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Minimalism, supervaluations and fixed points

Abstract

In this paper I introduce Horwich’s deflationary theory of truth, called ‘Minimalism’, and I present his proposal of how to cope with the Liar Paradox. The proposal proceeds by restricting the T-schema and, as a consequence of that, it needs a constructive specification of which instances of the T-schema are to be excluded from Minimalism. Horwich has presented, in an informal way, one construction that specifies the Minimalist theory. The main aim of the paper is to present and scrutinize some formal versions of Horwich’s construction.

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Notes

  1. 1.

    The symbols ‘<’ and ‘>’ surrounding a given expression e produce an expression referring to the propositional constituent expressed by e. Thus, when e is a sentence, ‘\(<e>\)’ means the proposition thate.

  2. 2.

    This characterization is not completely accurate. As Horwich admits, the theory should also have an axiom claiming that only propositions are bearers of truth (see Horwich 1998, fn. 7 on pp. 23, 43). Moreover, as we are going to see, some of the instances of the T-schema will not be included in the theory, due to the paradoxes of truth.

  3. 3.

    This is an exaggeration; strictly speaking, we will need other theories besides the theory of truth to explain all the facts about truth, because some of those facts will involve other phenomena. As Horwich says, Minimalism “provides a theory of truth that is a theory of nothing else, but which is sufficient, in combination with theories of other phenomena, to explain all the facts about truth” (Horwich 1998, pp. 24–25). See also the discussion below.

  4. 4.

    I assume that the reader is familiar with the Liar Paradox.

  5. 5.

    Whether bivalence is available to Horwich is a highly contentious matter. The fact that Minimalism does not contain the instance of the T-schema for the Liar sentence seems to be enough to block the derivation of this instance of bivalence from LEM (see, especially, Beall and Armour-Garb 2005, sec. 5.2 and Schindler 2018, sec. 4). For the sake of argument, in this paper I will suppose that Horwich can overcome this difficulty. One way to do that might involve understanding falsity as mere untruth. I will not pursue this issue further in this paper though. Thanks to two anonymous referees for prompting this clarification.

  6. 6.

    For further details see Horwich (1997, 2005).

  7. 7.

    See, for example, Horwich (2005, p. 102) and Armour-Garb (2005, p. 90 ff.) for more details of Horwich’s solution of the Liar Paradox.

  8. 8.

    Moreover, McGee also showed that none of the maximal consistent sets obtained is recursively enumerable and, hence, in this sense of ‘constructive’, the specification cannot be constructive. In what follows, I will suppose that Horwich’s does not expect his theory of truth to be recursively enumerable, which makes perfect sense, since he does not even think that it is a set! (See Horwich 1998, p. 20, fn. 4.) Thanks to Thomas Schindler for prompting this clarification.

  9. 9.

    This already raises some doubts concerning whether a deflationist can use the notion of groundedness in order to specify a theory of truth. For the moment, though, let us think of this construction as a mere technicality. I will return to this point later.

  10. 10.

    Given Lemma 2.1 it is enough to show that \(H_{\omega +1}\subseteq H_\omega \). To see this, suppose that \(\phi \in H_{\omega +1}\). In this case, by definition of \(H_{\omega +1}\), \(H_\omega \cup T_{H_\omega }\models \phi \). By compactness of first-order logic, there exists a finite \(\Delta \subseteq H_\omega \cup T_{H_\omega }\) such that \(\Delta \models \phi \). Since \(\Delta \) is finite, there will be a natural number n such that \(\Delta \subseteq H_n\cup T_{H_n}\). By monotonicity of first-order logic, it follows that \(H_n\cup T_{H_n}\models \phi \) and, hence, \(\phi \in H_{n+1}\subseteq H_\omega \), as desired.

  11. 11.

    This is a variant of a general problem Horwich has to face. For more details, responses and rejoinders on this issue see, for example, Gupta (1993a, b), Soames (1997, 1999), Armour-Garb (2004, 2010), Raatikainen (2005), Horwich (1998, p. 137), Horwich (2010b, pp. 43–45, 92–96) and Oms (2018).

  12. 12.

    See, for example, Horwich (2010a, p. 96, fn. 16).

  13. 13.

    If this were not the case, the construction would have to be adjusted via the addition of a fourth semantic value to represent the situation where there is no appropriate Y. Note that, if that fourth semantic value is not defined and there are no Y’s satisfying the condition \(\Phi \) and such that \(X\subseteq Y\subseteq \overline{D} - X^-\), then all sentences would trivially have the values 1 and 0. See Kripke (1975, p. 711) and Field (2008, p. 178) for more details.

  14. 14.

    For some criticisms to the use of that rule see Raatikainen (2005).

  15. 15.

    Question: is the inclusion proper?

  16. 16.

    What is important is that such principles do not use the truth predicate, for if they did, they would show that we need to go beyond the instances of the T-schema in order to explain all facts concerning truth.

  17. 17.

    Thanks to Elia Zardini for suggesting this line of thought.

  18. 18.

    Alternatively, we could define the set representing all we can know about truth with the restricted notion of logical consequence, \(\varvec{T}_\mathcal {N}=\{\phi : T_{\varvec{H}^*}\cup H^*_0\models _\mathcal {N}\phi \}\). In this case, \(\varvec{T}_\mathcal {N}=\varvec{H}^*\). That \(\varvec{T}_\mathcal {N}\subseteq \varvec{H}^*\) is proved as in Proposition 5.1. That \(\varvec{H}^*\subseteq \varvec{T}_\mathcal {N}\) is proved by showing that for every ordinal \(\sigma \), \(T_{\varvec{H}^*}\cup H^*_0\models H^*_\sigma \). The proof proceeds by induction on \(\sigma \). The base case and the limit case are clear; as for the successor case, suppose that \(T_{\varvec{H}^*}\cup H^*_0\models H^*_\sigma \). Take \(\phi \in H^*_{\sigma +1}\). By the definition of \(H^*_{\sigma +1}\), \(H^*_\sigma \cup T_{H^*_\sigma }\models _\mathcal {N} \phi \). Now, from the induction hypothesis and the fact that \(T_{H^*_\sigma }\subseteq T_{\varvec{H}^*}\), it follows that \(T_{\varvec{H}^*}\cup H^*_0\models _\mathcal {N} H^*_\sigma \cup T_{H^*_\sigma }\). Hence, \(T_{\varvec{H}^*}\cup H^*_0\models \phi \). For the purposes in this paper, I do not need to decide between \(\varvec{T}\) and \(\varvec{T}_\mathcal {N}\) because what I need is the result of Proposition 5.1, which holds in both cases.

  19. 19.

    This is not to say that \(\varvec{H}^*\) must be a maximally consistent set (which is not); \(\varvec{H}^*\) represents, as we have seen, the set of determinately true sentences (in Horwich’s epistemic sense) and what we are considering now is the set of true sentences simpliciter. Moreover, note that any model \(\mathcal {M}\) for \(\mathcal {L}^+\) with an extension of the truth predicate, \(Tr^\mathcal {M}\), such that \(\varvec{H}^*\subseteq Tr^\mathcal {M} \subseteq \overline{D} - \varvec{H}^{*-}\) satisfies \(T_{\varvec{H}^*}\)—which is the Minimalist theory of truth—and \(\varvec{H}^*\)—which contains all that can be known about truth. (This follows immediately from the fact that \(\varvec{H}^*=\varvec{VF}\).) Then, by the factivity of knowledge, it is reasonable to expect the extension of the truth predicate in the actual model to be a maximally consistent superset of \(\varvec{H}^*\). Thanks to a anonymous referee for prompting this clarification.

  20. 20.

    Thanks to José Martínez for suggesting this.

  21. 21.

    Not everything is good news, though; as Field (2008, p. 184 ff.) shows, some reasonable principles involving the quantifiers (e.g. \(\forall nTr\ulcorner \phi (n)\urcorner \rightarrow Tr\ulcorner \forall n\phi (n)\urcorner \)) will fail to hold even in \(\varvec{VF}^{mc}\). So the fixed point is not strong enough. Thanks to Lucas Rosenblatt for suggesting that point. Moreover, it is worth noticing that, as Schindler (2018) emphazises, some of the laws that hold in the theory based on maximally consistent extensions of the truth predicate do not hold if one moves to its classical close-off. This is a problem for Horwich’s demand that Minimalism be a classical theory of truth. Thanks to an anonymous referee for pressing that point.

  22. 22.

    It could be said that the construction for \(\varvec{H}^*\) implicitly uses compositional principles for truth in the relation of logical consequence and that, in consequence, Horwich should reject it too. This is not the case, though, as Horwich thinks that truth does not play any role in the foundations of logic (see Horwich 1998, pp. 74–76).

  23. 23.

    Thanks to Manuel García-Carpintero and Sven Rosenkranz for their helpful insights on this issue.

  24. 24.

    Thanks to José Martínez, Thomas Schindler, Elia Zardini and two anonymous referees for all their very stimulating comments and suggestions. I am also grateful to the audience at the LOGOS Seminar, especially to Manuel García-Carpintero, Lucas Rosenblatt and Sven Rosenkranz. During the writing of the paper, I have benefitted from the project FFI2015-70707P of the Spanish Ministry of Economy, Industry and Competitiveness on Localism and Globalism in Logic and Semantics.

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Oms, S. Minimalism, supervaluations and fixed points. Synthese 197, 139–153 (2020). https://doi.org/10.1007/s11229-018-1831-7

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Keywords

  • Minimalism
  • Deflationism
  • Paul Horwich
  • Saul Kripke
  • Fixed point constructions