥t✐❛❧✐s♠❀ ❛♥❞ ✭✐✐✐✮ ✐❞❡♥t✐t② st❛t❡✲
♠❡♥ts ❜❡t✇❡❡♥ r✐❣✐❞ ❞❡s✐❣♥❛t♦rs ✐♥ P✉t♥❛♠✬s s❡♥s❡ ♥❡❡❞ ♥♦t ❜❡
♥❡❝❡ss❛r✐❧② tr✉❡ ✭✐❢ tr✉❡ ❛t ❛❧❧✮✳
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❜② ❍✐❧❛r② P✉t♥❛♠ ❞✐✛❡rs ❝♦♥s✐❞❡r❛❜❧② ❢r♦♠ t❤❡ ♦♥❡ ✐♥tr♦❞✉❝❡❞ ❜② ❙❛✉❧
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❜❡ s❤♦✇♥ t❤❛t ✐t ✐s ♠✉❝❤ ❜r♦❛❞❡r t❤❛♥ ❑r✐♣❦❡✬s✳
❇❡✐♥❣ ❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r ✐s ❛ ♠♦❞❛❧ ♣r♦♣❡rt② ♦❢ ❝❡rt❛✐♥ ❦✐♥❞s ♦❢ ❡①✲
♣r❡ss✐♦♥ ❛♥❞ ✐t ✐s ✉s✉❛❧❧② ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ■t s❡❡♠s
t❤❛t ❛❧❧ ❞❡✜♥✐t✐♦♥s ♦❢ r✐❣✐❞✐t② ❞♦ ❥✉st✐❝❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥✿
❋♦r ❛♥ ❡①♣r❡ss✐♦♥✱ e✱ t♦ ❜❡ r✐❣✐❞ ✐t ❤❛s t♦ ❤♦❧❞ t❤❛t ✭✐✮ t❤❡r❡
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e ❞❡s✐❣♥❛t❡s✱ ♦r r❡❢❡rs t♦✱ o ✐♥ w❀ ❛♥❞ ✭✐✐✮ t❤❡r❡ ❛r❡ ♥♦ t✇♦
❞✐✛❡r❡♥t ♣♦ss✐❜❧❡ ✇♦r❧❞s✱ w1 ❛♥❞ w2 ✱ ❛♥❞ t✇♦ ❞✐✛❡r❡♥t ♦❜✲
❥❡❝ts✱ o1 ❛♥❞ o2 ✱ s✉❝❤ t❤❛t e ❞❡s✐❣♥❛t❡s✱ ♦r r❡❢❡rs t♦✱ o1 ✐♥ w1
❛♥❞ o2 ✐♥ w2 ✳✶
❚❤❡ r❡q✉✐r❡♠❡♥t ✭✐✮ ❣✉❛r❛♥t❡❡s t❤❛t e ✐s ♥♦t ❛ ♥❡❝❡ss❛r✐❧② ❡♠♣t② t❡r♠
✭♦t❤❡r✇✐s❡ ✐t ✇♦✉❧❞ ♥♦t ❜❡ ❛ ❞❡s✐❣♥❛t♦r✮✳ ❚❤❡ r❡q✉✐r❡♠❡♥t ✭✐✐✮ ❡①❝❧✉❞❡s
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✐t ✇♦✉❧❞ ❜❡ ❛ ♥♦♥✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r✮✳
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶✳
❤tt♣✿✴✴✇✇✇✳❦r✐t❡r✐♦♥✳❛t
❝ ✷✵✶✷ ❚❤❡ ❛✉t❤♦r
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶
✶✷
■♥ ✇❤❛t ❢♦❧❧♦✇s✱ ■ ❛♠ ✐♥t❡r❡st❡❞ s♦❧❡❧② ✐♥ ♣r♦♣❡r ♥❛♠❡s ❛♥❞ ❞❡✜♥✐t❡
❞❡s❝r✐♣t✐♦♥s✳
■ st❛rt ✇✐t❤ ❛ s❤♦rt r❡✢❡❝t✐♦♥ ❛❜♦✉t ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥
t❤❡ ✇❛②s t❤❡ t✇♦ ❦✐♥❞s ♦❢ ❡①♣r❡ss✐♦♥ r❡❢❡r ✭❙❡❝t✐♦♥ ✶✮✳ ❚❤❡♥ ■ ❞✐s❝✉ss
❑r✐♣❦❡✬s ❛♥❞ P✉t♥❛♠✬s ❞❡✜♥✐t✐♦♥s ♦❢ r✐❣✐❞✐t② ❛♥❞ ❛♣♣❧② t❤❡♠ t♦ ❞❡✜✲
♥✐t❡ ❞❡s❝r✐♣t✐♦♥s ❛♥❞ ♣r♦♣❡r ♥❛♠❡s ✭❙❡❝t✐♦♥ ✷✮✳
❋✐♥❛❧❧②✱ ■ ❞r❛✇ s♦♠❡
✐♥t❡r❡st✐♥❣ ♣❤✐❧♦s♦♣❤✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ t✇♦ ♥♦t✐♦♥s ✭❙❡❝t✐♦♥ ✸✮✳
✶
❙❛t✐s❢❛❝t✐♦♥❛❧ ❘❡❢❡r❡♥❝❡ ✈s✳ ❈♦♥✈❡♥t✐♦♥❛❧ ❘❡❢❡r❡♥❝❡
❚♦ ❜❡❣✐♥ ✇✐t❤✱ ❧❡t ✉s st❛rt ✇✐t❤ ❝❡rt❛✐♥ ✈✐t❛❧ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ♣r♦♣❡r
♥❛♠❡s ❛♥❞ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s✳ ❙✐♥❝❡ t❤❡ t✇♦ ❦✐♥❞s ♦❢ ❡①♣r❡ss✐♦♥ r❡❢❡r
t♦ s♦♠❡t❤✐♥❣ ✐♥ ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t ✇❛②s✱ t❤❡② ❛r❡ r✐❣✐❞ ✭✐❢ ❛t ❛❧❧✮ ✐♥
❞✐✛❡r❡♥t s❡♥s❡s✳
❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ❵(ιx)(F x)✬ ✭✐✳❡✳✱ ❵t❤❡ F ✬✮ r❡❢❡rs t♦ ❛♥ ♦❜❥❡❝t✱
o ✐s ❛ ✉♥✐q✉❡ ✐♥st❛♥❝❡ ♦❢ t❤❡ ♣r♦♣❡rt② (λx)(F x) ✭✐✳❡✳✱ t❤❡
♣r♦♣❡rt② ❜❡✐♥❣ ❋ ✮ t❤❛t ✐s ❡①♣r❡ss❡❞ ❜② t❤❡ ♣r❡❞✐❝❛t❡ ♣❛rt ♦❢ t❤❡ ❞❡s❝r✐♣✲
✷
t✐♦♥❀ ✐❢ o ✐s ❛ ✉♥✐q✉❡ ✐♥st❛♥❝❡ ♦❢ (λx)(F x)✱ ✇❡ ♠❛② s❛② t❤❛t ✐t s❛t✐s✜❡s
t❤❡ ❞❡s❝r✐♣t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❵(ιx)(F x)✬✳ ❚♦ ❤❛✈❡ ❛ ❤❛♥❞②
❧❛❜❡❧ ✇❡ ♠✐❣❤t s❛② t❤❛t ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s r❡❢❡r s❛t✐s❢❛❝t✐♦♥❛❧❧②✳ ◆♦✇
❚❤❡
o✱
♣r♦✈✐❞❡❞
✐t ✐s ❛ tr✉✐s♠ t❤❛t ♦❜❥❡❝ts ✐♥st❛♥t✐❛t❡ t❤❡✐r ♣r♦♣❡rt✐❡s ♦♥❧② ✐♥✱ ♦r ✇✐t❤
o1 ✱ t♦ ❜❡ t❤❡ ✉♥✐q✉❡ ✐♥st❛♥❝❡
(λx)(F x) ✐♥ ❛ ❣✐✈❡♥ ♣♦ss✐❜❧❡ ✇♦r❧❞✱ w✱ ✐t ❤❛s t♦ ❤♦❧❞ t❤❛t ✐❢ ❛♥② ♦❜❥❡❝t✱
o2 ✱ ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ (λx)(F x) ✐♥ w✱ t❤❡♥ o1 = o2 ✳ ❚❤✉s✱ t❤❡ r❡❢❡r❡♥❝❡
r❡s♣❡❝t t♦✱ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❋♦r ❛♥ ♦❜❥❡❝t✱
♦❢
r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ❛♥❞ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❤❛s t♦ ✐♥✈♦❧✈❡
❛♥♦t❤❡r ❛r❣✉♠❡♥t ♣❧❛❝❡ r❡s❡r✈❡❞ ❢♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞✳ ❚❤❡ r❡❢❡r❡♥❝❡ r❡✲
❧❛t✐♦♥ ❢♦r ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s ✐s ❛
t❤r❡❡✲♣❧❛❝❡
❛♥ ✐♥❞✐✈✐❞✉❛❧ ❛♥❞ ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞✳
◆❡①t✱ t✉r♥ t♦
♣r♦♣❡r ♥❛♠❡s✳
✸
♦♥❡ ❜❡t✇❡❡♥ ❛♥ ❡①♣r❡ss✐♦♥✱
❚❤❡r❡ ❛r❡ ❣♦♦❞ r❡❛s♦♥s t♦ s✉♣♣♦s❡ t❤❛t
❛ ♣r♦♣❡r ♥❛♠❡ r❡❢❡rs t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥
❛❧❧
♣♦ss✐❜❧❡ ✇♦r❧❞s ✇✐t❤♦✉t ❡①✲
❝❡♣t✐♦♥❀ t❤✐s s❤♦✉❧❞ ❤♦❧❞ ✐rr❡s♣❡❝t✐✈❡ ♦❢ t❤❡ ♦❜❥❡❝t✬s ❡①✐st❡♥❝❡ ✐♥ ❛ ❣✐✈❡♥
✇♦r❧❞✳ ❲❤②❄ ❆ ♣r♦♣❡r ♥❛♠❡✱
o
n❀✹
❛ss♦❝✐❛t✐♥❣ n
✇❛s ✏❜❛♣t✐③❡❞✑ ❜②
❝♦♥✈❡♥t✐♦♥
n✱
r❡❢❡rs t♦ ❛♥ ♦❜❥❡❝t✱
o✱
♦♥❧② ♣r♦✈✐❞❡❞
✐t ♠❡❛♥s t❤❛t t❤❡r❡ ✐s ❛ ❧✐♥❣✉✐st✐❝ ✭s❡♠❛♥t✐❝✮
✇✐t❤
o✳
◆♦ ♦t❤❡r ❝♦♥❞✐t✐♦♥ ✐s t♦ ❜❡ ♠❡t ❡✐t❤❡r
❜② t❤❡ ♥❛♠❡ ♦r ❜② t❤❡ ♦❜❥❡❝t✳ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ♠✐❣❤t s❛② t❤❛t ♣r♦♣❡r
♥❛♠❡s
r❡❢❡r ❝♦♥✈❡♥t✐♦♥❛❧❧②✳
❲❤❛t ✐s ✐♠♣♦rt❛♥t ✐s t❤❛t ❧✐♥❣✉✐st✐❝ ❝♦♥✈❡♥✲
t✐♦♥s ❛r❡ ✉tt❡r❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ P♦ss✐❜❧❡ ✇♦r❧❞s ❛r❡ ❜✉t
❛rt✐✜❝✐❛❧ t♦♦❧s ✉s❡❞ t♦ ❡①♣❧✐❝❛t❡ ❝❡rt❛✐♥ ♠♦❞❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❧❛♥❣✉❛❣❡✳
■t ❤❛s t♦ ❜❡ s✉♣♣♦s❡❞ t❤❛t t❤❡ ❧❛♥❣✉❛❣❡ ✐♥ q✉❡st✐♦♥ ✐s ❛❧r❡❛❞② ❛t ♦✉r ❞✐s✲
♣♦s❛❧ ❛♥❞ ✇❡ ❛r❡ ✐♥✈✐t❡❞ t♦ ❞❡s❝r✐❜❡ s♦♠❡ ♦❢ ✐ts ♣r♦♣❡rt✐❡s ✐♥ t❡r♠s ♦❢
♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡r❡ ✐s ❛ ❧❛♥❣✉❛❣❡ r❡❛❞② ❢♦r ♦✉r
▼❛r✐á♥ ❩♦✉❤❛r✿
❖♥ ❛ ❇r♦❛❞❡r ◆♦t✐♦♥ ♦❢ ❘✐❣✐❞✐t②
✶✸
st✉❞②✐♥❣ ✐♠♣❧✐❡s ✭❛♠♦♥❣ ♦t❤❡r t❤✐♥❣s✮ t❤❛t ❛❧❧ ✐ts ✭s✐♠♣❧❡✮ ❡①♣r❡ss✐♦♥s
❤❛✈❡ t♦ ❤❛✈❡ ♠❡❛♥✐♥❣s ✜①❡❞ ❜② s✉✐t❛❜❧❡ ❧✐♥❣✉✐st✐❝ ❝♦♥✈❡♥t✐♦♥s ♣r✐♦r t♦
❡①❛♠✐♥✐♥❣ t❤❡✐r ♠♦❞❛❧ ♣r♦♣❡rt✐❡s✳ ❈♦♥❝❡r♥✐♥❣ ♣r♦♣❡r ♥❛♠❡s✱ ✇❡ ❤❛✈❡
t♦ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❛r❡ ✜①❡❞ ❧✐♥❣✉✐st✐❝ ❝♦♥✈❡♥t✐♦♥s ✐♥ t❤❡ s❡♥s❡ t❤❛t ❢♦r
❡❛❝❤ ♣r♦♣❡r ♥❛♠❡ t❤❡r❡ ✐s s♦♠❡ ♦❜❥❡❝t ♦r ♦t❤❡r t❤❛t ✐s ❝♦♥✈❡♥t✐♦♥❛❧❧②
❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♥❛♠❡ ✐♥ q✉❡st✐♦♥✳ ❆s ❛ r❡s✉❧t✱ t❤✐s ❛ss♦❝✐❛t✐♦♥ ✐s
❡st❛❜❧✐s❤❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ♣♦ss✐❜❧❡ ✇♦r❧❞s ❛♥❞ ✇❡ ♥❡❡❞ ♥♦t ✐♥✈❡st✐✲
❣❛t❡ t❤❡♠ t♦ ✜♥❞ ♦✉t ✇❤❛t t❤❡ ♣r♦♣❡r ♥❛♠❡ ✐♥ q✉❡st✐♦♥ r❡❢❡rs t♦✳ ◆♦✇
❣✐✈❡♥ t❤✐s ❢❛❝t✱ ❛ ♥❛♠❡ ❤❛s t♦ r❡❢❡r t♦ t❤❡ s❛♠❡ ✐♥❞✐✈✐❞✉❛❧ ✐♥ ❛❧❧ ♣♦ss✐✲
❜❧❡ ✇♦r❧❞s ✇✐t❤♦✉t ❡①❝❡♣t✐♦♥ ❛♥❞ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧✬s ❡①✐st❡♥❝❡
✐♥ t❤❡ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐♥ q✉❡st✐♦♥✳ ■t ♠❛② ❤❛♣♣❡♥ t❤❛t t❤❡ ♥❛♠❡ r❡❢❡rs
t♦ ❛♥ ♦❜❥❡❝t ❡✈❡♥ ✐♥ t❤♦s❡ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ❢❛✐❧s
t♦ ❡①✐st✳✺ ▲✐♥❣✉✐st✐❝ ❝♦♥✈❡♥t✐♦♥s ❤❛✈❡ t♦ ❜❡ ✉tt❡r❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡
st❛t❡ ♦❢ ❛✛❛✐rs ♦❜t❛✐♥✐♥❣ ✐♥ ♣❛rt✐❝✉❧❛r ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❙✐♥❝❡ ❧✐♥❣✉✐st✐❝
❝♦♥✈❡♥t✐♦♥s ❛r❡ ♥♦t s✉❜❥❡❝t❡❞ t♦ ❛♥② ❞❡✈❡❧♦♣♠❡♥ts ✉♥❞❡rt❛❦❡♥ ✐♥✱ ♦r
✇✐t❤ r❡s♣❡❝t t♦✱ ♣♦ss✐❜❧❡ ✇♦r❧❞s✱ t❤❡ r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛ ♥❛♠❡
❛♥❞ ❛♥ ♦❜❥❡❝t ♥❡❡❞ ♥♦t ❜❡ s✉♣♣❧❡♠❡♥t❡❞ ✇✐t❤ ❛♥♦t❤❡r ❛r❣✉♠❡♥t ♣❧❛❝❡
r❡s❡r✈❡❞ ❢♦r ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❚❤❡ r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ ❢♦r ♣r♦♣❡r ♥❛♠❡s ✐s
♠❡r❡❧② ❛ t✇♦✲♣❧❛❝❡ ♦♥❡ ❜❡t✇❡❡♥ ❛♥ ❡①♣r❡ss✐♦♥ ❛♥❞ ❛♥ ♦❜❥❡❝t✳
✷
❑r✐♣❦❡❛♥ ❘✐❣✐❞✐t② ✈s✳ P✉t♥❛♠✐❛♥ ❘✐❣✐❞✐t②
❙❛✉❧ ❑r✐♣❦❡ ✐♥tr♦❞✉❝❡❞ ❤✐s ♥♦t✐♦♥ ♦❢ r✐❣✐❞✐t② ✐♥ t❤✐s ♠❛♥♥❡r✿
▲❡t✬s ❝❛❧❧ s♦♠❡t❤✐♥❣ ❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r ✐❢ ✐♥ ❡✈❡r② ♣♦ss✐❜❧❡
✇♦r❧❞ ✐t ❞❡s✐❣♥❛t❡s t❤❡ s❛♠❡ ♦❜❥❡❝t ❬✳ ✳ ✳ ❪ ❖❢ ❝♦✉rs❡✱ ✇❡ ❞♦♥✬t
r❡q✉✐r❡ t❤❛t t❤❡ ♦❜❥❡❝ts ❡①✐st ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❈❡rt❛✐♥❧②
◆✐①♦♥ ♠✐❣❤t ♥♦t ❤❛✈❡ ❡①✐st❡❞ ✐❢ ❤✐s ♣❛r❡♥ts ❤❛❞ ♥♦t ❣♦tt❡♥
♠❛rr✐❡❞✱ ✐♥ t❤❡ ♥♦r♠❛❧ ❝♦✉rs❡ ♦❢ t❤✐♥❣s ❬✳ ✳ ✳ ❪✳ ■♥ t❤❡ s❛♠❡
✇❛②✱ ❛ ❞❡s✐❣♥❛t♦r r✐❣✐❞❧② ❞❡s✐❣♥❛t❡s ❛ ❝❡rt❛✐♥ ♦❜❥❡❝t ✐❢ ✐t ❞❡s✲
✐❣♥❛t❡s t❤❛t ♦❜❥❡❝t ✇❤❡r❡✈❡r t❤❡ ♦❜❥❡❝t ❡①✐sts ❬✳ ✳ ✳ ❪✳ ✭❝❢✳ ❬✹✱
♣✳ ✹✽❪✮
❙♦✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❑r✐♣❦❡❛♥ r✐❣✐❞✐t② ✭♦r ❑✲r✐❣✐❞✐t②✱ ❢♦r s❤♦rt✮ ❝❛♥ ❜❡ ❞❡✜♥❡
✐♥ t❤✐s ✇❛②✿
❆♥ ❡①♣r❡ss✐♦♥✱ e✱ ✐s ❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r ✐✛ t❤❡r❡ ✐s ❛♥ ♦❜❥❡❝t✱
o✱ s✉❝❤ t❤❛t e ❞❡s✐❣♥❛t❡s o ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ o
❡①✐sts ✭❛♥❞ ❝❛♥♥♦t ❞❡s✐❣♥❛t❡ ❛♥②t❤✐♥❣ ❡❧s❡✮✳
❚❤✐s ❞❡✜♥✐t✐♦♥ ✐s s✐❧❡♥t ❛❜♦✉t ✇❤❛t ❤❛♣♣❡♥s ✐♥ t❤♦s❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ o
❢❛✐❧s t♦ ❡①✐st✳ ❆♥②✇❛②✱ ❢♦r t❤❡ r❡❛s♦♥s ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✇❡
✶✹
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶
♠✐❣❤t ❛❞❞ t❤❛t ✭✐✮ ✐❢ e ✐s ❛ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥✱ ✐t r❡❢❡rs t♦ ♥♦t❤✐♥❣ ✐♥
t❤♦s❡ ✇♦r❧❞s ❛♥❞ ✭✐✐✮ ✐❢ e ✐s ❛ ♣r♦♣❡r ♥❛♠❡✱ ✐t r❡❢❡rs t♦ s♦♠❡t❤✐♥❣ t❤❛t
❞♦❡s ♥♦t ❡①✐st ✐♥ t❤♦s❡ ✇♦r❧❞s✳ ❆❢t❡r ❛❧❧✱ t❤✐s s❡❡♠s t♦ ❜❡ s✉❣❣❡st❡❞ ❜②
❑r✐♣❦❡✬s ❞✐s❝✉ss✐♦♥ ❛❜♦✉t t❤❡ ♥❛♠❡ ❵❍✐t❧❡r✬❀ ✭❝❢✳ ❬✹✱ ♣♣✳ ✼✼❢❪✮
❍✐❧❛r② P✉t♥❛♠ s❡❡♠s t♦ ❛❞♦♣t ❑r✐♣❦❡✬s ♥♦t✐♦♥ ♦❢ r✐❣✐❞✐t②✿
❑r✐♣❦❡ ❝❛❧❧s ❛ ❞❡s✐❣♥❛t♦r ❵r✐❣✐❞✬ ✭✐♥ ❛ ❣✐✈❡♥ s❡♥t❡♥❝❡✮ ✐❢ ✭✐♥
t❤❛t s❡♥t❡♥❝❡✮ ✐t r❡❢❡rs t♦ t❤❡ s❛♠❡ ✐♥❞✐✈✐❞✉❛❧ ✐♥ ❡✈❡r② ♣♦s✲
s✐❜❧❡ ✇♦r❧❞ ✐♥ ✇❤✐❝❤ t❤❛t ❞❡s✐❣♥❛t♦r ❞❡s✐❣♥❛t❡s✳ ✭❬✼✱ ♣✳ ✷✸✶❪✮
❚❤❡r❡ ✐s ❛ r❡♠❛r❦❛❜❧❡ s❤✐❢t ✐♥ t❤✐s q✉♦t❛t✐♦♥ ❜❡❝❛✉s❡ ✐t ❞♦❡s ♥♦t ♠❡♥t✐♦♥
t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ♦❜❥❡❝t ❞❡s✐❣♥❛t❡❞✳ ❙♦✱ ✇❡ ♠✐❣❤t ♣✉t ❢♦r✇❛r❞ t❤❡
❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ P✉t♥❛♠✐❛♥ r✐❣✐❞✐t② ✭♦r P✲r✐❣✐❞✐t②✱ ❢♦r s❤♦rt✮✿
❆♥ ❡①♣r❡ss✐♦♥✱ e✱ ✐s ❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r ✐✛ t❤❡r❡ ✐s ❛♥ ♦❜❥❡❝t✱
o✱ s✉❝❤ t❤❛t e ❞❡s✐❣♥❛t❡s o ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ e
❞❡s✐❣♥❛t❡s s♦♠❡t❤✐♥❣✳
◆♦✇ t❤❡ q✉❡st✐♦♥ ✐s ✇❤❡t❤❡r t❤❡ ♥♦t✐♦♥ ♦❢ ❑✲r✐❣✐❞✐t② ✐s t❤❡ s❛♠❡ ❛s
t❤❛t ♦❢ P✲r✐❣✐❞✐t②✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ s❤♦✉❧❞ ✐♥q✉✐r❡ ✇❤❡t❤❡r t♦ s❛② t❤❛t
❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r r❡❢❡rs t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤
t❤❛t ♦❜❥❡❝t ❡①✐sts ✐s t❤❡ s❛♠❡ ❛s s❛②✐♥❣ t❤❛t ❛ r✐❣✐❞ ❞❡s✐❣♥❛t♦r r❡❢❡rs t♦
t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦ ❛♥②t❤✐♥❣ ❛t
❛❧❧✳
❋✐rst ♦❢ ❛❧❧✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣r♦♣❡r ♥❛♠❡s t❤❡r❡ s❡❡♠s t♦ ❜❡ ♥♦ ❞✐✛❡r✲
❡♥❝❡✳ ❋♦r t❤❡ r❡❛s♦♥s ❣✐✈❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥✱ ❡✈❡r② ♣r♦♣❡r ♥❛♠❡ ✐s
P✲r✐❣✐❞ ❜❡❝❛✉s❡ ✐t s✐♠♣❧② r❡❢❡rs t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳
■t ❤❛s t♦ ❜❡ ❛❧s♦ ❑✲r✐❣✐❞ ❜❡❝❛✉s❡ ✐❢ ✐t r❡❢❡rs t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧
♣♦ss✐❜❧❡ ✇♦r❧❞s✱ ✐t ❤❛s t♦ r❡❢❡r t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ❛❧s♦ ✐♥ ❛❧❧ t❤♦s❡ ✇♦r❧❞s
✐♥ ✇❤✐❝❤ t❤❛t ♦❜❥❡❝t ❡①✐sts✳
❈♦♥❝❡r♥✐♥❣ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s ✐t s❡❡♠s t♦ ❤♦❧❞ t❤❛t ✐❢ ❛ ❞❡s❝r✐♣t✐♦♥
r❡❢❡rs t♦ s♦♠❡t❤✐♥❣ ✐♥ ❛ ❣✐✈❡♥ ♣♦ss✐❜❧❡ ✇♦r❧❞✱ t❤❡ ♦❜❥❡❝t r❡❢❡rr❡❞ t♦ ❤❛s
t♦ ❡①✐st ✐♥ t❤❛t ✇♦r❧❞✳ ❋♦r ❡①❛♠♣❧❡✱ ❧❡t ✉s ❛ss✉♠❡ ✭❛s ❑r✐♣❦❡ ❤✐♠s❡❧❢
❞♦❡s✮ t❤❛t t❤❡ ♦r✐❣✐♥ ♦❢ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ✐s ❡ss❡♥t✐❛❧ ❢♦r ✐t✳ ■❢ ❇❛r❛❝❦ ❖❜❛♠❛
✐s t❤❡ ♦✛s♣r✐♥❣ ♦❢ t❤❡ ❣❛♠❡t❡s ❳ ❛♥❞ ❨ ✭✐♥ ♦✉r ✇♦r❧❞✮✱ ❵t❤❡ ♦✛s♣r✐♥❣ ♦❢
❣❛♠❡t❡s ❳ ❛♥❞ ❨✬ ❤❛s t♦ ❜❡ ❛ ❑✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r ✕ ✐t r❡❢❡rs t♦ ❖❜❛♠❛ ✐♥
❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ❤❡ ❡①✐sts ❜❡❝❛✉s❡ ✐♥ ❛❧❧ t❤♦s❡ ✇♦r❧❞s ❤❡ ❝❛♥♥♦t
❜✉t ❜❡ t❤❡ ♦✛s♣r✐♥❣ ♦❢ t❤❡ t✇♦ ❣❛♠❡t❡s✳ ❆♥❛❧♦❣♦✉s❧②✱ t❤❡ ❞❡s❝r✐♣t✐♦♥
✐s P✲r✐❣✐❞ ❜❡❝❛✉s❡ ✐t r❡❢❡rs t♦ ❖❜❛♠❛ ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐t
r❡❢❡rs t♦ ❛♥②t❤✐♥❣ ❛t ❛❧❧❀ ❢♦r ✐t ✐s ♦♥❧② ❖❜❛♠❛ ❤✐♠s❡❧❢ ✇❤♦ ✐s ❝❛♣❛❜❧❡ t♦
s❛t✐s❢② t❤❡ ❞❡s❝r✐♣t✐✈❡ ❝♦♥❞✐t✐♦♥✳ ❚❤✐s ❤♦❧❞s q✉✐t❡ ❣❡♥❡r❛❧❧②✿ ❛❧❧ ❑✲r✐❣✐❞
❞❡s❝r✐♣t✐♦♥s ❛r❡ P✲r✐❣✐❞ ❛s ✇❡❧❧✳
▼❛r✐á♥ ❩♦✉❤❛r✿
❖♥ ❛ ❇r♦❛❞❡r ◆♦t✐♦♥ ♦❢ ❘✐❣✐❞✐t②
✶✺
◆♦✇✱ ❛r❡ t❤❡r❡ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s t❤❛t ❛r❡ P✲r✐❣✐❞ ✇✐t❤♦✉t ❜❡✐♥❣
❑✲r✐❣✐❞❄ ❚❤❡ ❛♥s✇❡r ✐s ♣♦s✐t✐✈❡✱ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡q✉✐r❡♠❡♥t ❝❛♥ ❜❡ ♠❡t✿
●✐✈❡♥ t❤❛t d ✐s ❛ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥ ❛♥❞ o ✐s ❛♥ ♦❜❥❡❝t✱ ✭✐✮
❢♦r ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s✱ w✱ ✐♥ ✇❤✐❝❤ d r❡❢❡rs t♦ s♦♠❡t❤✐♥❣✱ ✐t
r❡❢❡rs t♦ o ✐♥ w❀ ❛♥❞ ✭✐✐✮ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ♣♦ss✐❜❧❡ ✇♦r❧❞✱
w′ ✱ s✉❝❤ t❤❛t o ❡①✐sts ✐♥ w′ ❜✉t d ❞♦❡s ♥♦t r❡❢❡r t♦ o ✐♥ w′ ✳
❆❝❝♦r❞✐♥❣ t♦ ✭✐✐✮✱ d ❢❛✐❧s t♦ r❡❢❡r t♦ o ✐♥ w′ ✐♥ s♣✐t❡ ♦❢ t❤❡ ❢❛❝t t❤❛t o ❡①✐sts
✐♥ w′ ✳ ❚❤✉s✱ d ❝❛♥♥♦t ❜❡ ❑✲r✐❣✐❞✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✭✐✮ ❣✉❛r❛♥t❡❡s t❤❛t
d ❜❡ P✲r✐❣✐❞✳ ❙✐♥❝❡ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s r❡❢❡r s❛t✐s❢❛❝t✐♦♥❛❧❧②✱ t❤❡ ♦❜❥❡❝ts
r❡❢❡rr❡❞ t♦ ❜② t❤❡♠ ❤❛✈❡ t♦ ❡①❡♠♣❧✐❢② ❝❡rt❛✐♥ ♣r♦♣❡rt✐❡s✳ ■❢ ❛ ❞❡s❝r✐♣t✐♦♥
✐s ❑✲r✐❣✐❞✱ ✐t ❤❛s t♦ ❡①♣r❡ss ❛ ♣r♦♣❡rt② t❤❛t ✐s ❡ss❡♥t✐❛❧ ❢♦r t❤❡ ♦❜❥❡❝t
r❡❢❡rr❡❞ t♦ ❜② ✐t ✕ t❤❡ ♦❜❥❡❝t ❤❛s t♦ ❜❡ t❤❡ ✉♥✐q✉❡ ✐♥st❛♥❝❡ ♦❢ s✉❝❤ ❛♥
❡ss❡♥t✐❛❧ ♣r♦♣❡rt②✳✻ ◆♦✇ ✐t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐❢ ❛ ❞❡s❝r✐♣t✐♦♥ ✐s r❡q✉✐r❡❞
t♦ ❜❡ P✲r✐❣✐❞ ✇✐t❤♦✉t ❜❡✐♥❣ ❑✲r✐❣✐❞✱ ✐t r❡❢❡rs t♦ ❛♥ ♦❜❥❡❝t t❤❛t ❡①❡♠♣❧✐✜❡s
❛ ♣r♦♣❡rt② t❤❛t ♥❡❡❞ ♥♦t ❜❡ ❡ss❡♥t✐❛❧ ❢♦r t❤❡ ♦❜❥❡❝t✱ ❜❡❝❛✉s❡ ✐t ♠✐❣❤t
❡①✐st ✇✐t❤♦✉t ❜❡✐♥❣ ❛♥ ✐♥st❛♥❝❡ ♦❢ t❤❡ ♣r♦♣❡rt② ✐♥ q✉❡st✐♦♥❀ ❛t t❤❡ s❛♠❡
t✐♠❡✱ t❤❡ ♣r♦♣❡rt② ❤❛s t♦ ❜❡ s✉❝❤ t❤❛t ✐t ❝❛♥♥♦t ❜❡ ❡①❡♠♣❧✐✜❡❞ ❜② ❛♥②
♦t❤❡r ♦❜❥❡❝t✳
❈♦♥s✐❞❡r ❛♥ ❡①❛♠♣❧❡✳ ■t ♥❡❝❡ss❛r✐❧② ❤♦❧❞s t❤❛t ❡✈❡r② ♦❜❥❡❝t ✐s s❡❧❢✲
✐❞❡♥t✐❝❛❧ ❛♥❞ ❝❛♥♥♦t ❜❡ ✐❞❡♥t✐❝❛❧ ✇✐t❤ ❛♥② ♦t❤❡r ♦❜❥❡❝t✳ ●✐✈❡♥ t❤❛t ✇❡
❤❛✈❡ ❛♥ ♦❜❥❡❝t✱ o✱ t❤❡r❡ ✐s t❤❡ ♣r♦♣❡rt② (λx)(x = o) ✭✐✳❡✳✱ t❤❡ ♣r♦♣✲
❡rt② ❜❡✐♥❣ ✐❞❡♥t✐❝❛❧ ✇✐t❤ ♦ ✮ t❤❛t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ♠❡r❡❧② ❜② o ✐ts❡❧❢ ✕
✐t ✐s ❡ss❡♥t✐❛❧ ❢♦r o✳ ❲❤❡r❡✈❡r ✭❛♥❞ ✇❤❡♥❡✈❡r✮ o ❡①✐sts✱ ✐t ❝❛♥♥♦t ❜✉t
❡①❡♠♣❧✐❢② (λx)(x = o)✳ ◆❡①t s✉♣♣♦s❡ t❤❛t ❛♥♦t❤❡r ♣r♦♣❡rt② (λx)(F x)
✐s ❡♠♣✐r✐❝❛❧✱ ✐✳❡✳✱ ✐t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ❜② ✈❛r✐♦✉s ♦❜❥❡❝ts ✐♥ ❞✐✛❡r❡♥t
♣♦ss✐❜❧❡ ✇♦r❧❞s ✕ ✐♥ s♦♠❡ ✇♦r❧❞s o ❡①❡♠♣❧✐✜❡s ✐t ❜✉t ✐♥ ♦t❤❡rs ✐t ❞♦❡s
♥♦t✳ ❲❡ ♠❛② ❛♣♣❧② t❤❡ ❇♦♦❧❡❛♥ ♦♣❡r❛t✐♦♥s t♦ t❤❡ t✇♦ ♣r♦♣❡rt✐❡s ❛♥❞
❢♦r♠ ❵❝♦♠♣♦✉♥❞✬ ♣r♦♣❡rt✐❡s✱ ❡✳❣✳ (λx)(x = o & F x)✳ ❚❤✐s ♣r♦♣❡rt② ❝❛♥
❜❡ ❡①❡♠♣❧✐✜❡❞ ❡①❝❧✉s✐✈❡❧② ❜② o ♣r♦✈✐❞❡❞ o ❡①❡♠♣❧✐✜❡s ❛❧s♦ t❤❡ ❡♠♣✐r✐❝❛❧
♣r♦♣❡rt② (λx)(F x)✳ ■♥ t❤♦s❡ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ o ❡①✐sts ✇✐t❤♦✉t
❜❡✐♥❣ F ✐t ❝❛♥♥♦t ❜❡ ❛♥ ✐♥st❛♥❝❡ ♦❢ (λx)(x = o & F x)✳ ❲❤❛t ✇❡ ❤❛✈❡
❤❡r❡ ✐s ❛ r❡♠❛r❦❛❜❧❡ ♣r♦♣❡rt② ✕ ✐t ✐s ♥♦♥✲❡ss❡♥t✐❛❧ ❢♦r ❛❧❧ ♦❜❥❡❝ts✱ ❡✈❡♥
❢♦r o ✐ts❡❧❢✱ ❜❡❝❛✉s❡ ❛❧❧ ♦❜❥❡❝ts ✭✐♥❝❧✉❞✐♥❣ o✮ ❝❛♥ ❡①✐st ✇✐t❤♦✉t ❡①❡♠♣❧✐✲
❢②✐♥❣ ✐t✱ ❜✉t ❛t t❤❡ s❛♠❡ t✐♠❡ ✐t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ❡①❝❧✉s✐✈❡❧② ❜② o✱ ✐❢
❛t ❛❧❧✳✼ ❈♦♥s❡q✉❡♥t❧②✱ ✇❡ ❤❛✈❡ ❤❡r❡ ❛ ❣❡♥❡r❛❧ ♣r❡s❝r✐♣t✐♦♥ ❢♦r ❣❡♥❡r❛t✐♥❣
♥♦♥✲❡ss❡♥t✐❛❧ ♣r♦♣❡rt✐❡s t❤❛t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞✱ ✐❢ ❛t ❛❧❧✱ ❥✉st ❜② ♦♥❡
❛♥❞ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❍❡r❡ ❛r❡ s♦♠❡ ❡①❛♠♣❧❡s✿
• (λx)(x = ❙♦❝r❛t❡s ✫ x ✐s ✇✐s❡) ✭✐✳❡✳✱
❙♦❝r❛t❡s ❛♥❞ ✇✐s❡ ✮❀
❜❡✐♥❣ ❜♦t❤ ✐❞❡♥t✐❝❛❧ ✇✐t❤
✶✻
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶
• (λx)(x = P❧❛t♦ ✫ x ❛✉t❤♦r❡❞ ❵❚❤❡ ❘❡♣✉❜❧✐❝✬) ✭✐✳❡✳✱ ❜❡✐♥❣ ❜♦t❤ ✐❞❡♥✲
t✐❝❛❧ ✇✐t❤ P❧❛t♦ ❛♥❞ ❛✉t❤♦r❡❞ ❵❚❤❡ ❘❡♣✉❜❧✐❝✬ ✮❀
• (λx)(x = ❆r✐st♦t❧❡ ✫ x ✐s ❆❧❡①❛♥❞❡r✬s t❡❛❝❤❡r)
✐❞❡♥t✐❝❛❧ ✇✐t❤ ❆r✐st♦t❧❡ ❛♥❞ ❆❧❡①❛♥❞❡r✬s t❡❛❝❤❡r ✮✳
❲❤❡♥ ✇❡ r❡♣❧❛❝❡ t❤❡
λ✲♦♣❡r❛t♦r
❜② t❤❡
ι✲♦♣❡r❛t♦r✱
✭✐✳❡✳✱
❜❡✐♥❣ ❜♦t❤
✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣
❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s ✭✐♥ t❤✐s ♦r❞❡r✮✿
• ❵(ιx)(x =
❙♦❝r❛t❡s ✫
x
✐s ✇✐s❡)✬ ✭✐✳❡✳✱ ❵t❤❡ ♦❜❥❡❝t t❤❛t ✐s ❜♦t❤
✐❞❡♥t✐❝❛❧ ✇✐t❤ ❙♦❝r❛t❡s ❛♥❞ ✇✐s❡✬✮❀
• ❵(ιx)(x =
P❧❛t♦ ✫
❛✉t❤♦r❡❞ ❵❚❤❡ ❘❡♣✉❜❧✐❝✬)✬ ✭✐✳❡✳✱ ❵t❤❡ ♦❜❥❡❝t
x
t❤❛t ✐s ❜♦t❤ ✐❞❡♥t✐❝❛❧ ✇✐t❤ P❧❛t♦ ❛♥❞ ❛✉t❤♦r❡❞ ❵❚❤❡ ❘❡♣✉❜❧✐❝✑✮❀
• ❵(ιx)(x =
❆r✐st♦t❧❡ ✫
x
✐s ❆❧❡①❛♥❞❡r✬s t❡❛❝❤❡r)✬ ✭✐✳❡✳✱ ❵t❤❡ ♦❜❥❡❝t
t❤❛t ✐s ❜♦t❤ ✐❞❡♥t✐❝❛❧ ✇✐t❤ ❆r✐st♦t❧❡ ❛♥❞ ❆❧❡①❛♥❞❡r✬s t❡❛❝❤❡r✬✮✳
❚❤❡s❡ ❞❡s❝r✐♣t✐♦♥s ❡①♣r❡ss✱ ❛s t❤❡✐r ❝♦♥❞✐t✐♦♥s t♦ ❜❡ ❢✉❧✜❧❧❡❞ ❜② t❤❡✐r
r❡❢❡r❡♥ts✱ t❤❡ ❛❜♦✈❡ ♣r♦♣❡rt✐❡s t❤❛t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ❥✉st ❜② ♦♥❡ ♦❜❥❡❝t
✭✐❢ ❛t ❛❧❧✮✳ ❋♦r ❡①❛♠♣❧❡✱ s✐♥❝❡ t❤❡r❡ ✐s ♥♦ ♦❜❥❡❝t t❤❛t ❝❛♥ ❜❡ ❜♦t❤ ❙♦❝r❛t❡s
❛♥❞ ✇✐s❡ ❡①❝❡♣t ❢♦r ❙♦❝r❛t❡s ❤✐♠s❡❧❢✱ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❵t❤❡ ♦❜❥❡❝t t❤❛t ✐s
❜♦t❤ ✐❞❡♥t✐❝❛❧ ✇✐t❤ ❙♦❝r❛t❡s ❛♥❞ ✇✐s❡✬ ♠❛② r❡❢❡r t♦ ❙♦❝r❛t❡s ❛♥❞ ♥♦
♦♥❡ ❡❧s❡❀ ❤♦✇❡✈❡r✱ ✐t ❢❛✐❧s t♦ r❡❢❡r t♦ ❤✐♠ ✐♥ t❤♦s❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ❤❡
❡①✐sts ✇✐t❤♦✉t ❜❡✐♥❣ ✇✐s❡✳ ❚❤✉s✱ t❤✐s ❞❡s❝r✐♣t✐♦♥ ❝❛♥♥♦t ❜❡ ❑✲r✐❣✐❞❀ ♦♥
t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐t ✐s P✲r✐❣✐❞ ❜❡❝❛✉s❡ ✐♥ ❛❧❧ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦
❛♥②t❤✐♥❣ ❛t ❛❧❧✱ ✐t r❡❢❡rs t♦ ❙♦❝r❛t❡s✳
✸ ❙♦♠❡ ❈♦♥s❡q✉❡♥❝❡s
▲❡t ✉s s❛② t❤❛t ❛ ♣r♦♣❡rt② ✐s
❡ss❡♥t✐❛❧ ❢♦r ❛ ❣✐✈❡♥ ♦❜❥❡❝t
♣r♦✈✐❞❡❞ t❤❡
♦❜❥❡❝t ❤❛s t♦ ❡①❡♠♣❧✐❢② ✐t ✇❤❡♥❡✈❡r ❛♥❞ ✇❤❡r❡✈❡r ✐t ❡①✐sts ❛♥❞ t❤❡ ♣r♦♣✲
❡rt② ❝❛♥♥♦t ❜❡ ❡①❡♠♣❧✐✜❡❞ ❜② ❛♥② ♦t❤❡r ♦❜❥❡❝t✳
❞❡s❝r✐♣t✐♦♥
❡ss❡♥t✐❛❧✐st
▲❡t ✉s ❝❛❧❧ ❛ ❞❡✜♥✐t❡
♣r♦✈✐❞❡❞ ✐ts ♣r❡❞✐❝❛t❡ ♣❛rt ❡①♣r❡ss❡s s✉❝❤ ❛ ♣r♦♣✲
❡rt②✳ ◆♦✇ t❤❡ ♥♦t✐♦♥ ♦❢ ❑✲r✐❣✐❞✐t② ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ♦♥❧② t♦ s✉❝❤ ❞❡✜♥✐t❡
❞❡s❝r✐♣t✐♦♥s t❤❛t ❛r❡ ❡ss❡♥t✐❛❧✐st ✐♥ t❤✐s s❡♥s❡✳ ■❢ ❛ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥
✐s t♦ r❡❢❡r t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ t❤❛t ♦❜❥❡❝t ❡①✐sts✱
✐ts ♣r❡❞✐❝❛t❡ ♣❛rt ❤❛s t♦ ❡①♣r❡ss ❛ ♣r♦♣❡rt② t❤❛t ✐s ❡ss❡♥t✐❛❧ ❢♦r t❤❛t
♦❜❥❡❝t✳ ❖t❤❡r✇✐s❡ t❤❡r❡ ✇♦✉❧❞ ❜❡ ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐♥ ✇❤✐❝❤ t❤✐s ♦❜❥❡❝t
❡①✐sts ✇✐t❤♦✉t ❡①❡♠♣❧✐❢②✐♥❣ t❤❡ ♣r♦♣❡rt② ✐♥ q✉❡st✐♦♥ ❛♥❞ ✇✐t❤♦✉t ❜❡✐♥❣
t❤❡ r❡❢❡r❡♥t ♦❢ t❤❡ ❞❡s❝r✐♣t✐♦♥ ✐♥ q✉❡st✐♦♥✳
❍♦✇❡✈❡r✱ t❤✐s ✐s ❢♦r❜✐❞❞❡♥
❢♦r ❑✲r✐❣✐❞ ❞❡s❝r✐♣t✐♦♥s✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ ❑✲r✐❣✐❞✐t② ✐s✱ t❤✉s✱ ❝❧♦s❡❧② ❝♦♥✲
♥❡❝t❡❞ ✇✐t❤
❡ss❡♥t✐❛❧✐s♠✳
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ♥♦t✐♦♥ ♦❢ P✲r✐❣✐❞✐t②
▼❛r✐á♥ ❩♦✉❤❛r✿
❖♥ ❛ ❇r♦❛❞❡r ◆♦t✐♦♥ ♦❢ ❘✐❣✐❞✐t②
✶✼
✐s ❢r❡❡ ❢r♦♠ s✉❝❤ ❛ ♠❡t❛♣❤②s✐❝❛❧ ♣r❡❝♦♥❞✐t✐♦♥✳ ❚❤❡ ♣r♦♣❡rt② ❡①♣r❡ss❡❞
❜② ❛ P✲r✐❣✐❞ ❞❡s❝r✐♣t✐♦♥ ♥❡❡❞ ♥♦t ❜❡ ❡ss❡♥t✐❛❧ ❢♦r ❛♥②t❤✐♥❣ ❛♥❞ s✉❝❤ ❛
❞❡s❝r✐♣t✐♦♥ ♥❡❡❞ ♥♦t ❜❡ ❡ss❡♥t✐❛❧✐st✳
❚♦ ❜❡ P✲r✐❣✐❞✱ ❛ ❞❡s❝r✐♣t✐♦♥ ❤❛s t♦ r❡❢❡r t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧
♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦ s♦♠❡t❤✐♥❣✳ ❚❤✐s ♦❜t❛✐♥s ❛❧s♦ ✐♥ t❤❡
❝❛s❡ ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦ t❤❡ s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ t❤❡ ♦❜❥❡❝t
❡①✐sts ❛♥❞ ❢❛✐❧s t♦ ❞❡s✐❣♥❛t❡ ❛♥②t❤✐♥❣ ✐♥ ❛❧❧ t❤♦s❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ t❤❡
♦❜❥❡❝t ❞♦❡s ♥♦t ❡①✐st✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ ❛ ❞❡s❝r✐♣t✐♦♥ ❞❡s✐❣♥❛t❡s t❤❡
s❛♠❡ ♦❜❥❡❝t ✐♥ ❛❧❧ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦ s♦♠❡t❤✐♥❣✱ ✐t ♠❛② ❤❛♣♣❡♥
t❤❛t t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐♥ ✇❤✐❝❤ ✐t r❡❢❡rs t♦ ♥♦t❤✐♥❣ ❜✉t
t❤❡ ♦❜❥❡❝t ✐♥ q✉❡st✐♦♥ ❡①✐sts t❤❡r❡✐♥✳ ❙✉❝❤ ❛ ❞❡s❝r✐♣t✐♦♥ ✇♦✉❧❞ ❜❡ P✲r✐❣✐❞
✇✐t❤♦✉t ❜❡✐♥❣ ❑✲r✐❣✐❞✳ ❚❤❡ ♥♦t✐♦♥ ♦❢ P✲r✐❣✐❞✐t② ✐s ❜r♦❛❞❡r t❤❛♥ t❤❛t ♦❢
❑✲r✐❣✐❞✐t② ❜❡❝❛✉s❡ t❤❡ ❢♦r♠❡r ♥♦t✐♦♥ ❛♣♣❧✐❡s t♦ ❛❧❧ t❤♦s❡ ❡①♣r❡ss✐♦♥s t❤❛t
❛r❡ ❑✲r✐❣✐❞✱ ❜✉t t❤❡ ❧❛tt❡r ♦♥❡ ❝❛♥♥♦t ❜❡ ❛♣♣❧✐❡❞ t♦ s♦♠❡ ❡①♣r❡ss✐♦♥s t❤❛t
❛r❡ P✲r✐❣✐❞✳ ❚❤✉s✱ ❛❧❧ ❑✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs ❛r❡ P✲r✐❣✐❞ ❛s ✇❡❧❧✱ ❜✉t ♥♦t ✈✐❝❡
✈❡rs❛✳✽
◆❡①t✱ ❑r✐♣❦❡ ✉s❡❞ ❤✐s ♥♦t✐♦♥ ♦❢ r✐❣✐❞✐t② t♦ ✈❛❧✐❞❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❦✐♥❞
♦❢ ✐♥❢❡r❡♥❝❡✿
d1 = d2 ✳
❵d1 ✬ ❛♥❞ ❵d2 ✬ ❛r❡ ❑✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs✳
❍❡♥❝❡✿ ◆❡❝❡ss❛r✐❧②✱ d1 = d2 ✳
■❢ ❵d1 ✬ ❛♥❞ ❵d2 ✬ ❛r❡ ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s✱ t❤❡ st❛t❡♠❡♥t ❵d1 = d2 ✬ ✐s ♥❡❝✲
❡ss❛r②✱ ✐❢ tr✉❡✱ ❜❡❝❛✉s❡ t❤❡ ❞❡s❝r✐♣t✐♦♥s ✭q✉❛ ❑✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs✮ ❛r❡
❡ss❡♥t✐❛❧✐st✳ ■❢ t❤❡ ♣r♦♣❡rt✐❡s ❡①♣r❡ss❡❞ ❜② t❤❡ ♣r❡❞✐❝❛t❡ ♣❛rts ♦❢ ❵d1 ✬
❛♥❞ ❵d2 ✬ ❛r❡ ♥❡❝❡ss❛r② ❢♦r t❤❡ s❛♠❡ ♦❜❥❡❝t✱ t❤❡ ✐❞❡♥t✐t② st❛t❡♠❡♥t ❝❛♥✲
♥♦t ❜✉t ❜❡ tr✉❡ ♦❢ ♥❡❝❡ss✐t② ✭✐❢ tr✉❡ ❛t ❛❧❧✮✳ ◆♦✇✱ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥
t❤❡ ♥♦t✐♦♥ ♦❢ ❑✲r✐❣✐❞✐t② ✐s r❡♣❧❛❝❡❞ ❜② t❤❛t ♦❢ P✲r✐❣✐❞✐t②❄ ❚❤❡ ❛♠❡♥❞❡❞
✐♥❢❡r❡♥❝❡
d1 = d2 ✳
❵d1 ✬ ❛♥❞ ❵d2 ✬ ❛r❡ P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs✳
❍❡♥❝❡✿ ◆❡❝❡ss❛r✐❧②✱ d1 = d2 ✳
✐s ❜② ♥♦ ♠❡❛♥s ❣❡♥❡r❛❧❧② ✈❛❧✐❞ ✭♣r♦✈✐❞❡❞ ❵d1 ✬ ❛♥❞ ❵d2 ✬ ❛r❡ ❞❡✜♥✐t❡ ❞❡✲
s❝r✐♣t✐♦♥s✮✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ❵d1 ✬ ♠❛② ❢❛✐❧ t♦ r❡❢❡r✱ ✐♥ s♦♠❡ ♣♦ss✐❜❧❡
✇♦r❧❞s✱ t♦ t❤❡ ♦❜❥❡❝t t♦ ✇❤✐❝❤ ❵d2 ✬ ❞♦❡s r❡❢❡r ✐♥ t❤❡ ✇♦r❧❞s ✐♥ q✉❡st✐♦♥
✭❛♥❞ ✈✐❝❡ ✈❡rs❛ ✮✳ ■♥ ❢❛❝t✱ t❤❡r❡ ❛r❡ ♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ♦♥❡ ♦❢ t❤❡s❡
❞❡s❝r✐♣t✐♦♥s ✐s ❡♠♣t② ❛♥❞ t❤❡ ♦t❤❡r ♦♥❡ r❡❢❡rs t♦ s♦♠❡t❤✐♥❣✳ ■♥ s✉❝❤
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶
✶✽
✇♦r❧❞s✱ ❵d1
= d2 ✬
✇♦✉❧❞ ♥♦t ❜❡ tr✉❡ ❛♥❞✱ t❤✉s✱ ✐t ❝❛♥♥♦t ❜❡ ♥❡❝❡ss❛r✐❧②
tr✉❡ ✭✐❢ tr✉❡ ❛t ❛❧❧✮✳
❚❤✉s✱ t❤❡ ♥♦t✐♦♥ ♦❢ P✲r✐❣✐❞✐t② ✐s ♥♦t s✉✐t❛❜❧❡ ❢♦r
❑r✐♣❦❡✬s ♥❡❡❞s✳
■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤✐s r❡s✉❧t ✐s ❛ s✐♠♣❧❡ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢❛❝t
t❤❛t t❤❡ ♥♦t✐♦♥ ♦❢ P✲r✐❣✐❞✐t②✱ ✉♥❧✐❦❡ t❤❛t ♦❢ ❑✲r✐❣✐❞✐t②✱ ✐s r❛t❤❡r ✐♥❞❡♣❡♥✲
❞❡♥t ♦❢ ❡ss❡♥t✐❛❧✐s♠✳ ❚❤❡ r❡❛s♦♥ ❢♦r s❛②✐♥❣ t❤❛t ❵d1
= d2 ✬
♥❡❡❞ ♥♦t ❜❡
♥❡❝❡ss❛r✐❧② tr✉❡✱ ✐❢ tr✉❡ ❛t ❛❧❧✱ ❝♦♥s✐sts ✐♥ t❤❛t t❤❡ ♣r♦♣❡rt✐❡s ❡①♣r❡ss❡❞ ❜②
t❤❡ ♣r❡❞✐❝❛t❡ ♣❛rts ♦❢ t❤❡ t✇♦ ❞❡s❝r✐♣t✐♦♥s ❛r❡ ♥♦t ❡ss❡♥t✐❛❧ ❢♦r ❛♥②t❤✐♥❣✳
❋✐♥❛❧❧②✱ ♦♥❡ ♠❛② ✇♦♥❞❡r ✇❤❡t❤❡r t❤❡r❡ ✐s ❛ s✐♠♣❧❡ ❑r✐♣❦❡✲❧✐❦❡ t❡st
t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞✐st✐♥❣✉✐s❤ P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs ❢r♦♠ ♥♦♥✲r✐❣✐❞ ♦♥❡s✳
❆❝❝♦r❞✐♥❣ t♦ ❑r✐♣❦❡✬s t❡st✱ ✏ d✑ ✐s ❛ ❑✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r ♣r♦✈✐❞❡❞ t❤❡ s❡♥✲
t❡♥❝❡ ✏ d ♠✐❣❤t ♥♦t ❤❛✈❡ ❜❡❡♥
♦✛s♣r✐♥❣ ♦❢ ❣❛♠❡t❡s
X
❛♥❞
t❤❡ ♦✛s♣r✐♥❣ ♦❢ ❣❛♠❡t❡s
♦❢ ❣❛♠❡t❡s
X
❛♥❞
Y✳
X
Y✑
d✑
❝❛♥♥♦t ❜❡ tr✉❡✳
❋♦r ❡①❛♠♣❧❡✱ ✏t❤❡
✐s ❑✲r✐❣✐❞ ❜❡❝❛✉s❡ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ t❤❛t
❛♥❞
Y
♠✐❣❤t ♥♦t ❤❛✈❡ ❜❡❡♥ t❤❡ ♦✛s♣r✐♥❣
❚❤✐s t❡st ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ t❤❛t ✇❡ ❤❛✈❡ t♦
❡①❛♠✐♥❡ t❤❡ ❛❝t✉❛❧ r❡❢❡r❡♥t ♦❢ ❛ ❞❡s✐❣♥❛t♦r ❛s t♦ ✐ts ♠♦❞❛❧ ♣r♦♣❡rt✐❡s✳
■t ✐s ❡❛s② t♦ s❡❡ t❤❛t P✲r✐❣✐❞✐t② ❝❛♥♥♦t ❜❡ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ s❛♠❡
✇❛②✳
❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✐❢ ✏ d✑ ✐s s✉♣♣♦s❡❞ t♦ ❜❡ ❛ P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r
✭✇✐t❤♦✉t ❜❡✐♥❣ ❑✲r✐❣✐❞ ❛t t❤❡ s❛♠❡ t✐♠❡✮✱ ✏ d ♠✐❣❤t ♥♦t ❤❛✈❡ ❜❡❡♥
d✑
✐s
✐♥ ❢❛❝t tr✉❡❀ ❢♦r t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐♥ ✇❤✐❝❤ t❤❡ ❛❝t✉❛❧
r❡❢❡r❡♥t ♦❢ ✏ d✑✱ t❤♦✉❣❤ ❡①✐st✐♥❣ ✐♥ t❤❛t ✇♦r❧❞✱ ❢❛✐❧s t♦ ❜❡ ❛ r❡❢❡r❡♥t ♦❢ ✏ d✑✳
❙♦✱ ❛r❡ t❤❡r❡ ❛♥② ♦t❤❡r t❡sts ❛✈❛✐❧❛❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤ P✲r✐❣✐❞ ❞❡s✐❣♥❛✲
t♦rs ❢r♦♠ ♥♦♥✲r✐❣✐❞ ♦♥❡s❄ ■♥❞❡❡❞✱ t❤❡r❡ ✐s ♦♥❡✳
✾ ■♥st❡❛❞ ♦❢ ❡①❛♠✐♥✐♥❣ t❤❡
❛❝t✉❛❧ r❡❢❡r❡♥t ♦❢ ❛ ❣✐✈❡♥ ❞❡s✐❣♥❛t♦r✱ ✇❡ ♠✐❣❤t tr② t♦ ❡①❛♠✐♥❡ ❛❧❧ ♦❜❥❡❝ts
♦t❤❡r t❤❛♥ t❤❡ ❛❝t✉❛❧ r❡❢❡r❡♥t ♦❢ t❤❡ ❞❡s✐❣♥❛t♦r ❛s t♦ t❤❡✐r ♠♦❞❛❧ ♣r♦♣✲
❡rt✐❡s✳ ■❢ ✐t ❛♣♣❡❛rs t❤❛t ♥♦♥❡ ♦❢ t❤❡♠ ❝❛♥ ❜❡ t❤❡ ❞❡s✐❣♥❛t♦r✬s r❡❢❡r❡♥t
✐♥ s♦♠❡ ✭♥♦♥✲❛❝t✉❛❧✮ ✇♦r❧❞ ✭✐♥st❡❛❞ ♦❢ ✐ts ❛❝t✉❛❧ r❡❢❡r❡♥t✮✱ t❤❡ ❞❡s✐❣♥❛✲
t♦r ✐s P✲r✐❣✐❞✳ ❚❤✉s✱ ✏ d✑ ✐s ❛ P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r ♣r♦✈✐❞❡❞ t❤❡ s❡♥t❡♥❝❡
✏s♦♠❡t❤✐♥❣ ♦t❤❡r t❤❛♥
d
♠✐❣❤t ❤❛✈❡ ❜❡❡♥
d✑
❝❛♥♥♦t ❜❡ tr✉❡✳ ❖❜s❡r✈❡
t❤❛t t❤❡ ✜rst ♦❝❝✉rr❡♥❝❡ ♦❢ ✏ d✑ ❤❛s ✇✐❞❡r s❝♦♣❡ ♦✈❡r ✏♠✐❣❤t✑ ✇❤✐❧❡ t❤❡
s❡❝♦♥❞ ♦❝❝✉rr❡♥❝❡ ❤❛s ❛ ♥❛rr♦✇ s❝♦♣❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✏t❤❡ ✐♥❞✐✈✐❞✉❛❧ t❤❛t
✐s ❜♦t❤ ✐❞❡♥t✐❝❛❧ t♦ ❙♦❝r❛t❡s ❛♥❞ ✇✐s❡✑ ✐s P✲r✐❣✐❞ ✭✇✐t❤♦✉t ❜❡✐♥❣ ❑✲r✐❣✐❞✮
❜❡❝❛✉s❡ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ t❤❛t s♦♠❡t❤✐♥❣ ♦t❤❡r t❤❛♥ t❤❡ ❛❝t✉❛❧ ✐♥❞✐✈✐❞✲
✉❛❧ t❤❛t ✐s ❜♦t❤ ✐❞❡♥t✐❝❛❧ t♦ ❙♦❝r❛t❡s ❛♥❞ ✇✐s❡ ♠✐❣❤t ❜❡ ❜♦t❤ ✐❞❡♥t✐❝❛❧
t♦ ❙♦❝r❛t❡s ❛♥❞ ✇✐s❡✳
✶✵
▼❛r✐á♥ ❩♦✉❤❛r✿
❖♥ ❛ ❇r♦❛❞❡r ◆♦t✐♦♥ ♦❢ ❘✐❣✐❞✐t②
✶✾
◆♦t❡s
✶ ❚❤❡ t❡r♠s ❵r❡❢❡r❡♥❝❡✬ ❛♥❞ ❵❞❡s✐❣♥❛t✐♦♥✬ ✭❛♥❞ t❤❡✐r ✈❡r❜ ❢♦r♠s✮ ❛r❡ ✉s❡❞ ✐♥t❡r✲
❝❤❛♥❣❡❛❜❧② t❤r♦✉❣❤♦✉t t❤❡ ♣❛♣❡r✳
✷ ■❢ (λx)(F x) ✐s ✐♥st❛♥t✐❛t❡❞ ❜② t✇♦ ♦r ♠♦r❡ ♦❜❥❡❝ts✱ t❤❡♥ ♥♦t❤✐♥❣ ✐s ❛ ✉♥✐q✉❡
✐♥st❛♥❝❡ ♦❢ t❤✐s ♣r♦♣❡rt②❀ t❤❡ ❞❡s❝r✐♣t✐♦♥ ❞❡s✐❣♥❛t❡s✱ ♦r r❡❢❡rs t♦✱ ♥♦t❤✐♥❣ ❛t ❛❧❧
✐♥ s✉❝❤ ❛ ❝❛s❡✳
✸ ■t ✐s s♦♠❡t✐♠❡s ❛ss✉♠❡❞ t❤❛t t❤❡ r❡❢❡r❡♥❝❡ r❡❧❛t✐♦♥ ❢♦r ❞❡✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥s ✐s
❛ ❢♦✉r✲♣❧❛❝❡ ♦♥❡ ✇✐t❤ t✐♠❡ ❛s ❛ ❢♦✉rt❤ ❛r❣✉♠❡♥t✳ ■ ✐❣♥♦r❡ t❤✐s ❝♦♠♣❧✐❝❛t✐♦♥
t❤r♦✉❣❤♦✉t t❤❡ ♣❛♣❡r✳
✹ ■ ❤❛✈❡ ♥♦ s♣❛❝❡ t♦ ❞✐s❝✉ss ❜❛♣t✐s♠s ✐♥ ❛ ♠♦r❡ ❞❡t❛✐❧❡❞ ♠❛♥♥❡r✳ ▲❡t ♠❡ ❣✐✈❡
❥✉st ❛ s✐♠♣❧❡ ✭❛♥❞ s✐♠♣❧✐st✐❝✮ ❝❤❛r❛❝t❡r✐③❛t✐♦♥✿ ●✐✈❡♥ t❤❛t t❤❡r❡ ✐s ❛♥ ♦❜❥❡❝t t♦
❜❡ ❜❛♣t✐③❡❞✱ o✱ ❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ✐s t♦ ❜❡ ❛tt❛❝❤❡❞ t♦ o ❛s ✐ts ♣r♦♣❡r ♥❛♠❡✱
n✱ ❛♥❞ ❛♥♦t❤❡r ❞❡✈✐❝❡✱ d✱ ✐❞❡♥t✐❢②✐♥❣ o ❢♦r t❤❡ ❜❛♣t✐③❡r✱ b ✭d ♠✐❣❤t ❜❡ ❛♥♦t❤❡r
❡①♣r❡ss✐♦♥ ❛❧r❡❛❞② r❡❢❡rr✐♥❣ t♦ o✱ ♦r ❛ ♣♦✐♥t✐♥❣ ❣❡st✉r❡ t♦✇❛r❞ o✱ ♦r ❛♥②t❤✐♥❣ ❡❧s❡
t❤❛t ❡♥❛❜❧❡s b t♦ ✐❞❡♥t✐❢② o✮✱ b ❞❡❝✐❞❡s t❤❛t o ✐❞❡♥t✐✜❡❞ ❢♦r ❤❡r ❜② d ✐s ❤❡♥❝❡❢♦rt❤
♥❛♠❡❞ n✳ ■❢ t❤❡ ❛❝t ♦❢ ❜❛♣t✐s♠ ✇❛s s✉❝❝❡ss❢✉❧✱ ❛ ❧✐♥❣✉✐st✐❝ ❝♦♥✈❡♥t✐♦♥ ❛ss♦❝✐❛t✐♥❣
n ✇✐t❤ o ❤❛s ❜❡❡♥ ❡st❛❜❧✐s❤❡❞✳
✺ ❋❛♠♦✉s❧②✱ t❤✐s ❢❛❝t ✇❛s ❤✐❣❤❧✐❣❤t❡❞ ❜② ❉❛✈✐❞ ❑❛♣❧❛♥ ✐♥ ❤✐s ❞✐s❝✉ss✐♦♥ ❝♦♥❝❡r♥✐♥❣
t❤❡ r❡❢❡r❡♥t✐❛❧ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♥❛♠❡ ❵◗✉✐♥❡✬ ✐♥ t❤♦s❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ◗✉✐♥❡ ❞♦❡s
♥♦t ❡①✐st❀ ✭❝❢✳ ❬✸✱ ♣✳ ✺✵✸❪✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❘✉t❤ ❇❛r❝❛♥ ▼❛r❝✉s ❛❞♠✐ts t❤❛t
❛ ♣r♦♣❡r ♥❛♠❡ ❞♦❡s ♥♦t r❡❢❡r t♦ ❛♥②t❤✐♥❣ ✐♥ t❤♦s❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ ✐ts r❡❢❡r❡♥t
❞♦❡s ♥♦t ❡①✐st❀ ✭❝❢✳ ❬✺✱ ♣✳ ✻✶❪✳
✻ ❚❤❡r❡ ❛r❡ ✈❛r✐♦✉s ♥♦t✐♦♥s ♦❢ ❡ss❡♥t✐❛❧ ♣r♦♣❡rt② ♦♥ t❤❡ ♠❛r❦❡t✳ ❲❤❡♥ ✇❡ ✇♦r❦
✇✐t❤✐♥ t❤❡ ♣♦ss✐❜❧❡ ✇♦r❧❞ s❡♠❛♥t✐❝s ❢r❛♠❡✇♦r❦✱ ✇❡ ❛r❡ ❛❧❧♦✇❡❞ t♦ s❛② t❤❛t ❛
♣r♦♣❡rt②✱ P ✱ ✐s ❡ss❡♥t✐❛❧ ♣r♦✈✐❞❡❞ ❢♦r ❛❧❧ ♦❜❥❡❝ts✱ x✱ ❛♥❞ ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s✱
w✱ ✐t ❤♦❧❞s t❤❛t x ✐♥st❛♥t✐❛t❡s P ✐♥ w ✐✛ x ❡①✐sts ✐♥ w✳ ❚❤❡ ♣r♦♣❡rt✐❡s ■ t❛❦❡ t♦
❜❡ ❡ss❡♥t✐❛❧ ✐♥ t❤❡ ♠❛✐♥ t❡①t ❛r❡ ❛ ❧✐♠✐t✐♥❣ ❝❛s❡ s❛t✐s❢②✐♥❣ t❤✐s ❞❡✜♥✐t✐♦♥✳ ❋♦r
t❤❡② ❛r❡ s♣❡❝✐✜❝ ✐♥ t❤❛t t❤❡② ❝❛♥ ❜❡ ✐♥st❛♥t✐❛t❡❞ ❥✉st ❜② ❛ s✐♥❣❧❡ ♦❜❥❡❝t ✐♥ ❛❧❧
♣♦ss✐❜❧❡ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ✐♥st❛♥t✐❛t❡❞ ❛t ❛❧❧ ❛♥❞✱ ❛t t❤❡ s❛♠❡ t✐♠❡✱ t❤❡
♦❜❥❡❝t ✐♥ q✉❡st✐♦♥ ❤❛s t♦ ✐♥st❛♥t✐❛t❡ t❤❡♠ ✇❤❡r❡✈❡r ✐t ❡①✐sts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡
♣r♦♣❡rt✐❡s ❜❡✐♥❣ ❤✉♠❛♥ ♦r ❜❡✐♥❣ s❡❧❢✲✐❞❡♥t✐❝❛❧✱ t❤♦✉❣❤ ❡ss❡♥t✐❛❧ ❛❝❝♦r❞✐♥❣ t♦ t❤❡
❛❜♦✈❡ ❞❡✜♥✐t✐♦♥✱ ❞♦ ♥♦t ❢❛❧❧ ✇✐t❤✐♥ t❤✐s ❧✐♠✐t❡❞ ❝❛t❡❣♦r② ❜❡❝❛✉s❡ t❤❡② ❝❛♥ ❜❡
✐♥st❛♥t✐❛t❡❞ ❜② ✈❛r✐♦✉s ♦❜❥❡❝ts✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❜❡✐♥❣ ✐❞❡♥t✐❝❛❧ ✇✐t❤ P❧❛t♦ ✐s
❛♥ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ ♣r♦♣❡rt②✳ ❙♦✱ ✇❤❡♥ ■ t❛❧❦ ❛❜♦✉t ❡ss❡♥t✐❛❧ ♣r♦♣❡rt✐❡s ■ ❤❛✈❡
✐♥ ♠✐♥❞ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s ❧✐♠✐t❡❞ s♦rt✳ ❚❤✐s ❢❛❝t ❜❡❝♦♠❡s ❝r✉❝✐❛❧ ✐♥ ❙❡❝t✐♦♥ ✸✳ ■
❛♠ ✐♥❞❡❜t❡❞ t♦ ❑❛t❤r✐♥ ●❧ü❡r✲P❛❣✐♥ ❛♥❞ P❡t❡r P❛❣✐♥ ❢♦r ❞✐s❝✉ss✐♦♥ ♦♥ ❡ss❡♥t✐❛❧
♣r♦♣❡rt✐❡s✳
✼ ❚❤❡r❡ ✐s ❛❧s♦ ❛♥♦t❤❡r ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt② t❤❛t ✐s ❵❝♦♠♣♦s❡❞✬ ♦❢ t❤❡ t✇♦ ♦♥❡s✿ ✐t ✐s
t❤❡ ♣r♦♣❡rt② (λx)(x = o ∨ F x)✳ ■t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ✭✐♥ ❛ ❣✐✈❡♥ ♣♦ss✐❜❧❡ ✇♦r❧❞✮
❡✐t❤❡r ❜② o ✭❛♥❞ t❤✐s ❤♦❧❞s ✐♥ ❛❧❧ ✇♦r❧❞s ✐♥ ✇❤✐❝❤ o ❡①✐sts✮ ♦r ❜② ❛♥② ✐♥❞✐✈✐❞✉❛❧
t❤❛t ✐s F ✭✐♥ t❤❛t ♣♦ss✐❜❧❡ ✇♦r❧❞✮✳ ■t ❝❛♥ ❜❡ ❡①❡♠♣❧✐✜❡❞ ❜② ♠♦r❡ ♦❜❥❡❝ts ✐♥ s♦♠❡
♣♦ss✐❜❧❡ ✇♦r❧❞s✳ ❲❤❛t ✐s str✐❦✐♥❣ ✐s t❤❛t ✐t ✐s ❡ss❡♥t✐❛❧ ❢♦r o ❜✉t ❢♦r ❛♥② ♦t❤❡r
♦❜❥❡❝t ✐t ✐s ♥♦♥✲❡ss❡♥t✐❛❧ ✭❡♠♣✐r✐❝❛❧✮✳ ■t ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❡♠♣✐r✐❝❛❧ ❡ss❡♥t✐❛❧
♣r♦♣❡rt②❀ s✉❝❤ ♣r♦♣❡rt✐❡s ✇❡r❡ t❤♦r♦✉❣❤❧② st✉❞✐❡❞ ✐♥ ❬✶❪✳
✽ ■t ✐s ✉♥❞❡rst❛♥❞❛❜❧❡ t❤❛t t❤❡ ♣❤✐❧♦s♦♣❤❡rs ✇❤♦ ✇♦r❦ ✐♥ t❤❡ ❑r✐♣❦❡❛♥ ❢r❛♠❡✲
✇♦r❦ ❛r❡ ♥♦t ✇✐❧❧✐♥❣ t♦ ❛❞♠✐t ❛s r✐❣✐❞ t❤♦s❡ ❞❡s✐❣♥❛t♦rs t❤❛t ❛r❡ ♠❡r❡❧② P✲r✐❣✐❞✳
❈❤r✐st♦♣❤❡r ❍✉❣❤❡s s✉❣❣❡sts ❞✐st✐♥❣✉✐s❤✐♥❣ r✐❣✐❞ ❞❡s✐❣♥❛t♦rs ❢r♦♠ ✐♥✢❡①✐❜❧❡ ♦♥❡s❀
❛♥ ✐♥✢❡①✐❜❧❡ ❞❡s✐❣♥❛t♦r ❵❝♦✉❧❞♥✬t ❤❛✈❡ r❡❢❡rr❡❞ t♦ ❛♥②t❤✐♥❣ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡
✷✵
❑r✐t❡r✐♦♥ ✕ ❏♦✉r♥❛❧ ♦❢ P❤✐❧♦s♦♣❤② ✭✷✵✶✷✮ ✷✻✿ ✶✶✕✷✶
t❤✐♥❣ ✐t ❛❝t✉❛❧❧② r❡❢❡rs t♦✬ ❬✷✱ ♣✳ ✷✵❪✳ ❍✉❣❤❡s✬ ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐♥✢❡①✐❜❧❡ ❞❡s✐❣♥❛t♦r
✐s ❵t❤❡ ❢❛t❤❡r ♦❢ ❆❜❡❧✬ ✕ ✇❤✐❝❤ ✇❛s ♦r✐❣✐♥❛❧❧② ✉s❡❞ ❜② ❈♦❧✐♥ ▼❝●✐♥♥ ✐♥ ❬✻❪ ✕ t❤❛t
r❡❢❡rs t♦ ❆❞❛♠ ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇♦r❧❞s ❛♣❛rt ❢r♦♠ t❤♦s❡ ✐♥ ✇❤✐❝❤ ❡✐t❤❡r ❆❞❛♠
❞✐❞ ♥♦t ❡①✐st ♦r ❤❡ r❡♠❛✐♥❡❞ ❝❤✐❧❞❧❡ss❀ t❤✉s✱ ✐t ✇♦✉❧❞ ❜❡ ❛ P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦r✳
❍✉❣❤❡s ✇♦✉❧❞ s❛② t❤❛t P✲r✐❣✐❞ ❞❡s✐❣♥❛t♦rs ❛r❡ ♠❡r❡❧② ✐♥✢❡①✐❜❧❡✳ ❆♥❛❧♦❣♦✉s❧②✱
❙❝♦tt ❙♦❛♠❡s ✐♥ ❤✐s ❞✐s❝✉ss✐♦♥ ❛❜♦✉t ♣❛rt✐❛❧❧② ❞❡s❝r✐♣t✐✈❡ ♥❛♠❡s ✐s ❡①♣❧✐❝✐t ❛❜♦✉t
t❤❡ ✐❞❡❛ t❤❛t ❡①♣r❡ss✐♦♥s s✉❝❤ ❛s ❵Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t②✬ ❛r❡ ❜② ♥♦ ♠❡❛♥s r✐❣✐❞
❜❡❝❛✉s❡ ❤❡ ✐❞❡♥t✐✜❡s r✐❣✐❞✐t② ✇✐t❤ ✇❤❛t ✐s ❝❛❧❧❡❞ ❤❡r❡ ❑✲r✐❣✐❞✐t②❀ ✭❝❢✳ ❬✽✱ ♣✳ ✺✷❪✮✳
❆♥②✇❛②✱ t❤❡ s✉rr♦✉♥❞✐♥❣ ❞✐s❝✉ss✐♦♥ ✐♠♣❧✐❡s t❤❛t ❵Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t②✬ ✇♦✉❧❞ ❜❡
P✲r✐❣✐❞✳
✾ ■ ❛♠ ❣r❛t❡❢✉❧ t♦ ■❧❤❛♥ ■♥❛♥ ✇❤♦ ❝❛♠❡ ✉♣ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡st ♦❢ P✲r✐❣✐❞✐t②✳
✶✵ ■ ✇✐s❤ t♦ ❡①♣r❡ss ♠② ❣r❛t✐t✉❞❡ t♦ P❛✈❡❧ ❈♠♦r❡❥✱ ■❧❤❛♥ ■♥❛♥✱ ❑❛t❤r✐♥ ●❧ü❡r✲P❛❣✐♥
❛♥❞ P❡t❡r P❛❣✐♥ ❢♦r ❤❡❧♣❢✉❧ ❝♦♠♠❡♥ts ❛♥❞ st✐♠✉❧❛t✐♥❣ ❞✐s❝✉ss✐♦♥s✳ ❚❤❡ ♣❛♣❡r
✇❛s s✉♣♣♦rt❡❞ ❜② t❤❡ ❱❊●❆ ♣r♦❥❡❝t ◆♦✳ ✶✴✵✵✹✻✴✶✶✱ ❙❡♠❛♥t✐❝ ▼♦❞❡❧s✱ t❤❡✐r
❊①♣❧❛♥❛t♦r② P♦✇❡r✱ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✳
▼❛r✐á♥ ❩♦✉❤❛r
❉❡♣❛rt♠❡♥t ♦❢ ▲♦❣✐❝ ❛♥❞ ▼❡t❤♦❞♦❧♦❣② ♦❢ ❙❝✐❡♥❝❡s
❋❛❝✉❧t② ♦❢ P❤✐❧♦s♦♣❤②
❈♦♠❡♥✐✉s ❯♥✐✈❡rs✐t②
●♦♥❞♦✈❛ ✷
✽✶✹ ✾✾ ❇r❛t✐s❧❛✈❛✱ ❙❧♦✈❛❦✐❛
❁♠❛r✐❛♥✳③♦✉❤❛r❅❣♠❛✐❧✳❝♦♠❃
▼❛r✐á♥ ❩♦✉❤❛r✿ ❖♥ ❛ ❇r♦❛❞❡r ◆♦t✐♦♥ ♦❢ ❘✐❣✐❞✐t②
✷✶
❘❡❢❡r❡♥❝❡s
❬✶❪ P✳ ❈♠♦r❡❥✳ ❊♠♣✐r✐❝❦é ❡s❡♥❝✐á❧♥❡ ✈❧❛st♥♦st✐ ✭❡♥❣❧✳✿ ❊♠♣✐r✐❝❛❧ ❡ss❡♥t✐❛❧
♣r♦♣❡rt✐❡s✮✳ ❖r❣❛♥♦♥ ❋✱ ✸✭✸✮✿♣♣✳ ✷✸✾✕✷✻✶✱ ✶✾✾✻✳
❬✷❪ ❈✳ ❍✉❣❤❡s✳ ❑r✐♣❦❡✿ ◆❛♠❡s✱ ◆❡❝❡ss✐t②✱ ❛♥❞ ■❞❡♥t✐t②✳ ❈❧❛r❡♥❞♦♥ Pr❡ss✱
❖①❢♦r❞✱ ✷✵✵✹✳
❬✸❪ ❉❛✈✐❞ ❑❛♣❧❛♥✳ ❇♦❜ ❛♥❞ ❝❛r♦❧ ❛♥❞ t❡❞ ❛♥❞ ❛❧✐❝❡✳ ■♥ ❏✳ ❍✐♥t✐❦❦❛✱
▼♦r❛✈❝s✐❦✱ ❛♥❞ P✳ ❙✉♣♣❡s✱ ❡❞✐t♦rs✱ ❆♣♣r♦❛❝❤❡s t♦ ◆❛t✉r❛❧ ▲❛♥❣✉❛❣❡✱
♣❛❣❡s ✹✾✵✕✺✶✽✳ ❘❡✐❞❧✱ ❉♦r❞r❡❝❤t✱ ✶✾✼✸✳
❬✹❪ ❙✳ ❑r✐♣❦❡✳ ◆❛♠✐♥❣ ❛♥❞ ◆❡❝❡ss✐t②✳ ❍❛r✈❛r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠✲
❜r✐❞❣❡✱ ▼❛ss✳✱ ✶✾✽✵✳
❬✺❪ ❘✉t❤ ❇❛r❝❛♥ ▼❛r❝✉s✳ ❊ss❡♥t✐❛❧ ❛ttr✐❜✉t✐♦♥✳ ■♥ ❘✉t❤ ❇❛r❝❛♥ ▼❛r❝✉s✱
❡❞✐t♦r✱ ▼♦❞❛❧✐t✐❡s✱ ♣❛❣❡s ✺✹✕✼✵✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❖①❢♦r❞✱
✶✾✾✸✳
❬✻❪ ❈✳ ▼❝●✐♥♥✳ ❘✐❣✐❞ ❞❡s✐❣♥❛t✐♦♥ ❛♥❞ s❡♠❛♥t✐❝ ✈❛❧✉❡✳ ❚❤❡ P❤✐❧♦s♦♣❤✐❝❛❧
◗✉❛rt❡r❧②✱ ✸✷✭✶✷✼✮✿♣♣✳ ✾✼✕✶✶✺✱ ✶✾✽✷✳
❬✼❪ ❍✳ P✉t♥❛♠✳ ❚❤❡ ♠❡❛♥✐♥❣ ♦❢ ❵♠❡❛♥✐♥❣✬✳ ■♥ ❍✳ P✉t♥❛♠✱ ❡❞✐t♦r✱ ▼✐♥❞✱
▲❛♥❣✉❛❣❡✱ ❛♥❞ ❘❡❛❧✐t②✳ P❤✐❧♦s♦♣❤✐❝❛❧ P❛♣❡rs✱ ✈♦❧✳ ✷✱ ♣❛❣❡s ✷✶✺✕✷✼✶✳
❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ✶✾✼✺✳
❬✽❪ ❙✳ ❙♦❛♠❡s✳ ❇❡②♦♥❞ ❘✐❣✐❞✐t②✿ ❚❤❡ ❯♥✜♥✐s❤❡❞ ❙❡♠❛♥t✐❝ ❆❣❡♥❞❛ ♦❢
◆❛♠✐♥❣ ❛♥❞ ◆❡❝❡ss✐t②✳ ❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❖①❢♦r❞✱ ✷✵✵✷✳
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