The Logic of Opacity

Bacon, Andrew; Russell, Jeffrey Sanford

doi:10.1111/phpr.12454

Cited by 23 publications

(29 citation statements)

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“…This is not necessarily a problem: In the context of attitude reports, in response to the so-called problems of "quantifying in", some authors have rejected UI (e.g. Bacon and Russell [2], Lederman [24]) and many have denied eβ1/2 (this is probably the best understanding of Kaplan [18], cf. Kaplan [19]; see Yalcin [43], Lederman [25]).…”

Section: Attitudesmentioning

confidence: 99%

Closed Structure

2021

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According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

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Section: Attitudesmentioning

confidence: 99%

Closed Structure

2021

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“…39. For applications in logic, language, and mathematics, see, e.g., Prior (1971: Chapter 3), Wright (1983), Boolos (1985), Williamson (2003), Wright (2007), Rayo (2012), Jones (2016), and Bacon and Russell (2017). For applications within metaphysics, see, e.g., van Cleve (1994), Dunaway (2013), Williamson (2013: 254-261), Dorr (2016), Jones (2017), andGoodman (2017).…”

Section: Realists Accept Claims Likementioning

confidence: 99%

How to Unify

Jones¹

2018

Ergo, an Open Access Journal of Philosophy

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This paper evaluates the argument for the contradictoriness of unity, that begins Priest's recent book One. The argument is seen to fail because it does not adequately differentiate between different forms of unity. This diagnosis of the argument's failure is used as a basis for two consistent accounts of unity. The paper concludes by arguing that reality contains two absolutely fundamental and unanalysable forms of unity, which are in principle presupposed by any theory of anything. These fundamental forms of unity are closely related to the unity of propositions and facts.

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“…(Goodman (2016) makes a parallel point in reply to those like like Stalnaker (2012) who maintain that

(λ x . (λ y . \neg \exists z (z = y)) x) a = (λ x . \neg \exists z (z = x)) a ⃗ (λ y . \neg \exists z (z = y)) a = \neg \exists z (z = a)

is a counterexample to Reduction Congruence for

a

that only contingently exist.) This is important to keep in mind, since below we will consider the view of Bacon & Russell (2017) according to which not only does this equivalence fail, but the above argument would fail only at step (iii) were the

λ

‐terms involved replaced with their quantification‐involving counterparts.…”

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confidence: 99%

“…They must say something systematic about which terms can be instantiated. Bacon & Russell (2017) suggest the following candidate Pure Instantiation

\forall x φ ⃗ φ [a / / x],

provided

a

is a pure term. But they show that this principle is inconsistent with opacity, Quantified Substitution, and

(λ x y Z . ⊤) = (λ x y Z . x = y ⃗ (Z x \leftrightarrow Z y))

, where

⊤

is any propositional tautology. An obvious generalization of their argument shows that if there are cases of opacity, then Pure Instantiation and Quantified Substitution imply: Pure Universal Distinctness

(λ x y Z . (x = y) ⃗ (Z x \leftrightarrow Z y)) \neq α

for every pure

α

such that

α x y Z \in T

for

x, y, Z

of appropriate types. Pure Universal Distinctness, however, has false instances in the most familiar, consistent theories of the fineness of grain of propositions, properties and relations.…”

mentioning

confidence: 99%

Classical Opacity*

Caie

Goodman

Lederman

2019

Philos Phenomenol Research

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It is an apparent truism that, for any things x and y, if x and y are identical, then x and y have the same properties. For if x and y are identical they are one and the same thing, and so it seems whatever properties the one has the other must have as well. In this paper we will explore views that deny this apparent truism. We begin with familiar examples like the following: Hesperus/Phosphorus: Hesperus is Phosphorus. But while the ancients knew that Hesperus was visible at night, the ancients did not know that Phosphorus was visible at night. This example presents a putative false instance of the following schematic principle: Substitution a ¼ b ! ðu $ u½b=aÞ. 1 We will call false instances of Substitution cases of opacity. Given the existence of cases of opacity, it follows, given classical quantificational logic, that there are false instances of: Quantified Substitution 8x8yðx ¼ y ! ðu $ u½y=xÞÞ.

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The Logic of Opacity

Cited by 23 publications

References 24 publications

Closed Structure

Closed Structure

How to Unify

Classical Opacity*

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