Abstract
Kripke’s theory of truth (Kripke, The Journal of Philosophy 72(19), 690–716; 1975) has been very successful but shows well-known expressive difficulties; recently, Field has proposed to overcome them by adding a new conditional connective to it. In Field’s theories, desirable conditional and truth-theoretic principles are validated that Kripke’s theory does not yield. Some authors, however, are dissatisfied with certain aspects of Field’s theories, in particular the high complexity. I analyze Field’s models and pin down some reasons for discontent with them, focusing on the meaning of the new conditional and on the status of the principles so successfully recovered. Subsequently, I develop a semantics that improves on Kripke’s theory following Field’s program of adding a conditional to it, using some inductive constructions that include Kripke’s one and feature a strong evaluation for conditionals. The new theory overcomes several problems of Kripke’s one and, although weaker than Field’s proposals, it avoids the difficulties that affect them; at the same time, the new theory turns out to be quite simple. Moreover, the new construction can be used to model various conceptions of what a conditional connective is, in ways that are precluded to both Kripke’s and Field’s theories.
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Notes
I ignore languages and theories that do not fulfill the syntactic requirements necessary for Tarski’s Theorem on the Undefinability of truth to hold for them. See Tarski [39].
By “laws” I refer to inference schemata without premisses, and by “rules” to the inference schemata that have premisses. The term “principle” refers to laws and rules alike.
The predicate T, syntactically, may apply to every \(\mathcal {L}_{T}^{\rightarrow }\)-term; in practice, though, I will be mostly interested in its applications to closed \(\mathcal {L}_{T}^{\rightarrow }\)-terms coding closed \(\mathcal {L}_{T}^{\rightarrow }\)-formulae – I will introduce the conventions about coding in a moment.
For \(\mathcal {L}\) and P A, see Kaye [27]. Adopting \(\mathcal {L}\) and P A is purely a matter of convenience and causes no loss of generality, many other languages and theories that are syntactically expressive enough would do.
Here and in what follows, I will use results from the theory of inductive definitions. For the general theory and the relative notational conventions see Moschovakis [35].
Kripke, in fact, considered expansions of models of \(\mathcal {L}\) to models of \(\mathcal {L}_{T}\), and not expansions of models of \(\mathcal {L}^{\rightarrow }\) to models of \(\mathcal {L}^{\rightarrow }_{T}\). However, I present the construction for \(\mathcal {L}_{T}^{\rightarrow }\) rather than \(\mathcal {L}_{T}\), since I will use it later to interpret →. This change induces no significant differences. For detailed analyses of Kripke’s construction, see Halbach [25], Horsten [26], Kremer [28], McGee [34], Soames [38].
I thank an anonymous referee for helping me to clarify this point. Other evaluation schemata considered by Kripke include supervaluationism and weak Kleene logic: I do not consider them since supervaluationism does not yield a paracomplete theory, while weak Kleene logic would not mingle well with my treatment of →. Some evaluation schemata for the logic vocabulary that admit (a generalized version of) Kripke’s treatment are studied in Feferman [13], applying results of Aczel and Feferman [1].
The clause “\(dec(t) \notin \mathcal {L}_{T}^{\rightarrow }\)” doesn’t make Definition 1 non-inductive, as \(\textit {SENT}_{\mathcal {L}_{T}^{\rightarrow }}\) is hyperelementary.
Of course Φ(S 1, S 2) is just a shorthand for Φ(〈S 1, S 2〉).
For K 3, see Blamey [5]. I use 1 (0, 1/2) for his ⊤ (⊥, ∗).
This is well-known: see, e.g., McGee [34], Corollary 5.11, p. 113.
The extent and depth of such “silence” are quite radical: see Field [18], p. 72. Clearly, C-Gaps are “visible” in Kripke’s theory from a purely set-theoretical standpoint, but I am considering only what (consistent) Kripke fixed points can yield. I thank an anonymous referee for pointing out to me the relevance of Horsten [26] on the question of the radical silence under discussion. See, for example, Horsten’s observations on the theory P K F (an axiomatization of Kripke’s theory), in Chapter 10.2.2, pp. 144-146.
For similar reasons, I do not consider explicitly the still different theory in Field [14].
Field [18], p. 267.
Field’s construction uses always countable ω-models of \(\mathcal {L}\), but there is exactly one such model (up to isomorphism). So, if the definition of ⊧ F quantifies over more than one evaluation, it must quantify over the functions \(|\,|_{F}^{\mathcal {M}}\) that are just like | | F with the exception of using a different Kripke fixed point at revision stages. Field is not explicit as to which Kripke fixed points can be used in his theory (see [18], p. 249), and it is natural to think that some Kripke fixed point will not do. However, we can suppose that the Fieldian theorist can place a restriction on the quantification over such Kripke fixed points, if needed.
Clearly, the schema φ ⇔ φ plus Intersubstitutivity of truth yields the unrestricted Tarski schema: \((TB)\;\;\varphi \leftrightarrow T \ulcorner \varphi \urcorner \), thus addressing problem (K2) of Kripke’s theory as well.
Beall [3], Preface, p. viii.
Due to naïve truth, we have to abandon several long-standing logics (classical, intuitionistic, and so on) with their well-established conceptual foundations: this makes particularly urgent the need to justify the rather special and limited set of principles validated by theories of naïve truth.
Various natural language counterexamples to conditional principles commonly accepted in many logics are known. It may be of interest to note that in the literature there is a remarkable gap between theories of conditionals for natural languages and accounts of conditional connectives in pure logic. An analogous gap between theories of truth for natural languages (or even broad views on truth, such as correspondence, coherence, deflationism, and so on) and formal theories of truth exists but is much narrower.
This is the case of Kripke’s theory, for example. Depending on the specific setting, this reading can accommodate also multiple designated truth-values: for example, if all the truth-values are linearly ordered and there is one greatest designated value, one can read the disjunction as yielding the truth-value that is closer to it, with the same proviso as above. However, I will not consider the case of multiple designated truth-values, as I am only concerned with semantics with one designated truth-value in the present work.
The truth-ordering in the four-valued lattice for First Degree Entailment is an example of such a case (see Belnap [4], pp. 13-16). Thanks to an anonymous referee for some useful remarks on this point.
For Kripke’s theory this is quite obvious; for Field’s theory, see [18], pp. 259-260.
Seeing connectives and quantifiers as providing order-theoretic information about truth-values lends itself quite well to some specific interpretations of the truth-values themselves, e.g. degrees of credence.
See p. 9, the comment after Theorem 9. A similar problem affects the theory in Field [20].
Field [18], p. 261.
See Subsection 6.1.
This can be remedied in a variant of Field’s theory where κ is treated as λ (see [18], p. 271). However, if one adopts naïveté and argues that λ and κ should be treated in the same way, then she presumably accepts that negating a sentence is equivalent to saying that from that sentence a falsity follows via the conditional, as T is treated naïvely. But this general fact does not hold in Field’s modified theory either: for some \(\varphi \in \mathcal {L}_{T}^{\rightarrow }\), the sentences φ→0≠0 and ¬φ get a different value in the modified construction. Moreover, the value space of the modified construction is still F, so the more general considerations involving the ordering relations in F carry over to the modified theory as well.
Famously, some authors claim that such rules give the meaning of connectives and quantifiers. This kind of view has been developed by authors such as Gentzen, Dummett, Prawitz, Tennant, Read.
See Field [18], pp. 281-283. Let me emphasize that I do not consider substructural notions of logical consequence, nor theories that accept inconsistencies.
Fischer, Halbach, Speck, Stern [21] contains a detailed analysis of adequacy criteria. As they show, the criterion adopted here has a preeminent role between other proposed characterizations of adequacy.
One surely wants these rules, at the very least, if she is to attempt a characterization of Field’s theory.
See Martin [32].
A position not far from the latter view is expressed in Martin [32] “[we don’t know] what theory [Field’s] construction yields a [consistency] proof for.” (p. 343); “[...] I don’t see how [Field’s conditional] is a generalization; that is, I don’t see what generalization it is supposed to be. In the end, all we are given is the model-theoretic construction and a (necessarily very partial) list of the laws and nonlaws. Contrast this with the connectives in the Kripke case. I would go so far as saying that Kripke’s disjunction and conjunction are the classical disjunction and conjunction.” (ibid., p. 345).
McGee [34], p. 87, footnote 1.
The starting hypotheses for values 1 and 0 cannot be eliminated. One can give all the conditions under which a sentence gets value 1 (0) in a K 3 evaluation using only clauses about value 1 (0) in it (see Halbach [25], pp. 202 ff.) but this does not hold for 1/2: e.g. not all the conditions under which φ∧ψ has value 1/2 in a K 3 evaluation can be given using only its subsentences or negated subsentences having value 1/2.
This answers Martin’s question “of what is the new conditional a generalization” for this construction.
For some epistemic and algorithmic aspects of Kripke’s theory, see Cantini [10], p. 69 and following.
Writing P ∞ alone is not meaningful, since to know what is in it we must know the other members of the fixed-point triple. However, when writing P ∞ (or Q ∞ , or R ∞ ) alone, the triple will always be clear.
A similar fact is given for evaluations based on any Łukasiewicz logic.
I thank an anonymous referee for pointing out that consistent fixed points of Υ show that one can use a conditional very similar to Łukasiewicz’ one and get ω-models validating Intersubstitutivity of truth.
Of course, I am not inciting to abandon schematic laws: I am arguing that, in addition to the principles that a theory of truth validates, there are also conceptually motivated criteria of philosophical significance that should play a role in deciding whether to accept a theory or not. When those criteria are not met (as I have argued that it is the case for Field’s theory), they should outweigh the importance of schematic laws.
See the discussion after Proposition 4.
Contrast this fact with item (a) in Subsection 3.3. Note that in every consistent fixed point of Υ, for every \(\varphi \in \mathcal {L}_{T}^{\rightarrow }\), the sentence ¬φ has the same value of the sentence φ→0≠0, while Field’s theory does not validate this general equivalence (see footnote 33). There are good reasons to accept this equivalence, such as its classical validity. If one agrees on this equivalence and interprets naïvely the truth predicate, it is natural to see λ and κ as equivalent between them, as λ is equivalent to ¬λ and κ is equivalent to κ→0≠0 (modulo Intersubstitutivity of truth).
Let me thank an anonymous referee for highlighting the importance of this point to me.
Clearly, the situation would be entirely different if we adopted a different interpretation of partiality, e.g. the view embodied by the weak Kleene scheme.
An anonymous referee suggested that the line of thought I follow here could be used to argue in favor of “another inductively defined gap” (and suitable extensions of the treatment of the conditional), “in addition to the P-gaps”, along the lines offered by Roy Cook in his [11]. Cook propounds an indefinite extensibility theory in order to address revenge paradoxes. Cook’s theory presents a progression of larger and larger languages, and of more and more truth-values. The referee, then, seems to indicate that my P-gaps could be subject to similar extensions. This suggestion is particularly interesting, and I would like to address it in a future work.
I thank an anonymous referee for some useful suggestions on this point.
In Definition 27, I consider only consistent fixed points of Υ, to avoid losing central pieces of semantic reasoning. There are sentences \(\varphi , \psi \! \in \! \mathcal {L}_{T}^{\rightarrow }\) and sets V of fixed points of Υ (including inconsistent fixed points) s.t., defining \(\models ^{\mathit {V}}_{\textit {{\L }}K}\) as in Definition 27, we have that \(\models ^{\mathit {V}}_{\textit {{\L }}K}\! \varphi \!\! \rightarrow \!\! \psi \) but \(\varphi \not \models ^{\mathit {V}}_{\textit {{\L }}K}\!\! \psi \), which upsets a fundamental relation between conditional assertion and assertion of the conditional. It is possible to liberalize Definition 27 to include some inconsistent fixed points of Υ without this disastrous consequence, but this does not seem very interesting, as we cannot use arbitrary sets of fixed points of Υ. Identical remarks hold for an analogue of Definition 27 for Kripke’s theory and the K 3 material conditional ⊃.
This may be not entirely uncontroversial, but I cannot discuss this point here.
To simplify the notation, let P = ∅ = Q: the general case is immediate from this one (the proof sketch continues on the next page).
Proof (Sketch)
$$\begin{array}{@{}rcl@{}} &&\langle \mathfrak{E}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow}| \left\{\begin{array}{l} \varphi \in \mathfrak{A}_{\Phi} \text{ or}\\ \psi \in \mathfrak{E}_{\Phi} \text{ or}\\ \varphi, \psi \in \mathfrak{H}_{\Psi} \end{array}\quad\right.\};\: \\ \mathfrak{I}_{\Upsilon}^{1} &=& \mathfrak{A}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow}|\varphi \in \mathfrak{E}_{\Phi}, \psi \in \mathfrak{A}_{\Phi}\}; \\ &&\mathfrak{H}_{\Psi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \left\{\begin{array}{l} \varphi \in \mathfrak{E}_{\Phi}, \psi \in \mathfrak{H}_{\Psi}, \text{or}\\ \varphi \in \mathfrak{H}_{\Psi}, \psi \in \mathfrak{A}_{\Phi} \end{array}\quad\right. \}\rangle\\ &=&\langle \mathfrak{E}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \neg \varphi \in \mathfrak{E}_{\Phi} \text{ or} \psi \in \mathfrak{E}_{\Phi}\};\: \mathfrak{A}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \varphi \in \mathfrak{E}_{\Phi}, \neg \psi \in \mathfrak{E}_{\Phi}\};\: \emptyset \rangle\\ &=&\langle \mathfrak{E}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \neg \varphi \vee \psi \in \mathfrak{E}_{\Phi}\};\: \mathfrak{A}_{\Phi} \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \varphi \wedge \neg \psi \in \mathfrak{E}_{\Phi}\};\: \emptyset \rangle. \end{array} $$
$$\begin{array}{@{}rcl@{}} &&\langle\mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow}| \left\{\begin{array}{l} \varphi \in \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \,\,\text{or}\\ \psi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha})\,\, \text{or}\\ \varphi, \psi \in \mathfrak{H}_{\Psi}(\emptyset,P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \end{array}\quad\right.\};\\ \mathfrak{I}_{\Upsilon}^{\alpha+1} &=& \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow}|\varphi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}), \psi \in \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha})\}; \\ && \mathfrak{H}_{\Psi}(\emptyset, P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow}|\left\{\begin{array}{l} \varphi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}), \psi \in \mathfrak{H}_{\Psi}(\emptyset,P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \,\,\text{or}\\ \varphi \in \mathfrak{H}_{\Psi}(\emptyset,P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}), \psi \in \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \end{array}\quad\right.\}\rangle\\ &=&\langle\mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \neg \varphi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \text{ or} \psi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha});\\ && \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \varphi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}), \neg \psi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha})\}; \emptyset \rangle\\ & =&\langle \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \neg \varphi \vee \psi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha});\\ && \mathfrak{A}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha}) \cup \{\varphi \rightarrow \psi \in \mathcal{L}_{T}^{\rightarrow} \,|\, \varphi \wedge \neg \psi \in \mathfrak{E}_{\Phi}(P^{\emptyset}_{\alpha},Q^{\emptyset}_{\alpha})\}; \emptyset \rangle. \end{array} $$The proof is routine, see Cantini [9], Theorem 30.13 and Lemma 31.2.
t is a variant of the so-called McGee function (see Halbach [25], p. 157). This process holds for larger ordinals and closes at some ordinal \(\leq \omega _{1}^{\textsc {ck}}\), however saying something more precise on this point would require to deal with some complicated and ultimately not relevant issues on ordinal notations.
Admittedly, the ŁK-construction does not overcome limitation (K1), namely the absence of schematic laws: I argued that this is an acceptable price to pay at the end of Section 5.
Feferman [13] notes that the consistency of the biconditionals \(\varphi \leftrightarrow T \ulcorner \varphi \urcorner \) for the Łukasiewicz 3-valued conditional restricted to \(\varphi \in \mathcal {L}_{T}\) can be proven along the lines of the consistency proof of similarly restricted instances of naïve comprehension given by Brady [6]. This result, however, is immediate from Proposition 34, which gives a very simple proof of it. Feferman urges also for some improvement on Brady’s unsatisfactory restriction. He proposes to supplement classical logic with an intensional conditional →⋆ and its standardly defined biconditional ⇔⋆, s.t. all ⇔⋆-Tarski biconditionals hold (see Aczel and Feferman [1]). The semantics of →⋆, however, is quite disappointing: e.g., not even modus ponens is validated for it (Feferman [13], §11). Fixed points of the form \(\langle A_{\infty },B_{\infty },C^{\textsc {a,b}}_{\infty } \rangle \) may be seen as an answer to Feferman’s instigation to recover more Tarski biconditionals, with a conceptually motivated restriction of the excluded instances (also less draconian than Brady’s one), and a better-behaved conditional.
For the theory P K F, see Halbach and Horsten [24] (see also footnote 37).
This would be interesting in order to associate a logic to some collection of fixed points of Υ (by Corollary 30, presumably we do not want to consider all the fixed points of Υ). In the case of consistent fixed points extending \(\mathfrak {I}_{\Upsilon }(\emptyset ,\emptyset , C^{\textsc {pkf}})\), for example, it is clear how to describe such calculus. The arithmetical and truth-theoretic parts of the theory would be given by the axioms and rules of P K F, and its logic would consist of all the axioms and rules of P K F for ¬, ∧, ∨, and ∀ plus, for the conditional, modus ponens and the following rule for introducing the conditional: the theory proves ⇒ φ→ψ whenever the theory proves φ ⇒ ψ and one of the following is the case: (i) the theory proves ⇒ φ∨¬φ, (ii) the theory proves ⇒ ψ, (iii) the theory proves both φ ⇔ ¬φ and ψ ⇔ ¬ψ.
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Acknowledgments
I would like to express my gratitude to Volker Halbach, for all his encouragement and support at various stages of this work, and for the numerous helpful and beneficial discussions on the material of this paper. I am also obliged to Andrea Cantini, Hartry Field, Kentaro Fujimoto, Ole Hjortland, Leon Horsten, Harvey Lederman, Graham Leigh, Hannes Leitgeb, Pierluigi Minari, Carlo Nicolai, Graham Priest, James Studd, Philip Welch, Tim Williamson, and Andy Yu for many useful comments on this work. I am grateful to two anonymous referees for several observations and suggestions that led to improvements. Let me also thank the audiences at the University of Florence, the University of Oxford, the University of Bristol, the LOGICA 2014 Conference, the Humboldt University in Berlin, and the Technical University in Vienna for their precious feedback. Finally, I gratefully acknowledge the support of the Art and Humanities Research Council and of the Scatcherd European Scholarship.
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Rossi, L. Adding a Conditional to Kripke’s Theory of Truth. J Philos Logic 45, 485–529 (2016). https://doi.org/10.1007/s10992-015-9384-4
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DOI: https://doi.org/10.1007/s10992-015-9384-4