On Hierarchical Propositions

Philosophy

Journal of Philosophical Logic https://doi.org/10.1007/s10992-019-09509-9 On Hierarchical Propositions Giorgio Sbardolini1 Received: 7 October 2017 / Accepted: 1 April 2019 / © The Author(s) 2019 Abstract There is an apparent dilemma for hierarchical accounts of propositions, raised by Bruno Whittle (Journal of Philosophical Logic, 46, 215–231, 2017): either such accounts do not offer adequate treatment of connectives and quantifiers, or they eviscerate the logic. I discuss what a plausible hierarchical conception of propositions might amount to, and show that on that conception, Whittle's dilemma is not compelling. Thus, there are good reasons why proponents of hierarchical accounts of propositions (such as Russell, Church, or Kaplan) did not see the difficulty Whittle raises. Keywords Paradoxes * Propositions * Type theory * Simple and ramified hierarchy 1 Intensional Paradoxes There are several paradoxes concerning propositions.1 Following Whittle [15], let us focus on two. The first is what we may call Whittle's paradox: suppose that R is a property that applies to propositions of the form F(a) (for some property F and object a) iff F does not itself apply to the proposition in question. We then consider a proposition R(b) (for some object b) and ask: does R apply to this proposition? Suppose first that it does. Then, by the definition of R, it must not apply to it. So R does not apply to this proposition. But then- by the definition of R again-R must apply to the proposition-contradiction! (Whittle [15, p. 217]) The second is an intensional version of the Liar paradox. Consider the proposition expressed by the sentence '∀p(Ap → ¬Tp)', which states that for any proposition p, if p is asserted by Epimenides, then p is not true. Suppose furthermore that 1The most famous include Russell [12, 13], Myhill [8], Prior [9], Kaplan [7].  Giorgio Sbardolini email@example.com 1 ILLC, University of Amsterdam, Science Park 107, 1098 XG Amsterdam, The Netherlands G. Sbardolini Epimenides asserts only the proposition expressed by that sentence. Is this proposition true or not? If it is, then, since it is asserted by Epimenides, it is not true. If it is not true, then since it is the only proposition asserted by Epimenides, it is true. Hence the proposition expressed by '∀p(Ap → ¬Tp)' seems to be both true and not true. We shall follow Whittle and assume that propositions are structured entities. This gives some indication of what Whittle means-in the passage above-when he says that propositions "have a form". However, it should be emphasized that paradoxes concerning propositions arise whether or not propositions are structured, and depend only on logical features of the notions of a proposition and of propositional quantification (as we shall see below). Having made this point, for the rest of the paper I shall be articulating a way to understand the idea that propositions live in a hierarchy, which is a popular strategy against paradoxes like the ones above. Whittle raises a dilemma for (some) hierarchical approaches, which I'll discuss below. I believe, however, that the dilemma does not point to a general difficulty for hierarchical accounts of propositions. Instead, it appears to arise from a conflation of two assumptions: (i) the idea that complex objects, like propositions, are built out of simpler objects, and (ii) the philosophical conception motivating some 'ramifiers', like Russell [13], Church [1], and Kaplan [7], which is presumably centered around something like a Vicious Circle Principle. Although these ideas might be connected, it is the ban against circularity that justifies the hierarchy, not the claim that propositions are structured. So there's no reason to think that the dilemma Whittle raises is compelling for the ramifiers. In the conclusion I shall raise some independent worries concerning Whittle's own hierarchical account of propositions. 2 The Dilemma Why go hierarchical? Intuitively, because requiring that in one way or the other propositions are arranged hierarchically helps us avoid a (vicious) circularity in paradoxical reasoning.2 In Whittle's paradox, circularity is hidden in the definition of R: a property that applies to a proposition F(a) iff F does not apply to F(a). In the intensional version of the Liar, more subtly, we assume that '∀p(Ap → ¬Tp)' expresses a proposition, and that this proposition is in the range of the universal quantifier that is used to express it. Reasoning from these definitions dramatically brings out the circularity: the property thus defined applies to some propositions if and only if it doesn't, and the proposition thus expressed is true if and only if it isn't. Famously, the most general ban against circularity is Russell's Vicious Circle Principle, which in the intentions of Russell offers philosophical justification for the Ramified hierarchy. The VCP is notoriously controversial, and its precise formulation is unclear.3 But weaker claims than the VCP justify less cumbersome theories 2It may not be that all paradoxes involve some circularity, but Russell certainly thought they all did. Let us set aside the question whether hierarchical accounts succeed in dealing with any paradoxes, and focus on what motivates them. 3See among others, Gödel [3], Quine [10], Goldfarb [4], Jung [6], Hodes [5]. On Hierarchical Propositions than the Ramified hierarchy, and these simpler theories sometimes enjoy wide acceptance. For instance, requiring that sets are arranged hierarchically helps us avoid a circularity hidden in the definition of r , a set whose members are any sets x such that x is not a member of x. The ban in ZFC against self-membered sets can be seen as a weak version of Russell's all-out ban against circularity, one that doesn't rely on the VCP, arises naturally from a well-motivated conception of sets, and provides justification for Zermelo's widely accepted Cumulative hierarchy. Conceptually, type theories and set theories are very different. A set theory is a description of the universe of sets, and sets are distinctive mathematical objects. Set theories are used in mathematics to address several questions of existence and consistency. Type theories have been devised to regiment certain formal features of languages (with a recursively specified syntax). They are used to categorize linguistic expressions according to their logical (i.e. combinatorial) features. Still, there is some continuity between sets and types. In an extensional semantics, the types of wellformed expressions correspond to certain kinds of sets in a precise way. To say that a name η is type e is to say that there is a set E of individuals and that the extension of η (its referent) is an element of E. To say that a generalized quantifier θ is type 〈〈e, t〉, 〈〈e, t〉, t〉〉 is to say that the extension of θ is a relation (i.e. still a set) T between two sets of individuals A and A′ (for example, if θ is the universal quantifier, T is the subset relation, thus asserting that all members of A are members of A′). Finally, to say that a predicate φ is type 〈e, t〉 is to say that there is a set F of sets of individuals and that the extension of φ is an element of F . And so on. All these claims of existence have to be licensed, of course, by a coherent conception of sets, which is typically the one offered by Zermelo's foundations. So, there is one general reason to go hierarchical, and this is to avoid some kind of circular reasoning that is present in the paradoxes-at least according to some authors. But then there are very different hierarchies, and very different conceptions of the hierarchically arranged objects (whether they are sets or propositions). Often, especially in work on the set-theoretic paradoxes, 'circular' reasoning is discussed under the heading "impredicative reasoning". There is widespread agreement that impredicative reasoning in mathematics is largely acceptable up to minimal constraints-such as those implemented by Zermelo's axioms (i.e. Separation in particular). It is much less common to think that impredicative reasoning is unacceptable across the board, as the VCP would seem to require. Just how much circularity (impredicativity) is acceptable may vary depending on the subject matter, and is a subtle point on which different philosophical conceptions, motivating different hierarchical accounts, can be brought to bear. As we shall see below in more details, the Ramified hierarchy is a subsystem of the Simple hierarchy: one in which certain order-theoretic restrictions are imposed. The Simple hierarchy itself is a subsystem of Zermelo's hierarchy: one in which certain other restrictions are imposed, as pointed out long ago by Gödel [3]. Let's suppose that Zermelo's hierarchy is the upper end of hierarchical systems-i.e. the most liberal we are willing to consider (within the realm of consistency). Hierarchical accounts of any kind of objects (propositions or sets) correspond to a more or less restrictive subsystem of Zermelo's hierarchy. Each of these accounts will have to G. Sbardolini find its conceptual motivation in the underlying metaphysics of the hierarchically arranged objects. The particular hierarchy discussed by Whittle appears to be sort of intermediate between Ramified and Simple: At level 0, one would start with objects that are neither properties nor propositions. At level 1, one would have properties that apply to these and propositions that are about them. At level 2, one would have properties that apply to the things that are levels 0 and 1, and propositions that are about them-and so on. (Whittle [15, p. 217]) Visually, we get the following picture, where I indicate the levels, the corresponding linguistic expressions as they appear in Whittle's paradox, and their type: Level 0 a, ... basic type Level 1 F, ... type of properties of things of level 0 Level 1 F(a), ... type of propositions about things of level 0 Level 2 R, ... type of properties of things of level 0 or 1 The sketch continues with another line labelled 'level 2', for expressions with the type of propositions about things of level 0 or 1, and so on. Whittle points out, correctly, that this hierarchy helps us against Whittle's paradox. However, he raises the following dilemma about it: On a hierarchical account, no property can apply to a proposition that is itself built out of that property. But what then about propositions of the form ¬¬p, for example? For here we have an operator-negation-applying to a proposition that is itself built out of that operator. But if, across the board, propositions of the form F(F(a)) are disallowed, then surely propositions of the form ¬¬p should be too. For why should negation be allowed to simply exempt itself from the hierarchical restrictions? ... On the other hand, applying these to negation would seem to eviscerate logic. For it is surely an important and central feature of our concept of negation that it can be iterated. ... We thus have a dilemma: either render the approach ad hoc (by simply exempting certain connectives and quantifiers from the hierarchical restrictions) or eviscerate logic (by not doing that!). Needless to say, neither option would seem to be very attractive. (Whittle [15, pp. 219–220]) The dilemma, if compelling, is a threat to hierarchical accounts of propositions: Whittle mentions Russell [13] and Church [1] as two theories for which this dilemma would arise. But I shall argue that the dilemma is not compelling, and indeed that there's a good reason why it does not arise for Russell or Church. The dilemma depends on the specific conception of the nature of propositions that Whittle adopts, and that justifies the hierarchy we sketched above. However, such conception is arguably not in the background of Russell and Church, nor of other famous ramifiers like Kaplan [7], and it does not motivate their theories-nor does Whittle claim that it would. As Whittle offers no independent argument for the particular conception he puts forward, there's no reason to think that ramifiers are affected by the dilemma. On Hierarchical Propositions 3 Hierarchies Whittle's paradox is in effect a property version of Russell's paradox: the role played by propositions in the argument is inessential. To see this, consider the property version of Russell's paradox. First, assume a Naıve Comprehension Principle NC1: ∃P∀Q(P (Q) ↔ φ(Q)) (NC1) where Q ranges over properties. The property version of Russell's paradox follows from consideration of the property R that applies to all and only those properties that don't apply to themselves. An instance of NC1 straightforwardly entails that: ∀Q(R(Q) ↔ ¬Q(Q)) i.e. a property R applies to all and only those properties Q that don't apply to themselves. Contradiction follows immediately by instantiating the quantified variable on R itself. Regarding Whittle's paradox, assume a Naıve Comprehension Principle NC2, which differs (inessentially) from NC1 in that the universally quantified variable is now interpreted as ranging over propositions: ∃P∀q(P (q) ↔ φ(q)) (NC2) On a different instance of φ, NC2 gives: ∀q(R(q) ↔ ∃F∃a(q = F(a) ∧ ¬F(q)) i.e. a property R applies to all and only those propositions q such that for some property F and object a, q is the proposition that F(a) and F does not apply to q. Contradiction follows in the way we already said. In a sense, these are different versions of Russell's paradox: whether the universally quantified variable ranges over sets, properties, or propositions, it makes little difference. As is well known, Russell's paradox does not arise in Zermelo's hierarchy, because Naıve Comprehension is not licensed by Zermelo's axioms and the existential claims displayed on the previous paragraph are shown to be false. A fortiori, the paradox does not arise in the Simple and Ramified hierarchies (which are stricter than Zermelo's). The philosophical justification for the Simple hierarchy can be put as follows: it is very natural to distinguish between predicates that apply to basic things (is a natural number), predicates that apply to sets of basic things (is equinumerous with the set of natural numbers), predicates that apply to sets of sets of basic things (is equinumerous with the set of subsets of natural numbers), etc. It is just as natural to distinguish between predicates of basic things, predicates of properties of them, predicates of properties of properties of them, etc. Simple type theory imposes these distinctions on logical languages: there are level 0 predicates (of basic things), level 1 predicates (of sets/properties of them), level 2 predicates (of sets/properties of sets/properties of them), etc. Violation of these distinctions leads either to nonsense or to necessary falsehood (e.g. outright contradiction, or inconsistency with Cantor's theorem). Since predicates can be divided in this way, classes of things to which they may apply can G. Sbardolini be divided accordingly. Thus we have a basic class of things to which level 0 predicates apply, a class of things to which level 1 predicates apply, and so on. Things divided into classes in this way are said to form a Simple hierarchy. It applies to sets, and to properties. Accordingly, I will be talking about levels of predicates of hierarchically arranged things, as well as levels of hierarchically arranged things. Recall that in Whittle's hierarchy, propositions about level n stuff live at level n+1. This might be motivated by the idea that propositions are built out of properties (plus some other things). Properties are directly tied to predication, for predicates express properties and have sets as extensions, and since predicates are typed, properties are arranged hierarchically. There's a broader philosophical question about the relation between predication and propositions-the objects of thought. A popular view is that properties are the middleman in this relation: they are expressed by predicates, but they are also parts of propositions. On this view, presumably propositions themselves will be arranged hierarchically: one does not run into a proposition that contains a property of level n unless one is at least at level n. However, this is one view among many, and the nature of the relation between predication and propositions remains controversial. The mereological model I just sketched of the relation between propositions and properties is by no means common ground. If properties are not parts of propositions, we have as yet no apparent reason to think that propositions are arranged hierarchically. Whittle's paradox by itself gives no reason to think that propositions are arranged hierarchically: formally, they could all live at any level of the Simple hierarchy (e.g. the lowest): paradox is avoided so long as we distinguish between properties of propositions (like F ) and properties of properties of propositions (like R)-as I remarked above, quantification over propositions in Whittle's paradoxical argument is not essential, as it is a property version of Russell's paradox. There are reasons to think that propositions are arranged hierarchically though- but in a different sense of 'hierarchy'. So far, we haven't touched on Russell's Ramified hierarchy. This theory is helpful if one is to account for other kinds of paradoxes, i.e. those that, as Russell knew well, might arise even within the restrictions of Simple type theory. The Intensional version of the Liar paradox, given above, is one such example. It is routine to check that the expression '∀p(Ap → ¬Tp)' is syntactically well-formed, by the lights of the Simple theory of types. Intuitively, there are no self-applying properties nor self-membered sets, which is what we were interested in avoiding, and yet the Intensional Liar paradox still follows. To handle this case we need ramification, at least according to Russell, Church, and Kaplan.4 What's the point of ramification? That's controversial, and it goes by different interpretations of the Vicious Circle Principle. But the general idea is that certain objects, like the proposition expressed by '∀p(Ap → ¬Tp)' are defined in a way that "presupposes" the collections of all objects to which it belongs -in this case, we define a proposition as the one expressed by a sentence that contains a quantifier 4However, the more popular view nowadays is that we don't need ramification at all. It was Ramsey [11] who suggested that these paradoxes don't have a solution in pure logic (or type theory). According to Ramsey, contradiction follows in these cases from some faulty assumption perhaps about the notions of truth and predication. On Hierarchical Propositions over all propositions (including the definiendum itself). According to the ramifiers, there is something bad about this definition, although it has proven hard to say what exactly. But the point that matters here is that whereas the Simple hierarchy is motivated by the idea that classes of things are "formed" in stages (to use the popular metaphor for Zermelo's hierarchy), the Ramified hierarchy is motivated by a philosophical conception of quantification, and of how certain objects are "defined" by means of the quantifiers.5 As before, it is helpful to think of hierarchies starting from thinking about predicates. Consider predicates of propositions such as is true or is asserted by Epimenides. Given the underlying conception of the Simple hierarchy I sketched, we may assume that all predicates of propositions have the same level. The class of things to which they apply is the class of propositions-there is one such class from the perspective of the Simple hierarchy. Both is true and is asserted by Epimenides express properties of propositions and they both have sets of propositions (subsets of the class of propositions) as extensions. If so, then it's plausible to think that slightly more complex predicates of propositions have the same level too. For instance, predicates like is true and asserted by Epimenides, and is such that if it is asserted by Epimenides then it's not true. Ramification comes in next. Along with the distinction of predicates into levels, we are going to distinguish each level into different orders. For an example (coming from Russell [14]; see also Giaquinto [2]), compare these two predicates: i. is a good military strategist. ii. has all the properties of a great general. We may suppose that Napoleon was a good strategist, and that he had all the properties of a great general. The point to notice is that (ii) expresses a property that Napoleon had only if he had the property expressed by (i). The opposite direction doesn't hold, for one may well be a good strategist without having all the properties of a great general. Predicates (i) and (ii) are of the same level since they both apply to Napoleon. But we will say they are of different orders. In particular, let's say that if is true, and is asserted by Epimenides, are order α, then so are the predicates is true and asserted by Epimenides, and is such that if it is asserted by Epimenides then it's not true. For a proposition is described by one of these predicates just in case it belongs to some subset or other of the class of all propositions. However, the predicates is true iff everything is false, is true iff nothing asserted by Epimenides is true, etc., are level α +1. Intuitively, a proposition is described by one of these second kinds of predicates just in case some condition holds of the class of all propositions "as a whole". One might see each predicate as being implicitly "context-sensitive", so that is true really should read: is of order α and true. Then a predicate such as is true iff every proposition of order α is false is implicitly of order α + 1, whereas is true iff every proposition of order α + 1 is false is implicitly of order α + 2. And so on. 5As I mentioned, there is controversy surrounding what exactly the VCP says and why should it be true. But it is not controversial that the VCP is a ban against "definitions" of something x that "presuppose" the class of things which x belongs to: Russell was at least that much clear. G. Sbardolini Notice that all these higher order predicates are still predicates of propositions, so they are all the same level. Yet, intuitively, they all "presuppose" sets of propositions of increasing orders. In other words, from the perspective of the Simple hierarchy, all propositions form a single class, i.e. all propositions live at the same level. But now, within that level, we distinguish orders of predicates of propositions. Hence propositions do not form a single class from the perspective of the Ramified hierarchy. There will be a basic class of propositions to which order 0 predicates apply, a class of propositions to which order 1 predicates apply, and so on. A hierarchy that combines the division into levels and into orders is called a Ramified hierarchy. In the Ramified hierarchy of propositions, the Intensional Liar paradox does not arise. For suppose that the predicate is such that if it is asserted by Epimenides then it's not true is order α. Suppose that q is the proposition expressed by '∀p(Ap → ¬Tp)', namely that q is described by a predicate of order (at least) α + 1: the predicate is true iff all propositions asserted by Epimenides are not true. It follows that q belongs to the class of propositions which predicates of order α + 1 apply to. Then a predicate of order α is such that if it is asserted by Epimenides then it's not true does not apply to q, for q does not belong to the class of propositions which predicates of order α apply to. The Simple hierarchy arises from consideration of the difference between predicating something of a thing, predicating something of a class of things, predicating something of a class of classes of things, and so on. Violation of restrictions imposed by the Simple hierarchy may lead to inconsistency, as we said. The Ramified hierarchy, in the way I presented it, arises from the consideration that certain acts of predication are, in some sense, quantificational: they presuppose a class of things. Violation of restrictions imposed by the Ramified hierarchy may lead to the intensional paradoxes. Of course there are many details I didn't discuss, and other conceptions of these two hierarchies are possible, but I have offered a plausible sketch of the philosophical considerations motivating them. I contend that the sketch I offered is, albeit vague, somewhat faithful to Russell's original intentions-see Goldfarb [4] and Hodes [5] for interpretations of Russell according to which ramification is motivated by the theory of higher order quantification. Hierarchies (of levels or of orders) are helpfully understood as ways of classifying predicates based on their logical features, and classes of things which these predicates are applied to are classified accordingly. Hierarchies do not (necessarily) arise from putting together complex objects out of simpler objects. I have presented conceptions of the Simple and the Ramified hierarchy on which neither is motivated by mereological composition. As we shall see, this undermines Whittle's dilemma.6 6To be sure, the conception of the Ramified hierarchy I sketched is underdescribed, and one should well wonder why propositions depend on syntactical features of their definitions, in the way the VCP seems to suppose. If propositions are structured objects, one might see why. But the assumption I have made to justify the Ramified hierarchy is one of metaphysical dependence, not one about the intrinsic nature of propositions. Hence, metaphysical accounts of propositions other than the structured account seem prima facie possible, which would support the Ramified hierarchy. On Hierarchical Propositions 4 Dilemma Revisited Whittle's dilemma arises from a difficulty that potentially targets any logical operator, but negation in particular. Since the proposition that ¬p is built out of negation and p, then intuitively ¬p is going to live upstairs, and p downstairs. Accordingly, the proposition that ¬¬p lives twice removed from p. That's a problem because the inner occurrence of '¬', since it applies to an object living downstairs, has to denote a different operator than the outer occurrence of '¬', since the latter applies to an object living upstairs. And so on. In effect, negation cannot be iterated-which, as Whittle points out, is tantamount to say that we don't really have a negation. Now, why should negation push one up in the hierarchy? The thought that increasing grammatical complexity forces one up the hierarchy is compelling, perhaps, if one goes up the stairs by way of composing propositions out of parts. However, the conceptions I sketched for the Simple and the Ramified hierarchies are not motivated by this thought. Suppose we want to make sense of the idea that ¬p lives upstairs, and p downstairs. Suppose that the floors in this metaphor are levels. We would be saying that ¬p is level n, and p is level n − 1. But that's not justified: in the Simple hierarchy of propositions, levels go up by consideration of sets (or properties) of things of a certain level. So we have propositions at one level, and sets or properties of propositions at the next level, and so on. Negation does not take from a thing to a set of things, and neither do conjunction, disjunction, etc. Thus, there's a principled reason why the logical connectives are exempted from the restrictions of the Simple hierarchy. Suppose instead that the floors in the metaphor are orders. We would be saying that ¬p is order β, and p is order β − 1. But that's not justified either, at least not on the quantificational reading of the VCP sketched above. In the Ramified hierarchy of propositions, orders go up by consideration of classes of propositions which we may presuppose in order to define more and more propositions. So we have propositions of an order, and propositions defined by presupposing the class of all propositions of that order at the next order, and so on. There's no sense in which negation "presupposes" classes of propositions, and neither do conjunction, disjunction, etc. It is only quantifiers that trigger presuppositions of this sort, because they seemingly require a domain of propositions for the variables to range over. So, even by the lights of the Ramified hierarchy, there's a principled reason why increasing grammatical complexity, at least by means of the connectives, does not push one up the hierarchy. In effect, absent an independent argument why the Simple or the Ramified hierarchy should be interpreted according to considerations of mereological complexity, there's no reason to assume that negation and the other connectives interact at all with either hierarchy. Whittle complains (p. 220) that Russell, and commentators such as Gödel and Church, didn't recognize that the exemption of the connectives from hierarchical restrictions is ad hoc, and didn't notice the dilemma he raised. But I made the case that Whittle's difficulty arises only for a specific conception of the nature of propositions, that apparently motivates the particular hierarchy Whittle is working with, on which the relation between levels is understood as partly a mereological matter. To be sure, whether Russell assumed (at various points in his career) that G. Sbardolini propositions have parts, and whether this view had some role in preparing his ramified approach to the antinomies, are matters for discussion. In any case, there's good reason why the logical operators are exempt from either hierarchy, why Russell's theory is not ad hoc (at least, not in the respects I addressed here), and why Gödel and Church didn't notice Whittle's dilemma (although they did notice other difficulties for Russell's theory). 5 Conclusion Whittle's difficulty concerning negation is real, if one adopts the particular conception Whittle is working with, but no compelling reason to do so is offered. To solve the difficulty, Whittle sketches a positive proposal in the second part of the paper, about which I shall now make a couple of brief comments. Whittle's suggestion is to build propositions out of functions, rather than properties (so, we are still under the assumption that propositions are structured objects): "rather than building the proposition that John is tall, for example, out of John, together with the property of being tall, we should instead build it out of John together with the 'tallness function', i.e. the function that sends tall things to the truth value t, and everything else to the truth value f " (pp. 220–221). Insofar as functions correspond to their characteristic sets, the Simple hierarchy applies to functions for the same reasons it applies to the extensions of predicates, and so Whittle is recommending a Simple hierarchy. As such, I doubt that it leads to a solution to all intensional paradoxes. In effect, Whittle does not discuss how his proposal is meant to address the Intensional Liar paradox. There is another, more basic problem. Functions are extensionally individuated, and propositions are not. Suppose, following the familiar example, that all and only animals with hearts are animals with kidneys. Then the function h that sends all and only animals with a heart to t is identical to the function k that sends all and only animals with kidneys to t, because h and t have the same extension. Hence h and t are the same function. Thus the mereological complex built out of the function h and Felix the cat (plus whatever else is needed), is the same mereological complex built out of the function k and Felix the cat (plus, etc.). But the proposition that Felix is an animal with a heart is distinct from the proposition that Felix is an animal with kidneys. Perhaps one should think of "functions" in Whittle's sense as intensionally individuated, i.e. something closer to propositional functions in Russell's sense. But Whittle explicitly rejects this move (see his footnote 12). So it's unclear how the proposal should lead to a plausible theory of propositions. The case for Whittle's proposal, in any case, is undermined by the main point of this paper. It is only by adopting a very peculiar conception of logical hierarchies, and one for which no motivation is in the offing, that any difficulties arise of the sort that lead to Whittle's dilemma. I offered plausible (if not well-motivated) conceptions of the Simple and Ramified hierarchies, arguably somewhat close to what the original ramifiers had in mind, and neither leads to the difficulty raised by Whittle. On Hierarchical Propositions Funding Information This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 758540) within the project From the Expression of Disagreement to New Foundations for Expressivist Semantics. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. Church, A. (1976). Comparison of Russell's Resolution of the semantical Antinomies with that of Tarski. Journal of Symbolic Logic, 41(4), 747–760. 2. Giaquinto, M. (2002). The search for certainty. Oxford: Clarendon Press. 3. Gödel, K. (1944). Russell's mathematical logic. In Schlipp, P.A. (Ed.) The philosophy of Bertrand Russell (pp. 125–153). New York: Tudor Publishing Company. 4. Goldfarb, W. (1989). Russell's reasons for ramification. In Savage, C.W., & Anderson, C.A. (Eds.) Rereading Russell: essays on Bertrand Russell's metaphysics and epistemology (pp. 24–40). 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Does sentience come in degrees?

Philosophy

I discuss whether "sentience" (i.e., phenomenal consciousness) comes in degrees.

Animal Sentience 2020.XXX: Lee on Mikhailovich & Powell on Invertebrate Minds 1 Does sentience come in degrees? Commentary on Mikhalevich & Powell on Invertebrate Minds Andrew Y. Lee School of Philosophy, Australian National University Abstract: I make some remarks about whether sentience, or what philosophers call 'phenomenal consciousness', comes in degrees. Andrew Y. Lee is a Postdoctoral Research Fellow in Philosophy at Australian National University. His research is on philosophical questions about the structure, ethics, and epistemology of consciousness. Website Mikhalevich & Powell (2020) (M&P) provide a wide-ranging and commendable discussion of some empirical and philosophical questions about invertebrate minds. My commentary aims to address a question that M&P mention mostly in passing, but which seems to me worth thinking about more carefully: does sentience come in degrees? This question is particularly interesting in the present context: if arthropods are sentient, we might wonder whether some arthropods (such as ants) are in some sense less sentient than other organisms (such as humans). Following this journal's terminological guidelines, I use 'sentience' to mean what philosophers call 'phenomenal consciousness', where an organism is phenomenally conscious just in case there is something it is like to be that organism. Now, it is obvious that the capacity for sentience comes in degrees. But does sentience itself come in degrees? In other words, given two organisms that are sentient (at a time), can we meaningfully say that one is more sentient than the other? The relevant passage from M&P (p. 8) asks whether sentience "is a discrete, rather than continuous property," where they say that the latter would mean that sentience "is continuously distributed in the animal world, with no clear-cutoff between experiencers and those that are incapable of experience," and where they state that the key issue concerns "what the lowest bounds of this spectrum might look like." Speaking metaphorically, the question concerns whether "the lights are either on or off" or whether there might also be a "dimmer switch." I think these remarks actually contain three distinct questions: Q1. Is sentience all-or-nothing or a matter of degree? Q2. Is sentience discrete or continuous? Q3. Is the concept of sentience vague or sharp? Animal Sentience 2020.XXX: Lee on Mikhailovich & Powell on Invertebrate Minds 2 From the context, I think it is clear that the main issue for M&P concerns Q1. But some of their remarks are also naturally read as asking Q2 or Q3. Let's begin with why these questions are distinct from one another: Q1 asks whether sentience is all-or-none or a matter of degree. In other words, are some organisms more sentient than other organisms? The metaphor of the light is illustrative here: If sentience is all-or-none, then the lights are either on or off; but if sentience comes in degrees, then the light can shine at varying levels of brightness. If the latter view is correct, then it is natural to ask what exactly would make an organism more or less sentient. We will return to this question in a moment. Q2 asks whether sentience is discrete or continuous. At first, this may strike some as equivalent to Q1. But taking sentience to admit of degrees is compatible with taking sentience to be either discrete or continuous. Imagine a light switch with a continuous dimmer versus a light switch whose dimmer has ten discrete brightness values ranging from very dim to very bright. Both switches generate differing degrees of light, but the former is continuous while the latter is discrete. In brief: to be all-or-nothing is to have only two possible magnitudes, on or off; to be degreed is to have many possible magnitudes; to be continuous is to have a continuum of magnitudes; to be discrete is to be not continuous. Given this, Q2 arises only if we have already answered 'yes' to Q1. Q3 asks whether the concept of sentience is vague or sharp. In other words, are there organisms for whom there is no determinate fact of the matter about whether they are sentient or not? This question about vagueness is independent of the question about degrees, since whether the concept of sentience is vague leaves open whether some organisms are more sentient than others. As an analogy, consider how both the concepts of "tall" and "scientist" are vague, but the former is a matter of degree (one person can be taller than another) while the latter is all-or-nothing (one person cannot be "more scientist" than another). On the other hand, it is hard to understand what it would mean for there to be indeterminate cases of sentience: even if there is barely something it feels like to be an organism, that organism would still count as sentient (Simon 2017). This suggests that the concept of sentience is sharp: the lights are either on or off, and even the dimmest light counts. Let's return now to Q1. In what sense might some organisms be more sentient than others? There are plenty of aspects of sentience that come in degrees: for example, the intensity, duration, complexity, or precision of an experience, the number of qualitatively distinct experiences an organism is able to have, or the degree to which information about a sentient experience is accessible for cognition (Pautz 2019). But do any of these give us reason for saying that sentience itself comes in degrees? It seems to me that the key question is whether we can make sense of greater or lesser quantities of sentience, in the same way that we can make sense of greater or lesser quantities of light. However, that question arguably depends on unresolved questions about how conscious experiences are structured. Given this, I think it is better to simply to talk about each of the aforementioned aspects directly rather than to invoke the nebulous notion of 'degree of sentience' (Bayne, Hohwy, & Owen 2016). And in light of this, I think it is best to avoid talk of degrees of sentience, at least until we have a better understanding of what that might mean. Animal Sentience 2020.XXX: Lee on Mikhailovich & Powell on Invertebrate Minds 3 References Bayne, Tim ; Hohwy, Jakob & Owen, Adrian M. (2016). Are There Levels of Consciousness? Trends in Cognitive Sciences 20 (6):405-413. Mikhalevich, Irina & Powell, Russell (2020) Minds without spines: Evolutionarily inclusive animal ethics. Animal Sentience 29(1) Pautz, Adam (2019). What Is the Integrated Information Theory of Consciousness? Journal of Consciousness Studies 26 (1-2):1-2. Simon, Jonathan (2017). Vagueness and zombies: why 'phenomenally conscious' has no borderline cases. Philosophical Studies 174 (8):2105-2123.


Philosophy

I here respond to Pietro Salisâs objections against my original critique of the Prosentential Theory of Truth. In addition, I clarify some points regarding the relationship between anaphoric relationships and âgeneral semantic notionsâ like âequivalenceâ, âconsequenceâ, and âsameness of contentâ, and make some further points about âs ability gto explain pragmatic and expressive features of âtrueâ.

Truth-Predicates Still Not like Pronouns: a Reply to Salis Arvid Båve1 Received: 18 October 2018 /Revised: 12 March 2019 /Accepted: 15 April 2019 # The Author(s) 2019 Abstract I here respond to Pietro Salis's objections against my original critique of the Prosentential Theory of Truth (PT). In addition, I clarify some points regarding the relationship between anaphoric relationships and Bgeneral semantic notions^ like Bequivalencê, Bconsequencê, and Bsameness of content^, and make some further points about (PT)'s ability gto explain pragmatic and expressive features of Btruê. Keywords Truth . Prosentential . Grover . Belnap . Brandom . Salis . Anaphor . Pronoun Pietro Salis's paper, BThe generality of anaphoric deflationism^ (2019) is devoted to criticizing the objections against the Prosentential Theory of Truth (PT) that I raised in Båve (2009a). I will here try to sustain my original criticisms, sometimes through modifications motivated by Salis's criticisms. The original prosententialists were Grover et al. (1975), but their theory was substantially developed by Brandom (1994). A common thread (among many differences) is that the word 'true' is taken to function in a way similar to anaphoric expressions like pronouns. For instance, the sentence, 'That's true!', as uttered in response to some statement, is taken to have that preceding statement as its anaphoric antecedent, just as 'Mary' is the antecedent of 'she' in 'If Mary is absent, then she is probably sick'. My original case against (PT) consisted mainly of BMaterial Criticisms^ and BPrincipal Criticisms^ (2009a: §II and §III, respectively). The first consist of a multitude of arguments to the effect that 'true' does not work like paradigmatic anaphoric expressions (like pronouns), neither in the way prosententialists allege, nor any other way. The second type of criticism is that even if 'true' were similar to paradigmatic anaphoric expressions like pronouns (so that the Bmaterial criticisms^ would not apply), saying so would merely be an Bidle analogy^ with no explanatory power. Rather, given an account of the functioning of anaphoric expressions, one would be able to derive from (PT) a Philosophia https://doi.org/10.1007/s11406-019-00081-9 * Arvid Båve firstname.lastname@example.com 1 Department of Philosophy, Stockholm University, Götgatan 92A, 118 62 Stockholm, Sweden theory of the functioning of 'true' lacking any notions of Banaphor^, Bantecedent^, etc. I took this to show that there is nothing Bessentially proformal^ about 'true'. I will not describe all of the specific claims about particular constructions made by the prosententialists that I discussed in the original paper, but only those discussed by Salis, and I will do so consecutively and with my own comments and replies interspersed as we march along. Salis first takes issues with my objections against Brandom's treatment of sentences like, (1) Goldbach's conjecture is true. Brandom takes (1) to indirectly inherit its content from. (2) Any even number is the sum of two primes. About the expression 'is true' in (1), Brandom says, B[i]t applies to a term that is a sentence nominalization or that refers to or picks out a sentence tokening. It yields a prosentence that has that tokening as its anaphoric antecedent^ (1994: 305). I argued that 'true', as Brandom characterizes it, differs from other, paradigmatic proforms, since no other proform has an antecedent which is determined by way of reference (in the sense of denotation, rather than anaphoric cross-reference) in this way. Salis objects that, BBåve here tendentiously reads the connection between the antecedent and the prosentence in terms of reference, rather than in terms of content inheritance. It is not clear why he thinks that content inheritance could be granted only by a non-deflationary notion of referencê (2019: 513). But, firstly, Brandom himself describes the relation between the antecedent and the prosentence as involving Breferring^ or Bpicking out^, by which he means denotation, rather than anaphoric cross-reference. Thus, Salis's suggestion is not faithful to Brandom. A further note about my use of Bparadigmatic^: a paradigmatic anaphor is an uncontroversial case of a anaphor. Everyone should agree that whereas, e.g., pronouns fit this descriptions, 'true' does not. But I am not presupposing that all anaphors are paradigmatic anaphors. That would be wildly question-begging against prosententialists. It is nevertheless important, for determining how plausible it is to say that 'true' is anaphoric, to see whether it resembles paradigmatic anaphors. Could we suppose, though, as Salis seems to suggests, that we instead take the relation between (1) and (2) to involve only anaphoric cross-reference, or Bcontent inheritancê? (This would be a modification of Brandom's view.) First of all, it is clear that on Brandom's account, (2) is determined as the antecedent of (1) only via the singular term, 'Goldbach's conjecture'.1 So, could we suppose that (2) becomes the antecedent of (1) in virtue of this term inheriting its content from (2), rather than by denotationally referring to it? No, for 'Goldbach's conjecture' cannot by any stretch be taken to inherit its content from (2). If it did, then they would have to have the same content (in this context, anyway).2 But, surely, 1 Salis actually takes Brandom to claim that (1) itself Bpicks out a sentence tokening^ (2019: 508). But the quote from Brandom above makes clear that it is BGoldbach's conjecturê, not (1) in its entirety, that picks something out. This is also in line with the commonplace idea that only singular terms refer to things, and sentences never do. 2 The two obviously do not have the same meaning. Two tokenings can, however, have the same content in a context, which is why it is suitable here to speak of inheritance of content, rather than meaning. Philosophia they don't. They belong to different syntactic categories, one being a singular term and the other being a sentence. They also differ markedly from uncontroversial cases of content inheritance via anaphoric dependency. This is more easily seen if we switch examples from (1) to. (3) The third sentence in the Bible is true. If (3) is supposed to inherit its content from the third sentence in the Bible, its content should be necessarily and a priori equivalent with the content of the third sentence in the Bible. This holds in paradigmatic cases of anaphoric dependency. More precisely, for every token paradigmatic proform p with antecedent a, a sentence token s containing p will have a content necessarily and a priori equivalent with the content expressed by a sentence differing from s only in containing a in place of p. If a token of 'thusly' has a token of 'by running' as its antecedent, then the contents expressed by the relevant tokens of 'She got there by running' and 'She got there thusly' are a priori and necessarily equivalent (indeed, theywould seem to be identical). This clearly does not hold in the case of (3) and the third sentence of the Bible, however. (Note that this argument also targets Brandom's claim that (2) is the antecedent of (1).) In view of these observations, it seems that 'Goldbach's conjecture' in (1) cannot be taken to inherit its content from (2) via anaphoric dependency. Thus, the only relevant way in which one can relate to (2) in one's use of 'Goldbach's conjecture' is via denotational reference, as Brandom himself claims. Another conceivable retreat for Brandom is a modification of his account on which 'Goldbach's conjecture' has as its sole function to determine an antecedent from which the content of (1) is inherited, and that the singular termmanages to do so without denoting (2) (or some token of it).3 But this proposal of course raises the question how 'Goldbach's conjecture' latches on to (2) at all. These two expressions can scarcely be taken to be related the way 'she' is related to 'Mary' in 'IfMary is not here, shemust be sick'. For 'Goldbach's conjecture' would have to be able to latch on to (2) also in contexts in which (2) has not been uttered, and in which, indeed, neither speaker nor hearer even knows that Goldbach's conjecture is. And claiming that 'Goldbach's conjecture' determines (2) as the antecedent of (1) while holding that there is no explanation of how it does so seems unsatisfactory. There may seem to be a way of reconciling Brandom's and Salis's claims about (1) and (2), namely, by claiming that this particular case of denotational reference is included among the anaphoric relations. However, this merely amounts to widening the notion of Banaphoric relations^. This terminological change would not save the account from the objection that Banaphoric relations^, in this special sense, are different from anaphoric relations, as we normally conceive of them. The reader may at this point have worried about a dubious assumption underlying Brandom's treatment of (1), namely, that 'Goldbach's conjecture' refers to a particular sentence tokening. It is certainly more natural and plausible to take 'Goldbach's conjecture' to refer to the proposition that any even number is the sum of two primes. This is of course a further problem with Brandom's view. And note that it cannot easily be modified so as to conform to the more plausible account of the referent of 3 A reviewer for this journal proposes that this is what Brandom actually means. I prefer my above interpretation of Brandom, but this exegetical issue is orthogonal to the main point, which is that Brandom's theory faces difficulties on both interpretations. Philosophia 'Goldbach's conjecture'. For Brandom cannot very well hold that 'Goldbach's conjecture' refers to the proposition and continue to hold that (2) is the antecedent of (1). For if he did, then the relationship between (1) and (2) would be even less direct, and (1) would now differ even more starkly from paradigmatic cases of anaphor-antecedent relationships. To wit, the alleged antecedent (2) would now be determined, not by being the referent of 'Goldbach's conjecture', but by being a sentence expressing the proposition that is its referent. Surely, no other proform relates to its antecedents in this way. To sum up: Brandom cannot explain the meaning of (1) merely by reference to anaphoric relations or to Bcontent inheritancê. In order to explain how (2) becomes the antecedent, he must say (as he actually does) that this happens through 'Goldbach's conjecture' denotationally referring to (2). And my original point was that no other proform works this way, which in turn casts doubt on (PT). Against this, it may objected that since Brandom gives an anaphoric account of (denotational) reference, it is not a problem for him to take anaphoric dependencies to involve denotational reference. But, on the contrary, it does not matter which view of reference Brandom adopts, since it remains the case that paradigmatic proforms do not determine antecedents by way of denotation. This is so whether or not we adopt an Banaphoric^ account of denotation. Just as it is crucial to keep the denotation/anaphoric cross-reference distinction apart from the substantial/deflationary distinction, we must also keep the relationship between anaphors and antecedent apart from the relationship that holds between B'Snow is white'^ and 'Snow is white'. The former always holds between linguistic expressions whereas the latter only sometimes does (as in the example just given). But the latter also holds between 'Aristotle' and Aristotle, the flesh-and-blood human, and he could never be an antecedent or an anaphoric expression. What about Salis's complaint that I read Brandom as appealing to a non-deflationary notion of reference? I did complain, in an ad hominem aside, which was not part of the argument we have been discussing, that Brandom Bmakes use of a non-deflationary notion of reference, contrary to his professed commitment to eschew such representational notions^ (2009a: 305). I waswrong, I believe, to add the qualification Bnon-deflationary^ here. For none of what Brandom says about (1) precludes him from adopting a deflationary account of denotational reference. But the real problem, which still stands, is that Brandom appeals to denotational reference at all, contrary to his stated commitment to Bturn the explanatory tables on the representationalist tradition^ (1994: 136), i.e., of giving an account of contentfulness, etc., in terms of inference rather than representational notions like reference. It does not help that he gives a Bdeflationist^ or Banaphoric^ account of denotational reference. At best, he could change his overarching goal to explaining meaning/content without appealing to a nondeflationary notion of reference. Whether such an amendment would be congenial to his general outlook, however, I cannot discuss here. Finally, Salis also seems to misunderstand my claim about (1) and (2) being material equivalent. He writes, BThe explanation, according to Båve, does not work in terms of anaphoric dependence: It works in terms of 'material equivalence' between two sentences, and no proform works in this way for the example abovê. But I did not attribute to Brandom any view at all about the material equivalence between (1) and (2). I merely said they are materially equivalent. This is not related to my objection that Brandom's theory makes 'true' different from other proforms. Philosophia Let us now turn to Salis's objections against my main critique of (PT) (both the original version and Brandom's). My point was the following: that certain sentences containing 'true' are equivalent with, are consequences of, or entail, other sentences, is part of the linguistic data that a theory about 'true' should explain. One of the data, for instance, is that 'What John said is true', 'John said that snow is white' jointly entail, 'Snow is white'. But since talk of anaphoric relationships can be cashed out in terms of such relations as equivalence or consequence, or other Bgeneral semantic notions^, the claims made by (PT) will merely Bre-describê the data, rather than explain them. Salis objects against this, claiming that the equivalence between two sentences may well be explicable by appeal to an anaphoric relation between them. He also suggests that the notion of anaphoric dependence can be taken as primitive and that the notions of equivalence and consequence are taken as derivative, as explained in terms of anaphoric dependencies. On one reading, however, this cannot be right (and Salis surely didn't mean it this way). Surely, not all cases of equivalences or consequence can be explained in anaphoric terms. Consider, John is a bachelor ⇒ John is male A and B ⇒ A If these cases of consequence were to be explained by recourse to anaphoric dependencies, the latter notion would have to be stretched beyond recognition. (Indeed, one of my main complaints against Brandom was precisely that he overstretches the notions of anaphoric dependency, antecedent, etc., although perhaps not as blatantly.) When claiming that the notions of equivalence and consequence are more general than that of anaphora, I had the above kind of case in mind: we need to appeal to these notions in our semantic theory anyway, but we need not necessarily appeal to any primitive notions of anaphoric dependency. Thus, in a choice of primitives, we should opt for equivalence and consequence, rather than anaphoric dependency. All cases of the latter kind can be Bexplained^ in terms of the former, but not vice versa. When I say that anaphoric relations can be Bexplained^ in terms of equivalence, I do not mean to say that we can explain why one thing anaphorically cross-refers to another by saying, Bbecause so and so are equivalent^. Rather, I mean that in order to define the notions of pronoun, antecedent, and similarly anaphoric notions, we must use the notion of equivalence, or some other more general semantic notion, like Bsameness of content^. This should be rather obvious. Surely, we want to say that a pronoun is the kind of expression that inherits its content from another expression (its antecedent), or an expression due to which the sentence in which it occurs becomes equivalent with some other sentence, or some such. (Salis himself seems to suggest something like this.) But if we do so, anaphoric notions are no longer primitive, but explained in terms of sameness of content. This is in line with the claim I made in my original paper, that anaphoric phenomena should be explained in more general semantic terms. Salis is nevertheless right, I think, that cases of equivalence can be explained by appeal to anaphoric relationships (e.g., that between a token of 'She is happy' and 'Mary is happy', where 'she' cross-refers to 'Mary'). Thus, there is another sense in which claims of equivalence or sameness of content are indeed posterior to claims about anaphoric relationships. For this reason, my original claim that (PT)merely re-describes the data is inaccurate. Philosophia But I think the gist of my original critique is still cogent, although the point must be made more carefully. I spoke as if the following are synonymous: (a) A is the anaphoric antecedent of B (b) A and B have the same content But they obviously aren't, since there are expressions which have the same content but do not involve anaphora. This means that it was misleading at best to say that prosententialists merely Bre-describe the datâ. This also connects with the issue of explanatory primacy. If the sentences above were synonymous, one could complain that explaining why two sentences are equivalent by recourse to an anaphoric dependency between them would be circular: replacing the anaphoric notions in the explanation by their definientia would result in something like: B 'John is happy' is truê is equivalent with (or: has the same content as) 'John is happy' because B 'John is happy' is truê is equivalent with (or: has the same content as) 'John is happy'. Here, 'is the anaphoric antecedent of' is replaced in the right-hand side of 'because' by 'is equivalent with' (or 'has the same content as'). So, I agree with Salis that one can explain why the sentences below are equivalent by recourse to anaphoric dependencies: Mary is happy She is happy [where the antecedent of 'she' is 'Mary'] Thus, the real problem with (PT) isn't quite that it merely re-describes the data. The real problem becomes apparent, however, if we replace the anaphoric terminology with the relevant definientia, using the right definition. I take the following to be a more plausible definition of 'being the antecedent of': A is the (anaphoric) antecedent of B =df B is a token expression which, due to a convention governing the type to which B belongs (e.g., the type 'she'), acquires the content of B because B stands in R to A in the context in which B is tokened. Here, R is the relation holding between two token expressions in a context in virtue of which one is the antecedent of the other in that context. We need not dwell on this relation, which is a matter of controversy. R might be defined in terms of speakers' and hearers' intentions, in terms of relevance or salience, or some other way. Anyone should agree that there must be something that accounts for one token expression being the antecedent of a given token proform, even if neither of the two examples just given fits the bill. Assuming that the definition above is at least along the right lines, we can see more clearly what is wrong with (PT). The problem is that the theory might as well have been formulated without using the notion of anaphora, and instead, more directly, by appeal Philosophia to the more general semantic notions used in the definiens above. That is, 'true' could have been characterized as an expression due to which sentences in which it occurs inherit their contents (or become equivalent with) certain other sentences, namely, those to which they stand in relation R. This account would of course have to be further specified in order to handle the different ways in which 'true' can occur in sentences, but those complications may be set aside here. My point was that anaphoric notions like Bantecedent^ become idle if they can be replaced by more general semantic notions, as suggested here. And we have now seen that this point can be made without making the implausible claim that (a) and (b) are equivalent, or that (PT) merely Bre-describes the datâ. One of my original points was that if the workings of 'true' could be accurately described using the terms, 'proform', 'antecedent', etc., then that description would be idle in the sense that an equally accurate description may as well have been given without appeal to anaphoric notions, but rather in more general semantic terms. I expressed this point by saying that there is nothing Bessentially proformal^ about 'true'. A separate and independent question is whether one can indeed accurately describe the workings of 'true' in anaphoric terms. To this, I responded in the negative, and supported this answer by appeal to the many discrepancies between the relevant sentences containing 'true' and sentences containing uncontroversial examples of anaphoric expressions. And we have just seen that Salis's objection against one of these alleged examples, concerning (1) above, fails. My point about the dissimilarities between 'true' and paradigmatic proforms thus stands: the workings of 'true' cannot be accurately described using terms like 'proform' and 'antecedent', at least not in any way resembling the way in which the original prosententialists or Brandom did it. I also took these discrepancies to be relevant for answering a further question. I asked what 'true' would have to be like in order for the analogy with paradigmatic proforms not to be idle, and answered that it would have to bear striking, unexpected resemblances with paradigmatic proforms, consisting in shared idiosyncrasies, etc. (2009a: 308). But the discrepancies between the two types of expression of course undermine this possibility, too. The word 'true' and paradigmatic proforms do not share many unexpected features, for they do not share many features at all. Considering the specific example of utterances of 'That's true', made in response to a preceding utterance, is there at least something anaphoric about them? I think not. At any rate, it is definitely not obvious that there is. The most natural analysis takes 'that' here to be a demonstrative (cf. Künne (2003: 78f.)). Typically, it would refer to the proposition expressed by the preceding utterance, since that will typically be the most salient entity that can be true. Given the standard propositional truth equivalence schema, we can then straightforwardly explain why 'That is true' comes to be equivalent, on this occasion, with the preceding utterance. Hence, standard Horwichian deflationism handles this type of case without any unobvious appeal to anaphora. (Note also that Brandom himself distinguishes between demonstrative reference to a non-linguistic entity and anaphoric cross-reference to a linguistic one (1994: 456), so he cannot say that the above treatment of 'That is true' conforms to his anaphoric account.) Brandom does claim, however, that demonstrative reference is a kind of anaphoric reference (1994: 460), and takes demonstratives to presuppose anaphora (1994: 460, 511). However, this does not affect my point that 'That is true' is not an obvious case of an anaphoric expression. For that claim is not immediately obvious by itself, and Philosophia neither is Brandom's view about the relationship between demonstratives and anaphora. Thus, my point stands, even lacking an argument against Brandom's claims about demonstratives being a kind of anaphora. Let me also add a further difference between 'true', as Brandom treats it, and paradigmatic proforms. Other proforms are, so to speak, purely proformal; they do not contain material that is not proformal. Consider, 'he', 'thusly', 'so'. But 'true' is supposed to be a mere part of the relevant proform (a prosentence), which can contain all manner of non-proformal expressions, like 'Goldbach's conjecture', 'Everything he said', or 'that snow is white'. On Brandom's view, these other expressions help determine the antecedent. But other proforms simply do not work this way. They immediately cross-refer back to their antecedents without any detour through other expressions (and, again, certainly not in virtue of other expressions' denoting the antecedents). Let us finally consider Salis's positive defence of (PT). He writes, Bwe should not forget that the anaphoric theory is also explanatorily more general [than a theory merely stating the equivalence schema] because it also explains pragmatic and expressive differences between anaphoric antecedents and prosentences^ (2019: 519). Against this, I would claim, firstly, that (PT) does not explain the pragmatic features of 'true'. The relevant Bpragmatic^ features of 'That's true', uttered in response to some claim, are supposed to include, [e]xpressing this claim, distinguishing the endorsement of this claim from the rejection of other claims, acknowledging what was said, doing it with less resources and avoiding the repetition of what was said, and many others––but there is also a general expressive difference. Prosentences are, in general, expressively more powerful than the corresponding sentences (2019: 509). However, as far as I understand these alleged properties of the relevant utterances, they are not clearly Bexplained^ by (PT). Many of them are properties one can easily observe independently of any anaphoric theory of 'true', and these observations could be added to any theory of 'true'. Consider, for instance, the fact that 'That is true' Backnowledges an antecedent^, a feature that would have been absent if the preceding utterance had merely been repeated (cf. Brandom 1994: 302). But the observation that 'That's true' refers to something extraneous is just independently obvious. It is not somehow derived from (PT), it can be made independently of (PT), and it can be added to any theory about 'true'. It may be objected that the above way of accounting for how the utterance Backnowledges an antecedent^ fails, if the latter notion is by definition something involving an anaphor. On the other hand, if so taken, it would be question-begging to demand of other, non-anaphoric theories of 'true' that it explain the feature (so understood). The uncontroversial datum is merely that Bsomething extraneous is acknowledged^, and this datum can be accommodated on other theories, given the analysis of 'That's true' as containing a demonstrative. As for expressive strengthening, it is not clear, firstly, that this feature can be explained by (PT). However, it can clearly be explained on the basis of a simple deflationist theory that takes the meaning of 'true' to be exhaustively given by the claim that it is introduced via the trivial equivalence between 'that p is true' and the Philosophia corresponding 'p' (cf. Båve (2006: 139f.), (2009b: §4), and (2015)). This simple theory also arguably has the resources to explain the meaning of all the different uses of 'true', together with independent assumptions (see Båve (2006: Ch. 4) and (2010)). Thus, given that it scores higher with respect to clarity, explanatory scope, and simplicity, and given how it avoids the various problems we have found with (PT), it is preferable. 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