Abstract
Consider the following argument written on the board in room 227: 1 = 1. So, the argument on the board in room 227 is not valid. This argument generates a paradox. The aim of this paper is to present a resolution of this paradox and related paradoxes of validity, including a version of the Curry paradox. The proposal stresses the close connections between these validity paradoxes and paradoxes of truth (such as the liar) and paradoxes of denotation (such as Richard’s, Berry’s and König’s). So a more general aim is to provide a unified response to semantic paradox. The positive proposal is in part inspired by a brief, tantalizing suggestion of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concept of validity, for example, is everywhere applicable except for certain singular points or singularities. A second central claim is that ‘valid’ is a context-sensitive predicate. The key notions of this contextual-singularity theory are presented and applied to a variety of cases. Any purported solution to paradox must deal with the phenomenon of revenge, and a response to revenge is outlined. The paper closes with remarks about the accommodation of Tarski’s claim that natural languages are universal.
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Notes
1. Gödel, in Schilpp (1944, p. 150).
Jeffrey Ketland has argued for a reasonable and consistent theory of logical validity (Ketland, 2012). Roy Cook has argued that there is no paradox of logical validity, because the rules governing logical validity are not themselves logically valid (Cook, 2014). I am grateful to an anonymous reviewer for these references, and for stressing the point that we have two different notions of validity, and so two different settings for validity paradoxes.
Paradoxes of denotation and of extension also generate repetitions. For more on repetitions, see Simmons (2018).
I’ve argued elsewhere that the semantic expressions ‘true’, ‘denotes’, and ‘extension’ are context-sensitive expressions. See Simmons (2018).
See Grosz and Sidner (1986).
Lewis (1979, p. 347).
Similarly with ‘true’ in the case of the Liar: L* is true, but not trueL. There is a shift in the extension of ‘true’.
In general, if we are assessing an argument by the validα-schema, we are assessing whether the argument is (necessarily) truthα-preserving. And for that we’ll need to determine whether the conclusion is trueα if the premises are trueα. One cannot assess an argument for validityα by assessing its premises and conclusion by a schema other than the truthα-schema. I’m grateful to the same anonymous reviewer for pressing on this point.
See MacFarlane (2014, p. 23).
Op. cit., p. 150.
It’s easy to check that the primary tree for A* is identical to a secondary tree for A. This indicates that A can be rehabilitated—like A*, A can be assessed, as validR, by the validR-schema.
An analogue of argument A is generated by the following sentence on the board in room 232: (S/) The argument from ‘1 = 1’ to the sentence on the board in room 232 is not valid.
A version of S is the main focus of Beall and Murzi (2013). Beall and Murzi explore a substructural approach which rejects Structural Contraction. According to the present proposal, classical logic is fully preserved.
In exact parallel with V, there is a validity Curry generated by B. Initially, we assume Q—that B is validQ, where the subscript is modest—and reason to ‘1 ≠ 1’. The distinctive Curry step is now to infer, given this reasoning, that B is validQ. But the analogue B* of V* establishes that no argument from Q to ‘1 ≠ 1’ is validQ.
There is a validity Curry generated by argument A too. Consider again A*.
Keep (1)–(6) of A* as they are. But now observe that we’ve just established the conclusion (6) from the premise at line (2). That is, we can establish ‘A is not validC’ from ‘1 = 1’. So A is validC.
Here is the Curry step, introducing the predicate ‘validC’. (The reasoning continues, using TRC and VALC, to the absurd conclusion that 1 ≠ 1.) It is the Curry step where things go wrong: A* shows that no argument from ‘1 = 1’ to ‘A is not validC’ is validC.
Read identifies Pseudo-Scotus as a fourteenth century author who considered an argument like A, and took it as an attack on VAL (see Read 2001, pp. 184–185). Read resists this attack on VAL, since the Curry reasoning here employs only one direction of VAL (from left to right), and there are other plausible culprits—for example, the self-referential character of A. Read suggests, and I agree, that any proposed solution should recognize the connection between A and other semantic paradoxes like the liar. Read also claims that whatever this solution is, it will undercut Pseudo-Scotus’s attack on VAL (see Read 2001, pp. 187, 194). But by the lights of the present account, Pseudo-Scotus has it right. The distinctive Curry step assumes that any deduction is validC—but A* shows otherwise. No argument from ‘1 = 1’ to ‘A is not validC’ can be assessed by VALC. It is the application of VALC to A that is the culprit.
This treatment of the validity-teller runs parallel to the contextual-singularity account of the Truth Teller:
(T) (T) is true.
Employing a modest subscript in the usual way, T is not assessable by the schema TRT, and T is identified as a singularity of ‘trueT’ by T’s primary tree. Once T is removed from the extension of ‘true’, T may be reflectively assessed as not true—just as VT is reflectively assessed as not valid. For the contextual-singularity account of the Truth-Teller, see Simmons (2018).
Woodbridge and Armour-Garb (2008) provide examples of both the inconsistency and the indeterminateness of validity, including the case of the loop we’ve just considered, which we took to generate inconsistency. They point out that in this kind of loop, one could maintain consistency by rejecting the claim that the arguments have the same validity status. Then we’re landed in indeterminateness, since it is indeterminate which argument is valid and which is not. It’s natural to argue here that the symmetry of the loop forces the two arguments to have the same validity status.
But then Woodbridge and Armour-Garb go on to provide the case of an asymmetric loop which generates inconsistency if the two arguments have the same validity status, and indeterminateness if they don’t—and where there is no symmetry to reduce the case to one of inconsistency (see Woodbridge and Armour-Garb, 2008, pp. 69–71). There are, then, two forms of resistance to an adequate semantic characterization of validity, which are “two symptoms of a single underlying condition, which prima facie calls for a single treatment of validity’s pathological features” (op. cit., p71). The suggestion of the present approach is that the underlying pathological feature is ungroundedness, and the treatment is to identify the singularities of the context-sensitive validity predicate.
A full, general theory of singularities for the notions of denotation, extension, and truth is provided in Simmons (2018, Chapter 6.) This provides a framework for a singularity theory of validity.
A fuller account of revenge for the notions of denotation, extension and truth, drawing on a fully specified singularity theory, is presented in Simmons (2018, Chapter 9).
A ‘Tarskian’ account of truth stratifies ‘true’, providing extensions of ‘true’ at distinct levels; in Kripke’s theory, the minimal fixed point is a model of the object language, providing the extension and anti-extension of ‘true’. There is nothing analogous in the language T of the singularity theory.
Compare: “Der Schnee ist weiss” is not an English sentence, but it is in the extension of the English predicate “sentence”.
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