Abstract
A Curry paradox about truth is generated by the following sentence, written on the board in room 101:
-
If the sentence on the board in room 101 is true then 1 ≠ 1.
A Curry paradox about validity is generated by the following argument, written on the board in room 102:
-
The argument on the board in room 102 is valid. Therefore, 1 ≠ 1.
Though the sentence and the argument generate Curry paradoxes, they also generate more basic paradoxes, in a sense to be made clear. I argue that if we solve these more basic paradoxes, we have solutions to both kinds of Curry paradox. The positive proposal is in part inspired by a brief remark of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concepts of truth and validity, for example, are everywhere applicable except for certain singular points or singularities. A second central claim is that 'true' and 'valid' are context-sensitive predicates. This contextual-singularity approach to the Curry paradoxes applies also to other paradoxes of truth, validity, denotation, and predicate-extension. So a more general aim of the paper is to provide a unified response to semantic paradox.
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Notes
Assuming SL, we may deduce p from the negation of if p then q.
Assuming SL, we may deduce if p then q from the negation of p.
Paradoxes of denotation and of predicate-extension also generate repetitions. Suppose I write on the board just these three expressions: ‘pi’, ‘six’, ‘the sum of the numbers denoted by expressions on the board’. If we assume that the third expression denotes a number, say k, we reach absurdity: k = π + 6 + k. So the third expression is pathological, and fails to denote. But then, since the third expression fails to denote, we may conclude: the sum of the numbers denoted by expressions on the board is π + 6. And here we have a repetition of the third expression which is not pathological—it denotes a number, namely π + 6.
Or suppose I write on the board just these two predicates: ‘moon of the Earth’, ‘one-membered extension of a predicate on the board’. Suppose the second predicate has a determinate extension. Then we reach absurdity: the second predicate has a one-membered extension if and only if it has a two-membered extension. So the second predicate is pathological, and fails to have an extension. So we may conclude: the extension of the first predicate is the only one-membered extension of a predicate on the board. Here, in our conclusion, is a repetition of the first predicate, and it isn’t pathological—its extension is the extension whose only member is the Moon.
See Simmons (2018).
Op. cit., p. 150.
John MacFarlane characterizes relativism about truth this way: “To be a relativist about truth is to hold that languages with assessment-sensitive expressions are at least conceptually possible”. And “To be a relativist about truth in English (or some other natural language) is to hold that some expressions of English are assessment-sensitive” (MacFarlane 2014, p. 65). The case of T—and other cases of pathology—may be seen as counterexamples to MacFarlane’s characterization of relativism about truth.
This then is in opposition to those who take the Curry to be a different kind of paradox from the liar. Graham Priest claims that the Curry is different, because it has nothing to do with negation—see Priest (1995, p. 186). Weber et al. take the Curry to be different from the liar because the Curry does not fit the inclosure schema—see Weber et al. (2014, p. 823).
So the notion of validity that I am working with is to be contrasted with what might be called ‘logical validity’, the Tarskian notion where, roughly, validity is truth-preservation in virtue of logical form.
For further analysis of this validity paradox, see Simmons (2021).
For more discussion of these validity paradoxes, see Simmons (2021). Woodbridge and Armour-Garb (2008) call 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.
Here’s a variant of the argument V. Suppose the following sentence is written on the board in room 103:
- (P/):
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The argument from the sentence on the board in room 103 to ‘1 ≠ 1’ is valid.
Let the argument to which P/ refers be V/. Then we can reason about V/ exactly as we originally reasoned about V. Take the original reasoning 1–11 of Sect. 5, replace ‘V’ by ‘V/’ and ‘P’ by ‘P/’, and we obtain an argument that establishes that V/ cannot be assessed by the validity /P/ -schema (with the modest subscript in place). We establish that V/ and P/ are pathologicalP/, V/ is not validP/, and P/ is not true /P/ .
Curry reasoning parallel to 1–10/ of Sect. 6 goes wrong at the corresponding point. A repetition of V/ is obtained, and then it is falsely assumed that this repetition is assessable by VAL /P/ , as valid /P/ .
A version of P/ is the main focus of Beall and Murzi (2013). Beall and Murzi explore a substructural approach which rejects Structural Contraction. According to the present singularity proposal, classical logic is fully preserved.
Tarski (1969, p. 89).
Again, see Simmons (2018) for a fully developed singularity theory of truth, denotation, and predicate-extension.
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