Paracompleteness and Truth Preservation: Does Hartry Field Redundantly Double-Solve the Liar Paradox? Ben Burgis Underwood International College Yonsei University Republic of Korea
[email protected] Abstract: Hartry Field is the primary contemporary champion of a ‘paracomplete’ solution to the Liar and related semantic paradoxes, according to which we should reject the relevant instances of Excluded Middle without accepting their negations. Like other non-classical solutions to the paradoxes, the paracomplete appraoch runs into trouble with Curry’s Paradox. (Classical solutions tend to work just as well for Curry as for the Liar.) Since the mere statement of the truth condition of a Curry conditional entails its consequent, simply rejecting the sentence and its negation gets us nowhere. Usually, the price tag of a non-classical solution to Curry is disunity. The paradox-solver is forced to solve two paradoxes that intuitively seem to be of the same type in two different ways—for example, by rejecting Excluded Middle (for the Liar) and weakening the conditionals in the resulting non-classical logic (for Curry). Field, however, has noted a further problem. The statement that instances of Modus Ponens involving Curry conditionals are truth preserving is equivalent to the conditionals themselves, and thus entails their consequents. Given his commitment to the detachability of the conditionals in his preferred paracomplete logic, Field is forced to reject the claim that validity is universally truth preserving. I argue that the price tag of this last move is steeper than mere disunity. After all, the Liar itself can be solved without having to reject any of the premises of standard ‘Liar reasoning’ by rejecting the claim that the paradoxical argument is truth preserving (without rejecting the claim that the argument is valid). Thomas Hofweber has proposed precisely this solution. Hofweber’s approach allows him to retain the full resources of classical logic (including Excluded Middle) as well as (like Field) retaining an entirely transparent truth predicate. How, then, can field justify his combined approach (whereby he denies both Excluded Middle and truth preservation)? This approach takes on board the central intuitive costs of Hofweber’s approach without helping itself to the benefits.
1. Introduction: Classical and Non-Classical Paradox The Liar Paradox centers on sentences like (1), below.
Approaches
to
(1) Sentence (1) is untrue. Intuitively, this looks like a conflict between the Law of Excluded Middle (LEM) and the Law of Non-Contradiction (LNC). Given the LEM, (1) must be either true or untrue. But if it’s true, it’s untrue, and vice versa, which is impossible given the LNC.
2 The paraconsistent dialetheist solves the paradox by accepting that there are true exceptions to the LNC, and that (1) is one of them. These paradox-solvers reject classical logic in favor of an inconsistency-tolerant (or “paraconsistent”) logical framework in which contradictions no longer entail triviality. Hartry Field has argued for the mirror image of this approach, rejecting classical logic in favor of a logical framework that is not paraconsistent but paracomplete. Instead of accepting that (1) is both true and untrue (and rejecting the inference from there to triviality), Field rejects the claim that (1) is either true or untrue. [I]t is initially natural to take ‘reject’ to mean ‘deny’, that is, ‘assert the negation of’. But if we assert the negation of a disjunction, we certain ought to assert the negation of each disjunct (since the disjunction is weaker than the disjuncts). So asserting ¬(A v ¬A) should lead us to assert both ¬A and ¬¬A. But to assert both a sentence (¬A) and its negation is […] to assert a contradiction. (Field 2005, p. 23)
The resulting paracomplete logical machinery is extremely elaborate (see Field 2008), but it springs from a simple distinction. Even in a total information context, we should reject certain instances of the LEM without accepting their negations. Both paracomplete and paraconsistent paradox-solvers run into trouble when they move from the Liar to Curry. Curry’s Paradox involves sentences like (2), below. (2) If sentence (2) is true, Coldplay is aesthetically superior to Led Zeppelin. A dialetheist who simply carried over her solution to the Liar to this new paradox would be immediately faced with a dilemma between rejecting Modus Ponens and accepting the aesthetic superiority of Coldplay. Things aren’t much better for the paracompletist. To avoid the unthinkable consequent of (2), merely rejecting (2) is insufficient. After all, intuitively, the motivation for rejecting the claim that (2) is true is precisely that if it were true, its consequent would be as well. But the italicized portion of that sentence merely re-states the content of (2)!1 Moving from an intuitive level to a precise one, the problem is that we can plug (2) into Tarski’s Biconditional Truth Schema (Tr〈α〉 ↔ α), and, after applying a few otherwise unobjectionable logical rules—notably Contraction, which allows us to obtain α→β from α→(α→β)—derive from the mere statement of (2)’s truth conditions the conclusion that Coldplay
1
It could be objected that this isn’t precisely true, because (2) is a material conditional and the italicized passage is counterfactual. But on anything even in the neighborhood of a standard account of counterfactuals, a counterfactual conditional will entail a material conditional with the same antecedent and consequent.
3 is aesthetically superior to Led Zeppelin. 2 As we’ll see below, some paradox-solvers are willing to reject the relevant instances of the Biconditional Truth Schema, but this wouldn’t be an attractive solution for either Field or his paraconsistent counterparts. After all, one of the primary motivations for such non-classical solutions is a commitment to the ‘transparency’ of the truth predicate, whereby we can freely move back and forth from α to Tr〈α〉 and vice versa. Thus, both kinds of non-classical paradox-solvers are forced by Curry to revise their logics in ‘extra’ ways, revisions not necessary merely to solve the Liar. Most jettison Contraction, as well as Conditionalization. (What you can do quickly with Contraction, you can do slowly with Conditionalization and Modus Ponens.) Some critics have been troubled by this disunity in these approaches to the paradoxes. The Liar and the Curry look like instances of the same type of paradox. Intuitively, the right solution to the Liar should work equally well for Curry. Classical solutions to the Liar typically do have this attractive feature. The simplest classical solution involves rejecting the relevant instances of the Biconditional Truth Schema. Gilbert Harman, for example, recommends that we treat the Schema “not as something that holds without exception but as something that holds ‘normally’ or ‘other things being equal.’” (Harman 1986, pp. 16-17) Without the (relevant instances of) the Schema, the classical paradox-solver is free to affirm that (1) is either true or untrue. It no longer follows from this affirmation that (1) is both true and untrue. Similarly, without the relevant Schema instance, (2)’s truth conditions (whatever they might be) no longer need entail its horrifying consequent. Thomas Hofweber has proposed another solution to both the Liar and Curry that doesn’t require the rejection of even a single rule of classical logic (see Hofweber 2007). Hofweber affirms that all of those rules are valid. He further accepts the orthodox claim that valid arguments are truth preserving. However, he rejects the orthodox interpretation of the claim that Valid arguments are truth preserving as a universal generalization holding for all arguments without exception. Instead, he takes it as a generic truth, like Bears are dangerous. Just as Bears are dangerous is not falsified by the occasional old toothless bear, Valid arguments are truth preserving is not falsified by the occasional valid argument from true premises to inconsistent (and therefore clearly untrue) conclusions. Thus, Hofweber can accept every instance of both the LEM and the Biconditional Truth Schema without exception. He can that (1) is either true or untrue, that it is true if it is untrue and vice versa, and that the inference from these facts to inconsistency is valid, but he can still reject the conclusion that (1) is both true and untrue. Similarly, he can accept 2
Presumably, even Coldplay fans would be uncomfortable with the idea that the conclusion can be established by this bit of logical hocus pocus.
4 that (2)’s truth conditions validly entail an aesthetic absurdity without having to accept the truth of that conclusion. No disunity required. 2. Hofweber and Field on Truth Preservation We’ve already seen that Field is forced by his position to reject Contraction. Since he’s committed to keeping the conditionals of his paracomplete logic detachable—that is, he’s not prepared to reject Modus Ponens—he gives up on Conditionalization as well. Nor is this the end of what he’s willing to give up to solve Curry. After all, even without Contraction or Conditionalization, at least one problem remains. If, like Field, a theorist is unwilling to give up on Modus Ponens—and if, contra Hofweber, she also insists that validity unrestrictedly preserves truth—then it looks like we have to say that, if the antecedent of (2) is true, so is the consequent. But to say that is just to say (2)! In light of this (and many other considerations that are orthogonal to the point of this paper 3 ), Field gives up on unrestricted truth preservation. I’m inclined to state my conclusion by saying that the validity of a rule does not require that it generally preserve truth. However, some may think that this simply violates the meaning of the term ‘valid’: ‘valid’, they may say, simply means ‘necessarily preserves truth’, or ‘necessarily preserves truth in virtue of logical form’, or some such thing. I don’t think it does mean this […] but I don’t want to fight about semantics: if one insists on using ‘val- id’ to mean that, then my point is that every serious theory of truth employs rules whose ‘validity’ (in this sense) it rejects (or else can’t express). This seems initially surprising, but becomes less so when one reflects that the rule might still preserve truth ‘when it matters’. (Field 2009, p. 266]
Indeed, he says, “there is no reason to doubt that all the rules preserve truth when it matters.” (Field 2009, pp. 265-266) So far, this sounds strikingly like Hofweber’s claim that Valid arguments are truth preserving is a generic truth like Bears are dangerous. Field argues that rather than defining validity in terms of something other than truth preservation, we should understand validity as a primitive notion that governs our reasoning practices. Hofweber doesn’t say much about how we should understand validity, but both the idea that it should be taken as a primitive and the further claim that the most important thing about validity is the way that it regulates our reasoning practices fit quite comfortably with the main thrust of what he does say. As we’ll see below, there are some important differences between their accounts of the importance of validity to good deductive reasoning. Before we get to that, though, let’s pause and ask an obvious question. 3
The main two other considerations involve (a) Field’s interpretation of Gödelian incompleteness, and (b) the way in which the standard moves non-classical logicians make in order to get around Curry undermine the argument for unrestricted truth preservation which Field finds most compelling. There’s a lot to say about all of this, but none of it is relevant here, since everything I say assumes that the point on which Field and Hofweber agree—that not all valid arguments are truth preserving—is correct.
5 Paradoxes arise when a set of claims that individually look too obvious to countenance giving up lead us to the apparent necessity of accepting a conclusion that seems too obviously false to even consider accepting. A solution necessarily involves giving something up (if only the intuition that the conclusion is unacceptable). To argue for the correctness of some solution is to get into the reflective equilibrium game, weighing the intuitive costs of what’s been given up against alternative proposals. Is it worse to accept that there are exceptions to the T-Schema or exceptions to Excluded Middle? Is it more counterintuitive to deny that Modus Ponens is valid or to deny that valid argument forms always preserve truth? On the face of it, it looks like Field has made a strange trade, giving up two things when one would have sufficed. If a pardox-solver offered up an intricately complex consistent solution to the Liar Paradox, and then turned around and postulated true contradictions (and rejected the validity of explosion) to solve some meta-theoretical difficulty he’d gotten himself into, we would ask why he didn’t just solve the Liar Paradox that way. The same question arises in an even starker way for Field, given the inferential resources he denies himself by staying nonclassical. If he’s going to give up what Hofweber gives up, shouldn’t he take back the things Hofweber never had to give up in the first place— Contraction, Conditionalization and the Law of the Excluded Middle? If the “paracompleteness + failures of truth preservation” package can be justified, it can only be by showing that it has some tremendous intuitive benefits that don’t accrue to “failures of truth preservation” a la carte. To the best of my knowledge, Field has never addressed Hofweber’s view in print. That said, he has made comments in other contexts that could be relevant here. For example, Field has claimed that the rejection of unrestricted truth preservation “is somewhat less counterintuitive for paracomplete theorists [than it would be for classical logicians] because they don’t declare their rules not to be truth preserving.” (Field 2015, p. 40) At first glance, since Howeber accepts Excluded Middle, this does look like a comparative advantage of Field’s approach. But is it? Compare sentences (3), (4), and (5), below. (3) Hartry Field believes in God. (4) Either Hartry Field believes in God or sentence (4) is false. (5) All instances of Modus Ponens preserve truth. Given that Field is not a theist, (3) is false. This in turn makes (4) a standard sort of ‘contingent Liar,’ a statement whose truth-value depends (but could have failed to depend) on the truth-value of its ungrounded disjunct.
6 Now, if I’m a philosopher with boringly standard logical views (or an introductory logic student who takes what I’ve been learning about validity and truth preservation to be obviously correct), and I’m thus deeply committed to the belief that (5) is true, will I really be even slightly mollified by Field’s reassurance that (5) should be rejected because it has the same status as (4), rather than because it has the same status as (3)? I don’t see how. What else does Field’s paracomplete approach to validity and truth preservation have going for it that other truth-preservation-denying frameworks (like Hofweber’s) might lack? Happily, Field has given us a checklist, laying out what remains for him of the traditional connection between truth preservation and validity. ….on my own approach to the paradoxes and some others, the following all hold: (A) Logically necessary truth preservation suffices for validity in the regulative sense; for example, if ⇒ True(⟨A⟩) True(⟨B⟩) then one’s degree of belief in B should be at least that of A. (B) The validity of sentences coincides with logically necessary truth: it is only for inferences with at least one premise that the implication from validity to truth preservation fails. (C) If an argument is valid, there can be no clear case of its failing to preserve truth. (D) If an argument is valid, then we should believe that it is truth preserving to at least as high a degree as we believe the conjunction of its premises. This collection of claims seems to me to get at what’s right in truth preservation definitions of validity, without the counterintuitive consequences. (Field 2015, pp. 41-42)
How many of these boxes could Hofweber check? Certainly, he has no reason to reject (B). To the extent that (A) is simply the claim that all unrestrictedly truth preserving inferences are valid, Hofweber should affirm that as well. (C) is simply the consideration we have already dismissed about whether (5) has the status of (4) or of (3). That leaves us with (D). 3. Mapping Out the Connections Intuitively, logic is epistemically normative because logically valid inferences preserve truth. If we know that α is true, and we know that α logically entails β, then we know that β is true as well. This picture of the relationship between the concepts can be represented like this: Validity ! Truth Preservation ! Normativity
7 Hofweber and Field agree that the first connection doesn’t always hold, and they even agree on at least one major case in which it fails to hold (Curry reasoning), but they disagree about how to describe the exceptions. Field thinks valid arguments for which truth preservation fails are cases in which it is logically impossible in any case for the premises to be true in the first place—hence, his acceptance of (D). Hofweber thinks that the valid arguments for which it fails are cases in which the premises are true. After all, if Hofweber denied one of the premises of one of the paradoxical arguments—for example, by denying Excluded Middle—he would be as guilty as Field of redundantly double-solving the paradoxes. In the standard picture of the relationship between the three concepts given above, both connections are airtight. The connections are transitive, and the connection between validity and normativity is thus airtight as well. Validity ! Normativity I said earlier that Hofweber should agree with Field’s claim that validity is a primitive concept whose interest chiefly lies in its role in regulating belief. However, the way it regulates it is a little different for Hofweber than it is for Field. For Hofweber, the chain of connections goes like this: Validity !* Truth Preservation ! Normativity That is to say, for Hofweber, the connection between validity and truth preservation is not airtight. Thus, even though the connection between truth preservation and validity is airtight, the indirect connection between validity and normativity is not. Validity !* Normativity On Hofweber’s view, if you know that an argument is valid and you know that the premises are true, this gives you a very strong reason to believe that you should believe the conclusion. But it’s a defeasible reason. By contrast, on Field’s view, the connections work like this: (i) Validity ! Normativity (ii) Truth Preservation ! Validity ! Normativity [This is his (A), above.] (iii) Truth Preservation ! Normativity [This follows from (ii).] What’s the relationship between validity and truth preservation for Field? We can’t represent it with a !* for a non-airtight connection,
8 because it doesn’t look like there is any universal connection between the two for Field, even a defeasible one. (D) on his checklist tells us this: (iv) (Validity + True Premises) ! Truth Preservation …but he gives us no information about what other valid arguments might not be truth preserving. 3. Conclusion: Hofweber, Field and the Normative Role of Logic Let’s assume that Field is right to think that the central importance of the primitive notion of validity lies in its role in governing our reasoning practices. Given this assumption, it looks like Field has a slight advantage over Hofweber, since it looks like he can preserve the intuition that the validity of an argument will always (and not just generally) be relevant to the question of whether it is rational to believe the argument’s conclusion given belief in the premises. It might seem odd to think that this normativity can float free of truth preservation—after all, the traditional reason for thinking that logic is normative is precisely that reasoning aims at truth and logically valid arguments can’t take us from truth to falsehood—but on Field’s account, validity and truth preservation only come apart in contexts where the normativity of logic isn’t at play. If we don’t think the premises of some argument are true, then the validity of the argument doesn’t give us a reason to believe the conclusion on any account. If, however, you believe the premises, then truth preservation kicks in just in time to play its normative role. We said that it looks like Field has a slight advantage. One might well ask whether such a slight advantage is worth giving up on the LEM for, but it turns out that question is moot, because the slight apparent advantage dissolves in any case on further investigation. Here’s why: The normativity of logic doesn’t exclusively concern questions about what’s actually true. It also concerns hypothetical questions about what would be true, and (more importantly) about what might or might not be true. We’re often in the position of knowing that α entails β, but not having any idea of whether or not α is true. The question we use logic to investigate under these (extremely common) circumstances is not, is β true, but rather, if α is true, is β true? On Hofweber’s account, it’s easy to see how we can (justifiably) come to hypothetical conclusions of this nature on the basis of information about logic, even when we lack information about whether α is true. The fact that α entails β always and everywhere gives us a reason to believe that if α is true, so is β. Granted, the reason is defeasible, but what of it? If we weren’t justified in believing conclusions arrived it by means of defeasible reasoning, something like 99.9999% of human reasoning would be unjustifiable.
9 By contrast, nothing Field says about the relationship between validity and truth preservation is at all helpful here. If we know that α entails β, but we don’t know whether α is true, Field tells us that we shouldn’t believe β to a lesser extent than we belive α, but this advice is beside the point. In this situation, we don’t believe α at all! (Or, to be careful and Bayesian about it, we at least don’t believe it any more than we believe ¬α.) We aren’t interested in the question of whether we should start believing that β is true. We’re interested in whether we should believe that it’s true if α is true. And on that question, Field’s account of the relationship between validity and truth preservation is silent. Of course, Field might counter by bringing up (D). Even if we have no idea whether α is true or false, we at least know that if it’s true, the inference to β is truth preserving. But even this is beside the point given Field’s commitment to non-classical logic. We’re interested in whether it’s the case that if α is true, β is true. To turn if α is true, then if α is true, β is true into an answer to that question, you’d need Contraction. References Beall, JC, “Transparent Disquotationalism,” in Beall and Armour-Garb 2005, p. 7-22 ___ and Brad Armour-Garb (eds.) 2005, Deflationism and Paradox (Oxford). ___ (ed.) 2007, Revenge of the Liar (Oxford). Field, Hartry, “Is the Liar Sentence Both True and False?” in Beall and Armour-Garb 2005, pp. 23-40 ___2008, Saving Truth From Paradox (Oxford) ___and Peter Milne, 2009, “The Normative Role of Logic” in Proceedings of the Aristotelian Society Supplementary Volume lxxxiii, pp. 251-258 ___2015, “What is Logical Validity?” in [the FLC volume…fill in info later] Harman, Gilbert 1986, Change in View: Principles Of Reasoning (MIT). Hofweber, Thomas, “Validity, Paradox and the Ideal of Deductive Logic” in Beall 2007, pp. 145-158
