Abstract
Are there ‘self-referential’ propositions? That is, propositions that say of themselves that they have a certain property, such as that of being false. There can seem reason to doubt that there are. At the same time, there are a number of reasons why it matters. For suppose that there are indeed no such propositions. One might then hope that while paradoxes such as the Liar show that many plausible principles about sentences must be given up, no such fate will befall principles about propositions. But the existence of self-referential propositions would dash such hopes. Further, the existence of such propositions would also seem to challenge the widespread claim that Liar sentences fail to express propositions. The aim of this paper is thus to settle the question–at least given an assumption. In particular, I argue that if propositions are structured, then self-referential propositions exist.
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21 March 2018
Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ‘Jack is short’: Jack (1975, p. 693).
21 March 2018
Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ���Jack is short���: Jack (1975, p. 693).
21 March 2018
Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ���Jack is short���: Jack (1975, p. 693).
21 March 2018
Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ���Jack is short���: Jack (1975, p. 693).
21 March 2018
Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ���Jack is short���: Jack (1975, p. 693).
Notes
Thus, most recent work on truth for sentences does indeed reject it: e.g. Kripke (1975), Gupta (1982), Herzberger (1982), McGee (1991), Gupta and Belnap (1993), Maudlin (2004) and Leitgeb (2005). (Although there are exceptions to this trend, such as Priest (1979, 1987/2006), Field (2008) and Beall (2009)).
For example, Sobel (1992) and Glanzberg (2001) maintain the truth-schema for propositions, and certainly do not mean to embrace classical inconsistency. Indeed, Glanzberg goes so far as to write: I doubt that anything that failed to validate [it] could count as a reasonable theory of propositions (2001, p. 228).
However, if there is a mismatch between the surface and the logical form of the sentence, then it is the structure of the latter that is mirrored.
The transitive closure of a set x is the set whose members are the members of x, the members of the members of x, the members of the members of the members of x, etc.
In support of the identification of N(s(0)) and N(1) one might note that it is natural to describe the sentences ‘s(0) is a number’ and ‘1 is a number’ as being ‘about the same thing’ (i.e. 1). However, there are many other cases where we would give a comparable description, but where we would certainly not want to say that the things in question are constituents of the propositions expressed. For example, it is natural to describe ‘all odd primes are \(\varphi \)’ and ‘all primes greater than two are \(\varphi \)’ as being ‘about the same things’, but we would not want to say that these infinitely many numbers are constituents of the propositions in question.
That is, f is an \(\omega \)-ary function, with a place for each natural number. If desired, one could give a similar example using a unary function from sequences of propositions.
The more complicated case, where each proposition says something about all subsequent ones, is handled as follows. Here is an example where each proposition says that all later ones are untrue (giving a propositional version of Yablo’s paradox).
$$\begin{aligned} q_n'= & {} \lnot T(f(I_{n+1},I_1,I_2,\dots ), f(I_{n+2},I_1,I_2,\dots ), \dots )\\ q_n= & {} \lnot T(f(q_{n+1}',q'_1,q'_2,\dots ), f(q'_{n+2},q'_1,q'_2,\dots ), \dots ) \end{aligned}$$Here \(\lnot T(p_1,p_2,\dots )\) is shorthand for: \(\lnot T(p_1) \wedge \lnot T(p_2) \wedge \dots \). The sequence \(q_1,\dots ,q_n,\dots \) is then as required. One could give a similar example without infinite conjunctions, but for reasons of space I omit the details.
See, e.g., Sobel (1992) and Glanzberg (2001). I should note that in that work Glanzberg represents propositions as sets of possible worlds. But he is quite clear that this is merely a simplifying assumption, and that the claims of the work are not supposed to make essential use of this. He writes:
The Liar paradox...is insensitive to issues of how finely structured propositions must be. Thus, we may take the possible worlds view of propositions as at least a simplifying assumption, regardless of whether the familiar arguments, such as those of (Soames 1987), ultimately show propositions to be structured entities (2001, p. 245).
This difficulty is mentioned in Kaplan (1989, p. 496). Kaplan does not say in any detail how it should be solved, but the solution below is in the general direction that he suggests.
For example, one can define the n-tuple of \(x_1\), ..., \(x_n\) as \(\{\llbracket 1, x_1\rrbracket , \dots , \llbracket n, x_n\rrbracket \}\), where \(\llbracket i, x_i\rrbracket \) is \(\{\{i\}, \{i, x_i\}\}\).
Note that the defined use of ‘constituent’ is slightly narrower than that of previous sections. For on the former 0 is not a constituent of the proposition \(\langle N, [0]\rangle \); rather, it is a constituent of a constituent of this proposition.
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Acknowledgments
For comments and discussion, I am grateful to Andrew Bacon, Agustín Rayo, Ian Rumfitt, Zoltán Szabó, an audience at the 2014 Eastern APA, and two referees for this journal.
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Whittle, B. Self-referential propositions. Synthese 194, 5023–5037 (2017). https://doi.org/10.1007/s11229-016-1191-0
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DOI: https://doi.org/10.1007/s11229-016-1191-0