Abstract
Radical glut and gap theorists deny—in opposite ways—that the liar sentence has exactly one of the two values true and not true. I describe a scenario where a signalman finds himself in a situation analogous to the liar paradox: if he lights a fire at a certain time, that is analogous to the liar being true, and if he does not, that is analogous to the liar not being true. It is obvious that he must make exactly one of those states of affairs come about. It is argued that there are no relevant differences between the liar and the signalman’s dilemma, implying that the glut and gap theorists are wrong about the former. A further point is that whether or not the liar is true/the signalman lights the fire, language/the signalman is misleading relative to the conditions under which the liar/the fire “ought” to be true/lit.
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Notes
In the case of the liar paradox, y ranges over sentences, x ranges over sets of sentences, \(\varphi (y)\) means that y is true, \(\psi (x)\) means that x is definable, and \(\delta \) is a function defined by diagonalisation in such a way that if x is definable then \(\delta (x)=\alpha \) where \(\alpha =\langle \alpha \notin x\rangle \), where \(\langle \phi \rangle \) now is a sentence that is true iff \(\phi \) is satisfied.
As is well known, there are also many other paradoxes that satisfy the Schema, but to keep things as simple and brief as possible, I will not generalize my conclusions regarding the liar to those other paradoxes.
Thus, my position has similarities to (but is not identical with) Mares’s (2004), who rejects metaphysical dialetheism while accepting the possibility of semantic dialetheism; and to Tahko’s (2009), who tries to formulate and defend a metaphysical version of the law of non-contradiction. Further, I think that the positions I am distinguishing between are represented by actual philosophers as follows. Radical glut: Priest (2006b), radical gap: Field (2008), moderate glut (as just explained): Beall (2009), moderate gap: Kripke (1975). However, differences of vocabulary and differences of opinion about related matters make this question of interpretation difficult.
While there is much I can agree with in Chihara (1979), the distinction between the truth property and the truth predicate can be used to show what is missing from his account. Chihara compares truth to glub, defined by “an animal is a glub if, and only if, it is not a mouse; and it is not a glub if, and only if, it is neither a mouse nor different from itself” (p. 592), and argues that they are similar: in particular, that they are both “inconsistent concepts”. But there is also a very important difference: the truth predicate is intended to correspond to what is a perfectly fine property (and only prevented from always doing so by self-reference). In contrast, there is no glub property.
However, you might call what I am defending a classical theory of states of affairs. And it does imply this weak thesis concerning classical logic: If a given set of declarative sentences contains only sentences with non-misleading truth conditions, then classical logic holds for that set.
It should be equally clear that there are certain details in the following that I cannot be certain of, and that their accuracy does not matter.
See Lewis (1969) for an account of how conventions can be instituted in the absence of antecedent language conventions, as well as a more in-depth account of what it is for a sentence to have the truth property.
To make it explicit: I am here contradicting deflationists like Beall (2009), who claim that there is no truth property independent of the truth predicate. I am also contradicting those who claim that truth is primitive (e.g., McGinn 2000); on the contrary, it is a complex property that depends both on facts about social conventions and facts about whatever those conventions have made the given sentence be about.
Such a language user would also misinterpret “he kicked the bucket” and most other idioms.
Such failures are consistent with English being computationally compositional; a property that may be used to explain the possibility of language acquisition. Computational compositionality does not require that the truth value of \(\lnot \phi \) or \(T(\ulcorner \phi \urcorner )\) be a function of the truth value of \(\phi \), only that the meaning of \(\lnot \phi \) and \(T(\ulcorner \phi \urcorner )\)—together with the meaning of any other of the potential infinity of sentences—can be “worked out” from knowledge of finitely many rules.
If so, one might question whether it should be called a “sentence”. However, I am using that term in a wide sense that tracks whether there is a compositional intention. I did something similar with “truth predicate”, “negation (operator)”, and “liar sentence” above.
A comparison with Russell’s barber may be helpful here. Usually, the solution to that paradox is expressed simply as “there is no such barber”. While this also fits with our other cases—there is no sentence that has as its actual truth conditions that it is not true; there is no instant of time t such that Kuvata actually lights a fire at t iff not—it is more illuminating if we think of the paradox of the barber slightly differently. Instead of defining the barber in the usual way, define him to be someone living in a village where all the inhabitants share a sincere intention to follow a convention according to which the barber shaves all and only those who do not shave themselves. Such a barber could exist. Then the solution is, instead, that there is a difference between the intended convention and the actual convention, which might be, e.g., that the barber shaves all and only those who do not shave themselves, except that he also shaves himself. All, or most of, the villagers may be ignorant of the fact that there is (and must be) a discrepancy between the intended and the actual convention, and that the latter is not uniquely determined by the former, but requires further psychological determiners. All of this is closely analogous to the situation of the liar. (It is slightly less closely analogous to the situation of Kuvata, but only because I allowed commands to replace conventions in that scenario in order to make it more concrete.)
References
Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.
Beall, J. (2011). Dialetheists against Pinocchio. Analysis, 71(4), 689–691.
Chihara, C. (1979). The semantic paradoxes: a diagnostic investigation. Philosophical Review, 88(4), 590–618.
Eldridge-Smith, P. (2011). Pinocchio against the dialetheists. Analysis, 71(2), 306–308.
Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.
Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72(19), 690–716.
Lewis, D. (1969). Conventions. Cambridge: Harvard University Press.
Mares, E. (2004). Semantic dialetheism. In G. Priest, J. Beall, & B. Armour-Garb (Eds.), The law of non-contradiction. Oxford: Oxford University Press.
McGinn, C. (2000). Logical properties. Oxford: Oxford University Press.
Priest, G. (1994). The structure of the paradoxes of self-reference. Mind, 103(409), 25–34.
Priest, G. (1995). Beyond the limits of thought. Oxford: Oxford University Press.
Priest, G. (2006a). Doubt truth to be a liar. Oxford: Oxford University Press.
Priest, G. (2006b). In contradiction (2nd ed.). Oxford: Oxford University Press.
Priest, G. (2010). Badici on inclosures and the liar paradox. Australasian Journal of Philosophy, 88(2), 359–366.
Tahko, T. (2009). The law of non-contradiction as a metaphysical principle. Australasian Journal of Logic, 7, 32–47.
Acknowledgements
I gratefully acknowledge the help and feedback I have received from Thomas Brouwer, Andreas Fjellstad, Martin Rønberg, Naja Rønberg, Øystein Linnebo, Filippo Ferrari, Gerard Hough, Crispin Wright, Robert Trueman, Shay Logan, James Shaw, Eirik Gjerstad, Eric Epstein, Luisa Venancio, Silvia Jonas, Olla Solomyak, Georgie Statham, Sharon Berry, Elena Fiecconi, Elvira Di Bona, Claire Benn, Daniel Telech, Leora Katz, and the anonymous reviewers, as well as audiences in Cambridge, Copenhagen, Aarhus, Ascona, Warwick, Krakow, Bologna, Jerusalem, Witwatersrand, Salzburg, and Vichy. The title of this paper was inspired by those of Eldridge-Smith’s (2011) paper about the Pinocchio paradox and Beall’s (2011) reply.
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Hansen, C.S. The signalman against the glut and gap theorists. Synthese 198, 10923–10937 (2021). https://doi.org/10.1007/s11229-020-02759-0
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DOI: https://doi.org/10.1007/s11229-020-02759-0