Abstract
We argue that distinct conditionals—conditionals that are governed by different logics—are needed to formalize the rules of Truth Introduction and Truth Elimination. We show that revision theory, when enriched with the new conditionals, yields an attractive theory of truth. We go on to compare this theory with one recently proposed by Hartry Field.
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Notes
^{1} Here and below, we treat quotation names as falling among the logical constants; their interpretation does not shift in assessments of validity.
^{2} The above stipulations concerning logical resources will remain in effect for all formal languages considered below. Note that the exclusion of function symbols implies no loss of expressive power, for an (n+1)ary predicate can do the expressive work of an nary function symbol. The elimination of function symbols is merely a convenience; nothing essential hinges on it. Note also that, here and below, we often use symbols autonymously.
^{4} Revision theories of truth were discovered, independently, by Anil Gupta and Hans Herzberger, with seminal contributions by Nuel Belnap. See Gupta [28], Herzberger [36], and Belnap [8]; see also Yaqūb [79] and Chapuis [16]. The idea that truth is a circular concept, of which Aristotle’s Rules are partial definitions, was first put forward in Gupta [29].
^{5} We are setting aside hierarchical/contextual conceptions of truth because we do not think that they are descriptively adequate, partly for reasons given in Gupta [28]. See Tarski [69–71], Parsons [51], Burge [12], Barwise and Etchemendy [6], and Glanzberg [27] for different articulations of this conception. Skyrms [65], Gaifman [25, 26], and Simmons [64] develop theories that are broadly contextual but rather different from the mainstream contextual theories.
^{7} The classical material conditional reading is problematic even when paradoxical selfreference is absent—that is, even when (TI) and (TE) yield no inconsistency. See Martin [43, p. 198].
^{8} For an account of the Strong Kleene scheme and for a more detailed presentation of fixedpoint theories, see Gupta and Belnap [34, ch. 2].
^{9} Martin and Woodruff showed the existence of fixed points for the Weak Kleene scheme; Kripke showed that fixed points exist for all monotonic schemes, including Strong Kleene and supervaluation schemes.
^{10} This point is a variant of an observation in Gupta [28].
^{11} If the truth theory is based on the least fixed point of the Strong Kleene scheme, then (A8) and (A9) are also ruled invalid, while the inference in (A10) from the TruthTeller to the Liar is deemed valid. Some of these flaws can be removed by shifting to different fixed points or by shifting valuation schemes, but only at the cost of generating other flaws. As far as we can tell, no fixedpoint theory fully meets the Descriptive Adequacy requirement. This point holds for the fixedpoint theories made available by Priest [53], Visser [72, 73], and Woodruff [77].
^{12} For dialetheic languages: these conditionals do not turn out to be solely true.
^{13} Martínez–Fernández [45] has characterized exhaustively the threevalued truthfunctional languages that admit the fixedpoint property.
^{15} The idea that the conditionals in (TI) and (TE) might be different ordinary conditionals is unmotivated, and we set it aside.
^{16} The review is necessarily compressed. For a fuller exposition of the theory, see Gupta and Belnap [34].
^{17} Revision theory provides schemes for interpreting systems of interdependent definitions of terms of all categories. Furthermore, it is straightforwardly generalizable to systems of partial definitions. For notational simplicity, we work with a system consisting solely of one definition that circularly defines a oneplace predicate G.
^{18} The notion of hypothesis we need is more complex than in earlier revision theories because of the forthcoming introduction of the conditionals. Our hypotheses must provide not only interpretations for the defined term, they must also provide bases for the evaluation of the new conditionals. If the hypotheses were not suitably restricted, the behavior of the conditionals would be unacceptable. We formulate a suitable restriction below. In the absence of the new conditionals, hypotheses may be identified with subsets of the domain D.
^{19} That is, A and C(x _{1},…,x _{ n }) are exactly like except perhaps for a difference in bound variables. We assume that substitution creates no clash of variables. So, x _{1},…,x _{ n } must be all and only free variables of A
^{20} In Gupta and Belnap’s other system, S ^{∗}, validity receives a simpler definition: a sentence is valid iff it is true under all recurring hypotheses. However, the logic is considerably weaker, and calculus C _{0} is no longer sound.
^{21} In reading these conditionals, it is a useful mnemonic to think of the lefthand side (which corresponds to the definiendum in a definition) as higher and the righthand side (which corresponds to the definiens) as lower. So, in (B→C) we are stepping down as we move from the antecedent to the consequent, whereas in (C←B) we are stepping up. Incidentally, we read (B→C) as “B rightarrow C,” “B stepdown C,” and also as “if B, C”; and (C←B) as “B leftarrow C,” “B stepup C,” and “C, if B.”
^{23} Note that every formula counts as a subformula of itself.
^{24} Because of the presence of step conditionals in \(\mathcal {L}^{+}\) , no definition—not even a noncircular one—yields finitely reflexive hypotheses. (See Example 3.2.2.) Hence, no definition meets the requirement for finiteness, as this requirement is set out in Section 3.1 . We can, by liberalizing the requirement, obtain a notion of finite definition suitable for \(\mathcal {L}^{+}\) (see Standefer [68]). Here, however, we have chosen to restrict the theorem to those definitions formulated in \(\mathcal {L}\) which count as finite definitions of \(\mathcal {L}\).
^{25} The full logic of circular definitions is highly complex (in the recursiontheoretic sense) and is not axiomatizable; see Kremer [38], Antonelli [2, 3], and Kühnburger et al. [40]. The complexity of revision theories of truth has been studied by, among others, Burgess [14], Löwe and Welch [42], and Welch [74].
^{26} The weaker sense is “validity in S _{ 0 }”; for the definition of this notion, see Gupta and Belnap [34, p.147]
^{27} See Standefer [68].
^{28} Here and in the rest of the present subsection, we assume that the ground language is classical.
^{29} The same is true of David Lewis’s counterfactual conditionals, though with these conditionals the force of ‘must’ depends in part on the antecedent. The entailment conditionals studied by Alan Ross Anderson, Belnap, and others also retain modal force. On the other hand, the conditional defended by Robert Stalnaker lacks modal force.
^{30} Note that □ carries no modal force. We use this symbol because many of the laws governing it have been studied in modal logic.
^{31} For example, all occurrences of □□F x y could be read as G x y and all occurrences of □F x y as H x y.
^{32} Here, and in the rest of this subsection, we assume that the ground language is classical.
^{33} We note that a distinction between two readings of ‘iff’ is also drawn in earlier discussions of revision theory. (See, Gupta and Belnap [34, p. 138].) The distinction drawn there is that between reading ‘iff’ as material equivalence and reading it as definitional equivalence. This distinction is certainly a significant one. It is, however, insufficient for the purposes of the theory of truth. It is insufficient to provide, for example, an interpretation under which is true. Neither the material reading of ‘iff’ nor the definitional one yields an interpretation that is true. For further examples, see (vi) below.
^{35} Revision theorists have claimed it as a virtue of their theory that truth behaves like a classical concept when there is no vicious reference in the language. For a helpful discussion of this claim, see Kremer [39] and Wintein [76]. We note that we are unable to accept the desideratum Kremer titles “MGBD” on p. 357 of his essay.
^{36} At the suggestion of an anonymous referee, we provide a couple of illustrations. Consider, first, argument (A9), which we may formalize thus: To establish the validity of this argument in \(\mathbf {C}_{\mathbf {0}}^{+}\), it suffices to derive (¬T l⊃(T l⊃T t))^{0} from (t=‘Tt’ )^{0} and (l = ‘¬T l’)^{0}. So, suppose ¬T l ^{0} and derive (T l⊃T t)^{0}. To do this, suppose T l ^{0} and derive T t ^{0}. Now, the availability of classical rules for formulas with the same index enables us to complete the derivation.
Consider, next, argument (A4), which we may formalize thus: To establish validity, suppose S ^{0}, and derive (T(‘E & S’)←E)^{0}. By ←I, it suffices to to suppose E ^{−1} and derive T(‘E & S’)^{0}. Since S is Tfree, by Index Shift, we obtain S ^{−1}. So, by & I, (E & S)^{−1}. The partial definitions governing truth now yield the desired T(‘E & S’)^{0}.
^{37} The step conditionals make available new axiomatic theories of truth that deserve study. Let us take note of three such theories. The background logic of all these theories is classical logic supplemented with the logical rules for the step conditionals.

(i)
The basic theory, T _{0}, consists of all the step Tbiconditionals.

(ii)
The next theory, \(\mathbf {T}_{1}^{\textnormal {BG}}\), is richer. It is formulated in an extended language that allows quantification in and out of quotes (see Belnap and Grover [9]), and it consists of all the step Tbiconditionals together with the generalization that says that all these biconditionals are true.

(iii)
The final theory, \(\mathbf {T}_{2}^{\textnormal {PA}}\), is formulated within arithmetized syntax and consists of the step Tbiconditionals and the generalization that says that all instances of (IT) are true.
These theories are of special interest because they are different expressions of the idea that the Tarski biconditionals are the fundamental principles governing truth. Furthermore, unlike many axiomatic theories of truth considered in the literature, these theories preserve the generalization function of truth in expressively rich languages (see Section 4.3 below).
For axiomatic theories of truth, see Friedman and Sheard [24], an early and important investigation. See also Halbach [35] and Horsten [37], which provide accessible and illuminating accounts of these theories. Peter Aczel, Andrea Cantini, and Solomon Feferman did pioneering work on axiomatic theories. See Cantini [15] for references and for a masterful overview of onehundred years of work on the paradoxes; see also Section 3.6 below.

(i)
^{38} The step Tbiconditionals do not imply generalized versions of these laws (e.g., “a conjunction is true iff its conjuncts are true”). They fail to imply also the semantic principle about universal generalization used in McGee’s ωinconsistency theorem. It is a matter of fundamental disagreement among theorists of truth whether semantic laws should be accepted even at the cost of ωinconsistency. We ourselves side in favor of the semantic laws. The point we wish to stress, however, is that one can endorse the step Tbiconditionals without endorsing the semantic laws that generate ωinconsistency. The readings we are offering of the conditional are neutral on how one responds to McGee’s theorem.
^{39} A further objection may be lodged against us. The very fact that we are introducing new conditionals, it may be said, is a problem for us. For we now need to provide an account of which ordinary uses of ‘if’ correspond to which conditionals—a task that is none too easy. Response: (i) If introducing new conditionals is a problem for us then it is a problem for all leading theories of truth, for they too find it necessary to introduce new conditionals. (ii) In ordinary uses, we do not distinguish—and, for the most part, it causes no trouble if we do not distinguish—the different conditionals. For if the constituent sentences are uniformly stable (i.e., stable in all revision sequences), the stepconditionals are equivalent to the material conditional. It is only in highly specialized contexts that there is a need to make distinctions, and it is the job of the theory of truth to spell out the distinctions and how they are to be deployed.
^{41} See Asmus [4].
^{42} Feferman points out that Heinrich Behmann informally explored as early as 1931 ideas that point toward these theories. See Feferman [22], section 14, which provides a brief history of typefree theories of truth and satisfaction.
^{43} Aczel and Feferman’s notational conventions are the opposite of ours. Their symbol for material equivalence is ⇔, and their symbol for the new nonclassical equivalence is ≡.
^{44} We wish to stress that Aczel and Feferman do not commit themselves to S(≡). Indeed, they themselves have expressed reservations about it.
^{45} That is, the inference rule “ A⇔B, B; therefore, A” is admissible in the sense explained above (Section 3.5.1(iii)). Note, however, that the rule does not hold unrestrictedly in hypothetical contexts.
^{46} Our presentation of Field’s ideas is based primarily on Field [23]. The study of Aristotle’s Rules (and, more generally, comprehension principles) interpreted via new conditionals within nonclassical logics has a long history. See Feferman [22] and Cantini [15] for references. See Dutilh Novaes [20] for a historical comparison. Recent contributors include Brady [10], Beall [7], Zardini [81], and Bacon [5].
^{47} Field imposes a “local determination” condition on hypotheses: if a hypothesis assigns different truth values to a formula A relative to different assignments, then the assignments must differ over one or more variables free in A. See Field [23, p. 242].
^{48} Field uses the construction just sketched for a consistency proof. He does not think that the construction spells out the real meaning of the truth predicate or of the conditional. Nonetheless, we focus on this construction because it is our best guide to Field’s account of the conditional. Field calls the construction the “Official Theory” in Section 17.5 of his book.
^{49} More precisely: a sentence B is valid by Field’s semantics iff, for all ground models M, there is an assignment v of values to variables such that where h ^{∗} is the reflection hypothesis for M. The notion of validity of arguments can be recovered, as before, from the notion just defined.
^{50} Field partly recognizes this point in the course of his response to a related objection of Yablo [78].
^{51} Certain unwanted validities are, however, forced by Field’s general framework. For more on this point, see Standefer [67].
^{53} We wish to stress that the above critical observations are narrow in scope. Even if correct, they show at best that Field’s theory fails to meet the Descriptive Adequacy desideratum from Section 1. They do not by themselves provide a reason for rejecting Field’s theory. For, arguably, there are other important desiderata, and it could well be that the failure of Descriptive Adequacy is amply compensated by the success of Field’s theory in meeting these other desiderata. It is not our aim to offer here a full critique of Field’s theory.
^{54} It is this principle that motivates the choice of the Strong Kleene scheme in Field’s construction.
^{57} Note that restricted versions of (IP) also hold in our theory. So, for example, if A contains no occurrences of the truth predicate, then A and T(‘A’) can be substituted for one another in all extensional contexts. For these kinds of substitutions, the distinction between (IP) and (UP) is of little significance and can be neglected. However, when selfreferential truth is in play, the distinction between (IP) and (UP) is of vital importance. One needs to be very careful in extending ideas that hold for nonselfreferential truth to truth in general.
^{58} This point is confirmed by classical revision theories. These theories provide languages with a wellbehaved conditional, but they do not provide conditionals suitable for interpreting Aristotle’s Rules.
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Acknowledgments
We presented some of the ideas of this paper in talks given at Carnegie Mellon University, MIT, University of Barcelona, University of Illinois at Chicago, and the UConn Logic Seminar. We wish to thank our audiences for their questions and comments. We are especially grateful to the anonymous referees for their helpful suggestions.
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Gupta, A., Standefer, S. Conditionals in Theories of Truth. J Philos Logic 46, 27–63 (2017). https://doi.org/10.1007/s1099201593933
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DOI: https://doi.org/10.1007/s1099201593933