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CIRCULARITY AND INFINITE LIARLIKE PARADOXES By JESSE BUTLER A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Jesse Butler ACKNOWLEDGMENTS I thank Michael Jubien and Gene Witmer for being on the committee for this thesis and Greg Ray for agreeing to undertake the project and helping me clarify my thoughts. Any remaining obnubilations are entirely my responsibility. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iii A B STRA C T ............... ....................................... .............. ................. v CHAPTER 1 YABLO' S PARADOX IS INFERENTIALLY CIRCULAR BUT NOT REFEREN TIALLY CIRCULAR ................................................................ .................. 1.1 R eh earsin g Y P ..................................................... ................ .. 3 1.2 Inferential C ircularity ............................................................. ....................... 4 1.3 R creating the D ialectic........................... ..... ........................................ ......6 1.4 Characterizing Referential Circularity with Directed Graphs ...........................10 2 TO CONCERN A CIRCULAR PREDICATE IS TO BE RECURSIVELY C O N ST R U C T IB L E ................................................................................. .......... 15 2.1 Graham Priest's Argument Considered More Carefully .................................15 2.2 A Different Formal Treatment of YP and Other Similar Sets of Sentences......... 18 3 THERE IS A PARADOXICAL SET OF SENTENCES THAT DOES NOT CONCERN A CIRCULAR PREDICATE .....................................................23 3.1 Technical Preliminaries and Presentation....................... .................... 23 3.2 Is The Foregoing Really Not Predicatively Circular?..........................................25 3.3 A Paradox That Is Referentially Circular But Not Predicatively Circular ...........30 3.4 C conclusion ............................................................... ...... .......... 31 APPENDIX A USING DIGRAPHS TO DETERMINE WHEN A SET OF SENTENCES FORM S A PARAD O X ................................................................. ............... 35 B AN UNDEFINABILITY THEOREM ............................................. .....................37 C AN SMULLYAN STYLE INCOMPLETENESS THEOREM ...............................38 R E F E R E N C E S ........................................ ........................................................... .. 4 1 B IO G R A PH IC A L SK E TCH ..................................................................... ..................42 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Arts CIRCULARITY AND INFINITE LIARLIKE PARADOXES By Jesse Butler December 2005 Chair: Greg Ray Major Department: Philosophy Before Stephen Yablo's "Paradox without selfreference" it was commonly held that what was loosely and vaguely termed "selfreference" was necessary for semantic paradoxes of the liarlike variety. Yablo claimed that the infinite, liarlike set of sentences he provided proved otherwise. In subsequent years, Beall, Bueno and Colyvan, Priest, Sorensen and Tennant (and others) have debated whether Yablo has in fact demonstrated that selfreference is not required for paradox by exhibiting a paradoxical, yet completely noncircular set of sentences. My aim in this paper is twofold. First, I try to clarify this debate over whether Yablo's set of sentences is selfreferential (in the same way that the traditional liar sentence is) as preparation for the assertion that it is not. I argue that the situation regarding sets of sentences that form candidates for liarlike paradoxes can be plausibly formalized using directed graphs and that this treatment bears out my assertion. Second, I address Priest's claim that even given the lack of this kind of selfreference, Yablo's sentences are in some sense selfreferential because they, in Priest's somewhat mysterious terminology, "concern a circular predicate," as do, according to him, all standard semantic and settheoretic paradoxes. I argue that if a set of sentence concerns a circular predicate, there is a kind of recipe to construct the sentences of that set. Even though the word "recipe" is still a bit too metaphorical to do much good, another formal treatment is available that captures this intuitive notion and yields a sufficient condition for the paradoxicality of infinite sets of liarlike sentences. Using this formal treatment, it is easy to show that there must be an infinite set of liarlike sentences that is not constructible from a recipe, and so in Priest's terms does not concern a circular predicate. Thus, while Yablo's paradoxical set of sentences is circular in Priest's sense, this is incidental. We will have proved that there is a variant of Yablo's example that is not circular in either the narrow referential sense nor in Priest's broader sense. CHAPTER 1 YABLO'S PARADOX IS INFERENTIALLY CIRCULAR BUT NOT REFERENTIALLY CIRCULAR Stephen Yablo (1993)1 billed his paradox referredd to as "Yablo's Paradox" or "YP" hereafter) as a demonstration that selfreference was not necessary for liarlike paradoxes.2 Of course, philosophers have argued that YP either is or is not self referential and have so argued as to whether it fails or succeeds as a demonstration that selfreference need not be involved in paradox. My first goal in this paper is to get clear about several of the arguments for and against YP's selfreferential (or to speak more generally indirectly selfreferential or circular3) character. In sorting out whether YP is or is not circular, and if so in what way, I assert that we can understand more generally the phenomenon of liarlike paradoxes comprising infinite sets of sentences if we try to "picture" graph theoretically the references made by each sentence. With this technique, we see that none of the sentences of YP are indirectly selfreferential with the references they make. So YP is not referentially circular in the way traditional finite liars are. The second part of the paper attempts to show that there is a variant of Yablo's Paradox that does not concern a predicate of the natural numbers with circular satisfaction conditions, 1 For all references to Yablo see (Yablo 1993). 2 I take liarlike paradoxes to comprise sets of sentences each of which only makes truth claims about itself or the other sentences. 3 The idea being that even though the two sentences: "The second sentence enclosed in double quotes on page 1 is true," and "The first sentence enclosed in double quotes on page 1 is untrue," are not themselves selfreferential (each makes no claim about itself) they exhibit indirect selfreference. The first makes a claim of the second which in turn makes a claim about the first. "Circularity" might describe this more general notion. Occasionally, I will use that term to speak of this general form of indirect selfreference. and so is not circular in the sense that Priest (1997)4 claims YP is. I will argue that an infinite list of sentences can concern a circular predicate only if there is a recipe to generate the sentences of the list. We do this by representing each sentence of the infinite, supposedly paradoxical list as a recursive function from a subset of the natural numbers to {0, 1), and placing minimal conditions on the behavior of each of those functions relative to others of the "family" of functions that represent the sentences of the infinite list. Once we have represented infinite lists of sentences that are paradoxical in this way, it is easy to see that the number of such lists is uncountable. The number of recipes for infinite lists is countable, so the paradoxial infinite liars must outstrip the paradoxical infinite liars that concern predicates with circular satisfaction conditions. We begin (1.1) with a short rehearsal of YP by exhibiting the infinite list of sentences that form the paradox, and (1.2) a short demonstration that a most general form of circularity (inferential circularity) must be involved in any sort of liarlike paradox. In 1.3, I will try to recreate the dialectic provided by the arguments of Tennant (1995)5, Priest, Sorenson (1998)6, Beall (2001)7 and Bueno & Colyvan (2003)8 regarding YP's selfreferential character. In 1.4, I will consider Tennant's argument in a bit more detail and elaborate on its shortcomings while tentatively endorsing his conclusion. In that section, I will propose what I believe to be a better method for demonstrating that YP is not even indirectly referentially circular. By using this method, we will be able to 4 For all references to Priest see (Priest 1997). 5 For all references to Tennant see (Tennant 1995). 6 For all references to Sorensen see (Sorensen 1998). 7 For all references to Beall see (Beall 2001). 8 For all references to Bueno and Colyvan see (Bueno and Colyvan 2003). more easily separate the issues of what Sorensen calls, "selfreference at the level of specification and [selfreference] at the level of content," and that this separation will guide us to clearer thinking about the present problem. In 2, I will consider in detail parts of Priest's paper, while explicating and assessing his argument, I propose that his observations may be reformulated in a more productive treatment of the general problem. This will lead to a method of formalizing, in some generality, liarlike paradoxes comprising infinite lists of sentences. In these terms, we will be able to see that Priest's charge that YP "concerns a circular predicate" can be reframed as a observation that there exists a certain recipe for constructing the sentences of YP. Once we have seen that both Priest's and Beall's argument that YP is circular stem from the fact that its the sentences can be so generated, and armed with the formal characterization of paradoxicality in terms of infinite liarlike sets (2.2), we will be in position to show (in 3) that there must be a "Yabloesque" paradox, the sentences of which cannot be generated by a recipe.9 Thus, even if Priest is right that YP is circular in a broad sense, we will have shown that such circularity is not essential to Yabloesque paradoxes. 1.1 Rehearsing YP Yablo's Paradox concerns a countably infinite list of sentences so, sj, s2, of the following sort: so: for all k > 0, sk is untrue. sl: for all k > 1, sk is untrue. 9 We cannot precisely describe it as it contains nonconstructive elements. These sentences clearly form a paradox in the sense that they have no stable assignment of truthvalues. If so is true then each of sk for k > 0 is untrue, so then each of Sk for k > 1 is untrue, but in this case, sl is true after all because that's just what it claims. So so cannot be true because it claimed that sj was untrue. On the other hand, if each of so, s1, S2, ... is untrue then in particular each of S, S2, ... is untrue, and since the latter circumstance is precisely the condition under which so is true, so must be true after all. For future reference, let the set S be this collection of Yablo sentences, { so, s1, S2, .. }. 1.2 Inferential Circularity Before addressing YP in more detail, let us notice a general point. As soon as we begin considering liarlike paradoxes, we notice that there is always some element of what we would commonly term "circularity" involved. More specifically, it is clear that there must be a certain logical relationship between sentences comprising a paradox in light of which we can make an assumption about the truthvalue of a certain sentence, draw inferences about other sentences, then return to (as if we had completed a kind of reasoning circle or loop, but not engaged in circular reasoning) and reassess the truth value of the original sentence. Insofar as any truth valuation of sentences of a liarlike paradox leads to inconsistency and we can demonstrate this inconsistency, it must be the case that those sentences bear certain logical relations to each other that allow for this demonstration. I want to call sets of sentences the members of which bear these logical relationships "inferentially circular" in light of the fact that these sort of sets enable us to draw inferences which are circular in so far as these inferences begin and end with assertions about truth value of the same sentence. For example, if we reason about a "chain" set of liar sentences and see that the set has no truth valuation, we go through a process resembling this: If the (say) ith sentence is true (or untrue), then i+ lt sentence is true (untrue), but this means that the i+2nd is true (untrue), but in that case the ith sentence must have been untrue. On the other hand, if the ith sentence is untrue, then the i+ 1st is untrue (true) and the i+2nd is untrue (true), and .. ., but in that case the ith sentence must have been true. In order for inconsistency to result from each assumed truthvalue of the ith sentence, each of the sentences must bear logical relations to the accompanying sentences. In the case of liarlike paradoxes, sentences make only truth claims of each other, and so it is only such truth claims that provide the logical relationships which lead to inconsistency and paradox. And so every set of liarlike sentences that we claim to be demonstrably paradoxical must, in virtue of that claim, be inferentially circular. YP provides no exception: it must be inferentially circular because we can reason to contradiction from the truth of any of its sentences. In any of the arguments I consider or offer myself, I assume that assertions that YP is or is not circular are not assertions about whether it is or is not inferentially circular. Inferential circularity is essential to any paradox, YP included. For the liarlike paradoxes that came before YP, inferential circularity was typically grounded in some kind of content (what we will call later referential) circularity. That YP has the requisite inferential circularity raises a justifiable suspicion that is must also feature some sort of underlying content circularity. This question is one target of our investigation. Another target is to determine how YP is predicatively circular and whether there other similar sets of sentences that are paradoxical yet are notpredicatively circular. This locution comes from Priest who claims that YP, although not circular in exactly the same way the wellknown liar sentence, is circular in that it "concerns" a circular predicate of the natural numbers and that use of this particular predicate is in some way crucial to understanding the logic of the paradox. The concerned predicate is circular in that its satisfaction by a certain number, n, depends upon whether it is satisfied by any natural number greater than n. The predicate's satisfaction conditions can be stated only in terms of the predicate, that is, only circularly. 1.3 Recreating the Dialectic The goal of this section is to try to recreate a particular dialectic surrounding YP and circularity. In laying out the dialectic it is important to notice first that often the participants talk past each other. Tennant claims that YP is not at all selfreferential, Graham Priest begins to mark a distinction between two different kind of circularity when he says, "Note that each sentence refers to (quantifies over) only sentences later in the sequence. No sentence therefore refers to itself, even in indirect, looplike fashion" (237). But then goes on to claim on the page that, "[his formal diagnosis of YP] show that we have a fixed point ... of exactly the same selfreferential kind as in the liar paradox The circularity [of YP] is not manifest." Priest's claim is that YP merely concerns a circular predicate, and that the predicate's involvement (rather than indirect selfreference) is the source of the paradox's inferential circularity. Much of the foregoing disagreement results because they do not appreciate the distinction between the two different kinds of circularity. To get a handle on things, we need first to get clear on what Tennant's claim is in light of the fact that inferential circularity seems necessary for any sort of liarlike paradox, and second to mark a further distinction in kinds of circularity to help clarify the conversation between Priest, Sorensen, Beall and Bueno and Colyvan. To the first end, if we compare a crude version of apair liar (see footnote 3) with YP, we're immediately struck by the difference: the pair liar contains sentences that make truth claims of themselves (albeit indirectly), whereas the sentences of YP do not seem to do this. These attempts to capture and explain the difference between these two sorts of paradox account for one strain of a dialectic concerning YP. One example I have in mind is Neil Tennant's. If Tennant would agree to our claim that all liarlike paradoxes are inferentially circular (and it seems that he would), we might attribute to him, instead of his more radical pronouncement above, the claim that YP's inferential circular is not rooted in any sort of content circularity, as is the inferential circularity of any finite liar. And so we can read one of Tennant's conclusions as being that inferential circularity need not be rooted in content circularity. I endorse this conclusion and will say more about it shortly. As far as the second issue goes, let us stipulate two further different kinds of circularity. Say that a set of sentences is referentially circular if a particular sentence of the set is either directly selfreferential (like the traditional liar sentence) or indirectly selfreferential. We can explain indirect selfreference by making precise what a sentence of a liarlike set, which is not directly selfreferential, refers to. Such sentence (call itp) makes use of restricted quantification to pick out a set (Q) of sentences (which is a proper subset of the set to which belongs because is not directly selfreferential) and then makes some truth claim of some members of Q. For instance, if the sentences are organized in an ordered list like those of the S of YP, then the sentence might be something like, "Some of the following sentences are untrue," and so Q would be the set of all the sentences that came after in the list. The sentence directly refers to all the sentences of Q. If there is a sentence, q (a member of Q) that directly refers to a set of sentences P, which includes, then we can say thatp refers indirectly to itself (is indirectly selfreferential) via the intermediary q. And so generally, we say that a set of sentences is indirectly selfreferential if a member of the set refers indirectly to itself via any number of intermediaries. For example, the pair liar sentences of footnote 3 clearly form a set that is referentially circular. On the other hand, a set of sentences that, in Priest's terms, concerns a circular predicate is, as we mentioned earlier, predicatively circular.10 With these terms in mind, we see that Priest claims that YP is predicatively circular but not referentially circular. We have seen that every liarlike paradox must be inferentially circular, Tennant has argued that referential circularity is not required for paradox (and more generally inferential circularity), and Priest has hinted that referential circularity is not required for predicative circularity. We still lack a detailed argument for independence of referential and predicative circularity, and whether every liarlike paradox is predicatively circular is an open question. If the distinction makes clearer Priest's claim, it is a bit surprising when Sorensen weighs in on YP that he begins to muddy the waters a bit (145149). Sorensen's intuition is that selfreference at the "level of content" (i.e. referential circularity) can be separated from selfreference (or circularity) at the "level of specification" (i.e. predicative circularity). It seems that Sorensen thinks, as does Priest, that YP is not referentially circular but is predicatively circular, and thinks that predicative circularity can feature in paradoxes the sentences of which are not referentially circular, all of which makes the following claim seem very odd: "Priest is committed to saying that that the infinite, indirect liar is doubly selfreferential once at the level of specification, again at the level of content" (148). Whether or not Sorensen thinks something in Priest's method commits 10 Priest's locution seems a bit mysterious. In 2, I try to fill out a bit of what Priest may intend. him to identifying referential and predicative circularity, the above quote does say that Priest claims that YP is both referentially and predicatively circular. Whatever the reason for Sorensen's misreading of Priest, a confusion begins here which, I believe, runs the whole dialectic offtrack. Beall responds to Sorensen's claim that YP's predicative circularity has no bearing on its referential circularity. In the following quote Beall seems to argue that predicatively circular sets of sentences must also be referentially circular, and cites Priest's demonstration of YP predicative circularity as evidence for its referential circularity (my emphasis; it is meant to show where Beall hints that predicative circularity is not separable from referential circularity). If the foregoing is correct, then the issue becomes clear: if we have fixed the reference of 'Yablo's paradox' at all, then we have fixed the reference of 'Yablo's paradox' via (attributive) description. But, now, the upshot of Priest's point is plain: Priest has shown that any description we employ to pick out (or otherwise define) a Yabloesque sequence is circular; this much Sorensen concedes. From here, however, it is a small step to the circularity of the sequence itself We are fixing the reference of 'Yablo's paradox' via (attributive) description, which means that 'Yablo's paradox' denotes whatever satisfies the given reference fixing description. The situation, however, is this: that the satisfaction conditions of our available referencefixing descriptions require a circular satisfier a sequence that involves circularity, selfreference, a fixed point. Given all this, it follows that the reference of 'Yablo's paradox' is circular... (180) Finally, in response to Beall, Bueno and Colyvan argue that there are denoting terms (like "the integers" or "the supremum of set A") whose referents can only be fixed by the use of a circular recursivee) description but whose referents are not themselves circular in any sort of obvious way. According to them, the situation with YP is analogous. I think that the tedious attention to whether reference to S can be fixed by demonstration plus description or by description alone is misplaced. We could have avoided this problem by paying closer attention to Priest's original paper (or ignoring Sorensen's misreading of it), and by using a more intuitive explanation of exactly why the set of sentences that form YP is not referentially circular. 1.4 Characterizing Referential Circularity with Directed Graphs We want to show ultimately that referential circularity is entirely independent of predicative circularity. A first step toward that is to take a position with regard to the dialectic I have just presented. Specifically, I want to argue that even ifYP is predicatively circular it is not referentially circular. To this end, we will develop a general method for detecting referential circularity independent of predicative circularity, and use it to show that YP is not referentially circular. But before doing so I feel I need to justify the more specific goal. My worry is that Tennant has already accomplished that which I have just articulated, and if only we adverted to his method and conclusion, there would be no need to pursue another way of seeing that YP is not referentially circular. In response to this concern, I will briefly summarize a relevant part of Tennant's paper and then say why his method is not suited for the purposes of the present work. Tennant believes that he's provided a prooftheoretic diagnosis of paradoxicality in previous work which makes selfreference neither necessary nor sufficient for paradox (of the liar sort), and sees YP as a test for his account (199). One conclusion he reaches is that, "[YP's] apparent lack of selfreference ... is reflected, on my account in the non looping character of the reduction sequence .. ." (206). Toward the end, he claims that his method yields a condition for selfreference of liartype sentences, "I shall make so bold as to suggest that it is precisely when the nonterminating reduction procedures enter loops that selfreference is involved [in the paradoxicality of the set of sentences in question] And when they do not enter loops as with Yablo's example then self reference is not involved" (207). Proofs of absurdity in his system from traditional liars do enter loops and thus are selfreferential. These conclusions can be drawn because appropriate proofs of absurdity are available. It is the examination of the structure of proofs from different sets of sentences that allow us to detect selfreference. His paper is concerned specifically with YP, so he makes use of two "id est" inference rules: first, from TSn infer (Vk>n)TSk, and, second, from (Vk>n)TSk infer TSn (203). Since there is a inference rule for each sentence, the proof theoretic procedure is dependent upon knowing (truth functional equivalents of) the truth claims of each sentence of the set. What we should notice is that this sort of proof theoretic approach to detect referential circularity seems to require that a set of sentences be predicatively circular. To show that contradiction results from the supposed truth of any particular sentence, Tennant's proof of absurdity make use of the precise truth claims of each sentence of the set. At least in the kind of proofs that Tennant presents, the id est inference rules require that the absolute (nonrelative) truth claims are made by each and every sentence are available to us. For an infinite set, to know these absolute truth claims, we must have a recipe for what the sentence of the set are or at least for their absolute truth conditions. As foreshadowing, I assert that a set of sentences is predicatively circular only if there is some sort of "recipe" to construct each of the sentences of the set, so a set cannot be predicatively circular if there is no recipe that tells precisely what a sentence of a particular index claims. On the face of things, it seems that Tennant's method for showing circularity depends upon the id est inference rules, or at least on predicatively circularity. Since we want to show that YP is not referentially circular independently of its predicative circularity, there is at least prima facie motivation for us to search for another way to see this. In sum, I agree with Tennant's conclusion and do not want to point out any mistake in his reasoning11, but rather to suggest that we need a different method for detecting referential circularity that is not bound quite so closely to the predicative circularity of a set of sentences. To begin, recall that all such sets liarlike set that are paradoxical are inferentially circular. So in any finite set, we can always "start" at some particular sentence (say i) and by "following" the direct references work our way back to i, so that we see that sentence i was indirectly selfreferential. Intuitively, this is just what referential circularity comes to. If we could somehow "picture" the referential structure of the paradox independently whether the set of sentences concerns any circular predicate, and instead make use only of how each sentences restrictedly quantified over others as an initial step toward making truth claims about those sentences, and then use a specific feature of this picture to assess whether the sentences of the paradox were referentially circular, we would have another way to show that referential circularity was independent from predicative circularity. This method for detecting referential circularity would not be so closely bound to any predicate that the set concerns. So in order to show that YP lacks referential circularity with appeal only to the restricted quantification presented by each of it is sentences12 (rather than explicit appeal to it is particular predicate that the set 11 There is, however, a problem with Tennant's presentation; when he begins the analysis of YP in terms of his formal system, he renders generic members of S "formally and uncontroversially as follows: Sn: V k>n T(Sk)" (203). Greg Ray points out that Tennant makes a use / mention mistake. "s," is the name of a sentence of S, so the "n" in the preceding quoted string is a substring a constituent of that name. It does not make sense to quantify over such a thing. Priest addresses this issue, and I discuss his presentation in the next chapter. 12 Appeal just to this restricted quantification does not hurt our chances for demonstrating a set of sentences not constructible from a recipe because, to foreshadow again, the size of the set of lists of concerns), I develop a graph theoretic test for referential circularity of an arbitrary (finite or infinite) list of sentences that is a candidate for being a liarlike paradoxical set. As a preliminary, a bit about directed graphs digraphss). They can be visualized as geometric figures consisting of vertices (Vi, V2, .) and directed edges (El, E2, .) which connect pairs of vertices by runningfrom one to the other. Apath through a graph is a vertex, edge, vertex, edge, edge, vertex sequence such that if"Vi, Ej, Vk" is an arbitrary portion of the sequence (where possibly, i = k), Ej is an edgefrom Vi to Vk. A cycle is a path that has the same first and last vertex. From a set of sentences that is a paradox candidate, we can construct a directed graph (G) that will have the same number (n) of vertices as the candidate has sentences (l sentence refers or quantifies over thejth, kth, ..., /th ... sentences in the truth claims it makes, draw an edge from Vi to each of Vj, Vk, ... V .. 13 The list of sentences is graphically circular iff its corresponding directed graph, G, has a cycle. sentences each of whom restrictedly quantifies over all those who follow it outstrips the size of the set of lists of sentences that are recursively constructible (alternatively: "concerning a circular predicates"). We will find one that is a member of the first set and not a member of the second. And then, we will be almost all the way to finding a referentially circular set of sentences that is not predicatively circular. 13 The intuitive idea is to connect with edges all those vertices which correspond to the sentences of which the ith sentence restrictedly quantifies over in order to makes truth claims. At first, it may be troubling that two sets comprising dissimilar sentences may have exactly the same corresponding digraph; two sets of sentences will have the same corresponding digraph exactly when corresponding sentence (sentences with the same index) of the respective sets restrictedly quantify over the same sets. For example, consider two infinite sequences of sentences (S' and S") as candidates for a liarlike paradox. The first comprises only sentences of the form, "s',: at least two of the following are untrue," and the second comprises only sentences of the form, "s",: at least fortyeight of the following are untrue." Every sentence of S' and S" quantifies over all the sentences that come "after it" in S' or S". And so, the digraphs corresponding to each of the candidates will have the same edge / vertex structure, i.e. Edges from VI to each of V2, V3, V4, .. , edges from V2 to each of V3, V4, V5,.. That the corresponding graphs Gs, and Gs,, have exactly the same structure is not troublesome because, to consider the question most generally, we realize after some consideration that the truth conditions of s', and s", depend upon any of the sentences of S' or S" respectively that have index higher that i. Graphical circularity of a set of sentences guarantees that the corresponding graph G is "circular" in the sense that we could traverse G (starting and ending at some Vi) if we were allowed to "step" from one vertex to another iff there was an edge from the former to the latter. Since the vertex / edge structure of G mimics the truth claim structure of the corresponding set of sentences at least one of the sentences of the graphically circular set must be directly or indirectly selfreferential. And so I claim that a set of sentences of a candidate for a liarlike paradox is referentially circular iff the corresponding graph is graphically circular.14 Of course, it is possible for a set of sentences to be referentially circular and not form a traditionally paradoxical set (for example a truth teller). If a detection method for referentially circular paradoxicality is wanted, it is easily accomplished by considering algorithms for coloring our directed graph in an appropriate way. I hope that this technique for detecting paradoxes will go some way toward vindicating our characterization of referentially circular sentences in terms of corresponding graphically circular graphs. We provide a formal method for so deciding when a candidate forms a paradox in Appendix A. As expected, the set of sentences that constitute YP turn out not referentially circular on our characterization. If we form the associated graph and call it Gyp, notice that for each i, all the edges from Vi (edges which mimic the truth claims of the ith sentence) go to only vertices with index greater than i, so it is impossible for Gyp to contain a cycle and so impossible for it to be graphically circular. 14 Something else to notice is that if we require that a sentence from language L1 can make truth claims about a sentence from L2 iff L is a metalanguage for L2, then sets of sentences whose corresponding graphs are circular are disallowed. If the 1st sentence claims something about the truth of the 2nd sentence, the 2nd about the 3rd, then if we're observing the metalanguage / object language distinction, no sentence is allowed to make truth claims of a sentence whose number is less than the number of the former. It is just this (forbidden) kind of structure that allowed for circular digraphs. CHAPTER 2 TO CONCERN A CIRCULAR PREDICATE IS TO BE RECURSIVELY CONSTRUCTIBLE I have argued that a simpleminded graph theoretic technique shows us that YP is not referentially circular, and so have endorsed a claim made by Priest. I turn now to his assertion that YP is in some sense circular because it concerns a predicate with circular satisfaction conditions (in my terminology predicatively circular). I think the intuition is right YP does in some sense concern a predicate with circular satisfaction conditions. I also think that this is not an essential feature of Yablotype paradoxes, but to see this we must first get clear about Priest's argument. 2.1 Graham Priest's Argument Considered More Carefully Recall that the sentences of S clearly form a paradox in the sense that they have no stable assignment of truthvalues, and we can see this because we can run through the reasoning like that presented in 1.1. Graham Priest maintains that YP involves self reference, but is not referentially circular. Formalizing the sentences with a truth predicate, T, we have that for all natural numbers, n, sn is the sentence Vk > n, Tsk. Note that each sentence refers to (quantifies over) only sentences later in the sequence. No sentence, therefore, refers to itself, even in an indirect, looplike, fashion (237). He goes on to argue that YP concerns a circular predicate. He begins by proposing a formalization of the reasoning to contradiction from an arbitrary sentence of S, 1. Tsn < (Vk > n), Tsk 2. : T Tsn+1 3. Tsn < (Vk > n), Tsk 4. (Vk> n+1), Tsk 5. > Tsn+1 6. X (from (2) and (5)) 7. > Tsn 8. > (Vk) Tsk 9. (Vk > 0) Tsk 10. Tso 11. X (from (8) and (10)). Where (8) must be justified by something akin to universal generalization for Priest's semiformal scheme, since the "n" of the previous lines was arbitrary. The observation is that since we generalize away the "n" in step (8) it must play the role of a variable in the above treatment. In that case, "sn+1" cannot be the name of a sentence. Rather, Priest says, "s" must be understood as a name of a predicate which applies to natural numbers. But in that case, we cannot just apply the truth predicate to occurrences of, e.g., "sn+'". We have to make use of the twoplace satisfaction (SAT) relation between numbers and predicates, and s, a oneplace predicate of the natural numbers, and rewrite as follows.1 12. SAT(n, S) <:> (Vk > n), SAT(k, S) 13. > SAT(n+l, S) 14. SAT(n, S) <:> (Vk > n), SAT(k, S) 15. (Vk > n+l), SAT(k, S) 16. SAT(n+l, S) 17. X (from (13) and (16)) 18. ,SAT(n, S) 19. > (Vk)SAT(k, S) 20. (Vk > 0)SAT(k, S) 21. SAT(0, S) 22. X (from (19) and (21)) 1 Priest does not carry out the following derivation but merely suggests what it might look like. Priest's assessment of the situation is a little hard to follow2, but we can see that the original formulation began (1) with premise stating the truth conditions of an arbitrary sentence of S. The second formulation starts with a statement of the satisfaction conditions of the predicate S. We see from (12) plainly that the predicate has circular satisfaction conditions. I think that something in Priest's claim rings true. In his terms, predicate s has a fixed point, and if his treatment succeeds in capturing what turns the trick in YP, then there is a sort of circularity involved. One difficulty, however, is to see from his presentation precisely what role he thinks predicate s plays in YP. What does it mean for a set of sentences to "concern" a circular predicate? Priest claims to have uncovered and made explicit the logical form of the sentences of the Yablo sequence, and, in so doing, he's shown that each of these logical equivalents refer to the predicate s. So the natural understanding of "concern" is that each sentence of the set mentions that predicate. The set is predicatively circular because the predicate so referenced has circular satisfaction conditions.3 2 Priest claims on 237 that "the fact that s = '(Vk > x)( SAT(k, s))' shows that we have a fixed point, s, here of exactly the same selfreferential kind as in the liar paradox. In a nutshell, s is the predicate 'no number greater than x satisfies this predicate'. The circularity is now manifest." Professor Ray points out that it is a bit misleading to claim that s is '(Vk > x)(SAT(k, s))' because s could be any predicate which had the same satisfaction conditions. 3 It might also be argued that each sentence of the Yablo sequence uses the predicate s, and that a set is predicatively circularity if each member uses a predicate with circular satisfaction conditions. In this case, there are arguments exactly parallel to the one in chapter 3 aimed to show that there is a paradoxical sequence that is not predicatively circular. 2.2 A Different Formal Treatment of YP and Other Similar Sets of Sentences We're aiming to show that there is a set of sentences that is paradoxical but which is not predicatively circular. To do this, we will try to show that there is a class of sentences for which there can be no recipe we cannot tell precisely what the truth conditions for an arbitrary sentence are but which is clearly paradoxical. If we can show that there is such a class, then we can reason that Priest won't be in a position to assert that each sentence of any member of this class must reference a predicate with circular satisfaction conditions. There are some sentence sets that are paradoxical that needn't be predicatively circular. At first glance, our task may seem impossible: if a set of sentences is not predicatively circular, we cannot know the truth conditions of an arbitrary member of the set. How are we to know that the sequence is paradoxical? The only method we have seen so far for determining paradox is the sort of reasoning in sentences (12)(22). That treatment made use of a predicate that was mentioned by each sentence of a particular list to give truth conditions in terms of the truth or untruth of each of the subsequent sentences. In this section, we will argue that we can characterize a list of sentences without appeal to a single, certain circular predicate that is mentioned by sentences. If certain restrictions can be placed on the truth conditions set forth by an arbitrary sentence of the set, perhaps only relative to the truth conditions of another sentence, then it is possible to argue that the set forms a paradox without reasoning that makes use of precisely specified truth conditions for an arbitrary sentence. The motivation behind this technique should become clear as we develop the method in the next few pages. Recall that s,, the ith sentence of S, is "For all k > i, sk is untrue." If we let 0 go proxy for "untrue", 1 for "true", and in some loose way (to be made clear) associate the output of a function from a subset of the natural numbers to {0,1 at a particular value, n, with the truth claim made about sentence with index n, by a the sentence corresponding to this function, then for each i,f imitates s,, where for each i,J has domain {y: y > i} and for all n > i,f(n) = 0. Briefly, just as YP's ith sentence claims that each of the succeeding sentences is untrue, each is zero over its whole domain (natural numbers greater than i). Working with these sort of "imitators" is the first step toward our more general characterization of Yabloesque liars. To capture more of what was at work in YP, recall that the first part of its bite came from the fact that, for instance so and sl made exactly the same truth claims about each sentence whose index was greater than 1, so if the first were true then we could reason that the second would have to be true also, but the first claimed the second was untrue, so, on pain of contradiction, the first must have been untrue after all. Similar reasoning could be repeated for any of the sentences of S. The second part of the bite was that one of S had to be true, otherwise, since what (say) so claims to be the case would in fact be the case, and so would have been true after all. Another way in which inconsistency might occur in an ordered list sentences which only make truth claims about each other is if sentence x claims that both sentences and z are true, buty claims that z is untrue.4 Of course, the last was not a feature of the sentences of YP. With this in mind, we can create a model of infinite sets of sentences each of which makes only truth claims about those of greater index, for which there is a formal analog of paradoxicality. 4 There are more ways in which the truth of one of such a list of sentences could lead to inconsistency. For instance, if sentence with index 6 (call it "s'") claimed that s7 was true and that s9 was untrue, that s7 claimed that s8 was true and s8 claimed that s9 was true. I do not consider this sort of pathology because the aim is to show that if we have a set of sentences which spawn inconsistency only by the two ways mentioned, then the paradoxical sets "outstrip" those which concern a circular predicate. To do this it will be enough to consider the two kinds of pathology just mentioned. The type of functions we need for this task will be doubly indexed with subscripts and collected into sets with an order induced by their subscripts. I will call the ordered sets into which our functions are gathered "families"5 from now on for convenience. The constituent functions themselves, like (23) above, are supposed to imitate the ordered, infinite member sentences of a candidate for a liarlike paradox and each family will represent a single candidate for paradox. For example, if such a family is D1 = {dil, d,2, d,3 ... }, then d1,n has as domain the natural numbers greater than n and as range {0, 1}. The domains of the members of a family are so constrained because the sets of English sentences to be based on these families have a corresponding constraint each member of the sort of sets we will be considering makes claims only of sentences of greater index. In other words, the sets under consideration are not referentially circular. Recall that the members of the range will be proxies for "true" (1) and "untrue" (0). Call a member function, d,;, nullish iffdf for all x in its domain, dl,,(x) = 0. Call a member function, d;,, seminullish iffdf for some x > i, for all y > x, dl,(y) = 0. Say that member function, d1,, triangulates member function d,j (j > i) iffdf there is some x >j such that di,(j) = 1, and di,,(x) di,j(x). A family of such functions, D1, is Riff each of the member functions {d1,1, di,, dL3 . S} is recursive and is Piff the family contains an infinite number of nullish member functions, and for each member function which is not nullish either it is seminullish or it 5 One might be concerned that "family" is a poor choice of terminology for what are essentially ordered lists of functions. I think "family" is a better, more descriptive term that is appropriate because an order is induced on a family by the fact that each of its members has a different domain a function whose domain is a subset of another can be said to follow the latter in order in which functions are collected into families. triangulates some other member function. Call a family RP if it is both Rand P.6 In these terms, a family each of whose members is nullish imitates YP, and such a list is RP. Lemma 2.1: Each member of an Rfamily expresses a set that can be used to provide the truth conditions for a sentence of an infinite liarlike set of sentences. Proof of Lemma 2.1: Since each member is recursive, the set of ordered pairs that is the graph of that function is recursive and so is the set of ordered pairs that results if, in that graph, we replace each natural number n in the first position of each ordered pair with "sentence with index n" and replace each 0 in the second position of each ordered pair with "is untrue" and replace each 1 in the second position of each ordered pair with "is true." The set resulting from these substitutions is exactly the kind suitable for giving the truth conditions for one of an infinite set of liarlike sentences. Since the member functions of an Rfamily are recursive, they can be expressed by sentences in the language of arithmetic, and sentences of the language of arithmetic can be translated into English sentences. Claim 2.2: With each RP family there can be associated an infinite set of English sentences that forms a paradox. Proof of Claim 2.2: From lemma 2.1, we see that each member of an RP family can be translated into an English sentence that gives the truth conditions for a sentence of an infinite liarlike set. So if, for each i, we translate the ith member function 6 Intuitively, there must be some sort of restriction on the functions that are the members of a family because we want those functions to correspond to (and so be translatable into) English sentences. So the member functions of a family should at least be restricted to the class of recursively enumerable functions because, roughly, those functions are specifiable with a finite string of characters. So, such functions could be translated into a finite sequence of English words. On the other hand, it might be a bit more realistic to either take the range of each d, to be {0,1,2} corresponding to untrue, true and neither true nor untrue (meaning that the if sentence makes no claim about thej] sentence if d,(Q) = 2) or require that each d, be only recursively enumerable rather than recursive. In the later case, the domain of each d, would not be necessarily all of the natural numbers greater than i, we could say that d, is defined atj iff ith sentence makes a truth claims of thejth sentence. Or finally, we could say that each d, was only r.e. and that the range of the each d, was {0,1,2}. I believe the non predicatively circular paradox I try to demonstrate in the sequel and the method I use to construct it would go through, with slight modifications, with these strengthened models. But more importantly for my argument, I consider a non predicatively circular paradox by trying to show that the number of P families (a subset of each of the other families canvased in this footnote) is uncountable. It follows that the whole set of families considered here is uncountable and so a non predicatively circular paradox would exist among the extended set of families. A related worry is that I have given only sufficient conditions for paradoxes of this sort. There are sets of sentences that do not correspond to an p list but are in fact paradoxical. We will see an example in the conclusion. I do not think I have to address this issue in my argument because I'm trying to show that there is non predicatively circular paradox even if we restrict our attention to a single kind ofsentences which might make up a list of sentences of a semantic, liarlike paradox, so of course it follows that there will be a noncircular paradox if we consider more the exotic kinds of sentences of which these paradoxes could be comprised. 22 into an English sentence that gives these truth conditions (which is the sentence with index i), we will have an infinite liarlike set of sentences. This set is paradoxical because there is no truth valuation for each of its sentences. To see this, note first that none of the sentences that correspond to seminullish member functions can be true without contradiction by Yabloesque reasoning. The only remaining sentences to consider are those which correspond to the triangulating member functions. None of these can be true without contradiction because if the sentence with (say) index i, were true, then for somej (> i) and k (>j) the sentences with indicesj and k would both be true, but if the sentence with index were true then the sentence with index k would be untrue, a contradiction. Each of the sentences of the set cannot be untrue, because, again by Yabloesque reasoning, if dn is a nullish member, the sentence with index n would be true. There is no truth valuation for this set. 0 CHAPTER 3 THERE IS A PARADOXICAL SET OF SENTENCES THAT DOES NOT CONCERN A CIRCULAR PREDICATE Priest claims that the "situation involved in Yablo's paradox, however formulated, is intrinsically circular, in exactly the same way that those involved in more familiar paradoxes of the family are." (Priest, 240) In this chapter, we take the final steps in showing that there is a liarlike paradox that is not predicatively circular. The paradox is unfamiliar and must remain so because it avoids Priest's intrinsic circularity charge by having a structure that is, roughly speaking, not recursive. This rough characterization will be made precise shortly; we will use the machinery we have set up in the previous section. 3.1 Technical Preliminaries and Presentation Our plan is to show that there is a set of sentences that is paradoxical but not predicatively circular. I argued in the last chapter that any such set of sentences could equivalently be characterized as constructible from a recipe. If for every infinite set of sentences that were paradoxical, there were a method to construct the members of that set, and we help ourselves to Church's Thesis the claim that for every algorithm or recipe there is a corresponding recursive function we could claim of an arbitrary family, Da, there is recursive function (fa) such thatfa somehow encodes Da. Since Da is itself essentially a list of functions, one way thatfa could encode Da is by encoding it is member functions: for each natural number n,f(n) = number code, in an appropriate godel coding, of the function d,,). Given this encoding of Da, we could characterize the entire set of paradoxical infinite sets of sentences with a set of recursive functions. To spell out more thoroughly the present suggestion, say that for any RP family, Da, if is a recursive function,fa, such that for each i,fa(i) = to be the recursive description of RP family Da. And so there is a set, F, of recursive functions that includes an encoding function for each RPfamily. Since the set of recursive functions is countable1, and F is a subset of this set, F is countable. Our assumption is that each member ofF encodes an RP family, and there is one member of F for each RP family, so the set of RP families must be countable. So to show that there is a family that is RPthat is not specified by a recursive function2, it suffices to show that the set of lists that are RPis not countable. That's what we turn to next. Lemma 3.1: For any n, if {D1, D2, D, Dn} is a partial list of the RP families, then there is a partial list, {D1, D, Ds,, D, n, Dn+ }, of the P families such that the n+1st member of Dn+ is not nullish. Proof of Lemma 3.1: Consider a set (D) of families each of whose members are nullish except the n+1st. Each of D is RPbecause the n+1st member must be either a triangulator or seminullish, and D is infinite (because there are infinitely many appropriate triangulators). For any n, if D, D2, ... D, is a partial list of the RP families, then we can form the partial list {D1, D2, D,, Dn+ by letting Dn+ be a member of D. We are guaranteed that there is such a family because the first partial list is finite. E Claim (3.1) The set of RPlists is uncountable. Proof of Claim (3.1): Assume otherwise for contradiction. Since the set of RP families is countable and because of lemma 3.1, we can assume that they can be listed D1, D2, ... so that for each m, D2m is a family such that it is 2mth member is not nullish. We will demonstrate D* = {d*l, d*2, d*3, } an RPfamily that is 1 Because every recursive function can be paired with the appropriate Turing Machine, each of which can be uniquely encoded by an natural number. 2 1 will call this sort a "nonrecursive RP family". not among the proposed list. For even n, let d*n be nullish, and for odd n, if dn, (the nth member function of the nth family of our proposed countable set) is nullish, let d*n(n + 1) = 1 and for all x > n + 1 let d*n(x) = 0, otherwise let d*n be nullish. D* is RPbecause there are an infinite number of nullish members, and every member is seminullish. D* is not one of D, D2, ... because for all n, d*n: dn. Contradiction, the set of RPfamily is not countable. U Corollary 1: There are uncountably many infinite sets of English sentences that form liarlike paradoxes. Corollary 2: There is an infinite set of English sentences that form a semantic, liar like paradox that is not underwritten by a circular predicate. Corollary 1 follows by reasoning similar to that of the proofs of Lemma 2.1 and Claim 2.2. Corollary 2 follows by corollary 1 and the reasoning presented just before Lemma 3.1. To point towards a Yabloesque paradox that does not concern a circular predicate (because it is not recursively constructible), we can gesture toward what D* might look like were a corresponding list of English sentences given in the manner of Claim 2.2. The paradox is an infinite list of indexed sentences {So, sl, such that, (ifn is even) Sn is "For k > n, sk is untrue." (ifm is odd) Sm is "For k > m, sk is untrue." (OR) "s,m+ is true, and for k > m + 1, sk is untrue." 3.2 Is The Foregoing Really Not Predicatively Circular? With the hint of the set of sentences based on D* in mind, we now argue that the class of Yabloesque sequences which are not recursive are not predicatively circular. Our strategy is the following. I will present an informal argument that there is no truth valuation for the set of sentences based on D*, then I will try to mimic Priest's method to show that this set concerns a circular predicate. I hope we will see that it is not the case that there must be such a predicate. Argument that sentence set based on nonrecursive RP list D* is paradoxical. A. For arbitrary n, sentence with index n claims either that every sentence with index greater than n is untrue, or, that sentence with index n + 1 is true and each sentence with index greater than n + 1 is untrue. B. For an even > n + 1 sentence with indexp claims that every sentence with index greater thanp is untrue. C. Sentence n cannot be true. If it were, then sentence p would be untrue, but every sentence with index greater than p would also be untrue. This exactly what sentence p claims, so sentence p would be true after all. D. Index n was arbitrary so similar reasoning could be used to show that contradiction followed from the truth of any sentence of the set. E. It cannot be that each of the sentences of the set is untrue, because in that case, since there is a sentence (let it is index be s) that claims that every sentence of higher index is untrue, and by hypothesis each of the these sentences would be untrue, sentence s would be true. F. (Conclusion): the set is paradoxical. First notice that in this argument (a similar argument could be run for any non recursive sequence that met the RP condition), names of particular sentence never occur; only statements about the truth conditions of arbitrary sentences. Nevertheless, we can try, as Priest does, to "uncover" the logical form of the sentences. Instead of trying to formalize the argument with something similar to (1)(11), let us note that were we to do that, we would have to do something like quantify over the "n" of string such as "sn", and so we would reason, as Priest has, that this "n" must have been playing the role of variable. So I will skip straight to an attempt at carrying out Priest's method for discovering the circular predicate that is concerned by the set of sentences. 23. SAT(s, n) > (k > n + 1) SAT(s, k) (from (A)) 24. for an even > n + 1, SAT(s, p) <:> (k > p)SAT(s, k) (from (B)) 25. SAT(s, n) > SAT(s, p) (from (23)) 26. SAT(s, n) :> (k > p) SAT(s, p) (from (23)) 27. SAT(s, p) (from (24) and (26)) 28. SAT(s, n) (from (25) and (27)) 29. (n)SAT(s, n) (gen. based on (23)(28)) 30. (k > p) SAT(s, k) (reasoning about "for all") 31. SAT(s, p) (from (24) and (30)) 32. X ((29) and (31)) Now the most we can say about this supposedly concerned predicate is that its satisfaction conditions can be represented as: 33. SAT(s, n) <> (k > n)SAT(s, k) iffn is even 34. SAT(s, n) (k > n + 1)SAT(s, k) iffn is odd There couldbe a predicate here of whose satisfaction conditions we can assert (33) and (34), but we cannot give necessary and sufficient satisfaction conditions for it, because of the way that a nonrecursive RP list was defined. So by Priest's lights, we cannot assert that the set concerns any specific predicate because, according to his presentation it seems that we must know the necessary and sufficient satisfaction conditions (or at least logical equivalents) to determine that a circular predicate is concerned by the set in question. In the present case, we do not know what the predicate is, and so our claim that whatever it is it must be concerned seems arbitrary. Perhaps we should a different argument to the effect that there are two predicates (one corresponding to the nullish member functions and one corresponding to the semi nullish member functions) one of which must be concerned by every member of the sentence set. A first try using two predicates might be the following. 35. SAT(s, n) iff (k > n) SAT(s, k) and 36. SAT(s*, n) iff SAT(s*, n + 1) & (k > n + 1)SAT(s*, k) This approach won't work because the sentences were meant to quantify over each of those that followed. In the case of (35) and (36), it seems that sentences which concern s, for instance, do not refer to those which refer to s*. Perhaps there is another work around to make the predicate disjunctive in the following sort of way. 37. SAT(s,n) iff (k>n) SAT(s,k) or SAT(s,n+l)&(k>n+l)SAT(s,k) This approach does not work either because we need s to have the nondisjunctive satisfaction condition (for at least some natural numbers) "SAT(s, n) iff (k > n) SAT(s, k)" for step (24) of the reasoning presented in (23)(32) to go through.3 The possibility of a "closed form" presentation of the satisfaction conditions of predicate s is not available because the nature of the RP family on which this sentence set is based.4 Perhaps, as a final attempt, we could claim that, in principle, there exist sentences that give the satisfaction conditions of s in the form of an infinite (nonrecursive) list such as the following: 38. SAT(s, 0) iff (k > 0)SAT(s, k) 3 Any hope for the the sort of predicate whose satisfaction conditions are given by (43) is lost because each family can include triangulating member functions. In any one RP family there can be infinitely triangulating member functions such that (if we think about functions set theoretically) no triangulator function is a subset of another. In that case, there is no hope for a closed form satisfaction condition for the predicate such a family might concerned. 4 That is, we could not list such satisfaction conditions in a sort of closed form way because (first) there could be an (plist that contained an infinite number of different triangulators as in the previous footnote, and (second) D* is nonrecursive. 39. SAT (s, 1) iff SAT(s, 2) & (k > 2)SAT(s, k) 40. SAT (s, 2) iff (k > 2)SAT(s, k) 41. SAT (s, 3)iff(k> 3)SAT(s,k) 42 . Though this sort of list might give satisfaction conditions of a predicate which could be mentioned by each sentence of the set based on D*, it is important to recall that the construction of D* showed that there were an uncountably many nonrecursive Yabloesque sequences which were paradoxical. We can reason that each of these sequences form a paradox in a manner similar to (A)(F). Any attempt to uncover the logical form of these sentences a la Priest must rest on the assumption that the sentences of this sequence have a specific form (i.e. include names of the sentences that follow). But that such sentences have this form is left underdetermined by the argument we have presented, as names for specific sentences do not appear in it. So the assertion that each sentence refers to a single predicate is only hypothesis it cannot be shown that each of the sentences of a member of this class must make reference to a single predicate with circular satisfaction conditions. As a matter of fact, we can stipulate that there is a sequence based on a nonrecursive RP family each sentence of which refers to a different predicate, as reasoning similar to that presented in (A)(F) will go for such a sequence. So in the case of a sequence based on a arbitrary nonrecursive RP family, we have, so far, no reason to claim that the set is predicatively circular. Indeed, we have evidence that there are such sequences that are not predicatively circular. Finally we should consider whether the RP condition itself could be the thing concerned by a liarlike set of sentences which is based on an P family. The answer seems to be no. A liarlike set of sentences could not concern a predicate (if any there be) that described generally the properties of an RPlist if we were to be able both to reason to a contradiction and if the sentences were to make only truth claims of each other, because the truth conditions of sentences that concerned that sort of predicate would not be the truth or untruth of other sentences of the set, but would rather be the truth conditions of their successors. This is the case because the RP condition was meant to constrain the truth conditions of a given sentence relative to the truth conditions of other sentences rather than specifying the truth conditions of particular sentences outright. It is not obvious that a set of sentences that concerned the predicate formulation of the RP condition could constitute a liar paradox.5 3.3 A Paradox That Is Referentially Circular But Not Predicatively Circular One concern of the dialectic is the general concern over whether circularity is necessary for paradox. I have tried to show that there are infinite liarlike paradoxes that are not referentially circular that are also not predicatively circular. One last question may be whether the two types of circularity are independent of each other. We could answer this question affirmatively if we could show that there is a paradoxical infinite set of sentences that is referentially circular and not predicatively circular. To show that notice that if we change the sentence s2 of the paradox based on D* from "For k > 2, Sk is untrue." To "For k > 2, Sk is untrue, and sl is untrue." Now the corresponding digraph has a cycle, and the set is referentially circular. 5 Each sentence would not claim truth or untruth of it is companions, but would make claims about their truth conditions sentences of this sort are not the appropriate kind for liarlike paradoxes. Finally, we may also notice now that sets of sentences which are either referentially circular or predicatively circular, both or neither can fail to form paradoxes.6 It seems that even though inferential circularity is required, paradox is independent from the referential and predicative circularity we have discussed in the preceding. 3.4 Conclusion We tried to show in the third section of the first chapter that every liarlike paradox must be inferentially circular, that is, each sentence of must bear certain logical relationships to the other sentences. Because we have restricted our discussion to liar like paradoxes, we know that these logical relations must be borne in virtue of a sentence's making truth claims about other sentences of the paradox. Sentences which bear the appropriate relations may be generated by a paradox with a referentially circular structure, which means that a sentence of the set refers indirectly to itself, by means of its truth claims and the attributions made by sentences of which it makes truth claims. Given that the "punch" of liarlike paradoxes is not a result of vagueness or other vagaries of natural language, there exists a graph theoretic tool to detect referential circularity and another to detect paradoxicality of referentially circular sets of sentences. I believe that so separating the search for indirect selfreference and the conditions under which a set of sentences forms a paradox is generally more informative than a proof theoretic method like Tennant's. Sets of sentences which make only truth claims about each other might be paradoxical without being referentially circular and vice versa. 6 Examples of non paradoxical sets of sentences that are (1) predicative, nonreferential: a set like Yablo's except substitute "true" for "untrue"; (2) referential, nonpredicative: a set of sentences based on a non recursive Rfamily that contains only one seminullish member function (which is also nullish) and that is slightly modified such that its graph has a cycle; (3) both: a set exactly like the predicative, nonreferential set above except that one sentence is modified such that the graph has a cycle; (4) neither: a set of sentences based on the nonrecursive Rlist above in (2) without the modification that caused the cycle in the graph. It turns out on the present suggestion that YP is not referentially circular, but is, as Priest argues predicatively circular. The process of assessing the methods of Priest's argument that YP is underwritten by a circular predicate helps us see that if a set of sentences is predicatively circular, that set of sentences must be constructible from a recipe. Since we can imagine what's minimally required for a set of sentences to lack an assignment of truthvalues, we can formalize with families of recursive functions the behavior of a subset of infinite liarlike paradoxes. Once we have seen how this might come off, it is a short step to showing that there must be a nonrecursive list of functions which meets the formal requirements for paradox because such families form an uncountable set. One result of the whole adventure is that we can provide the sketch of a paradox the sentences of which is not predicatively circular. And so it does not seem that predicative circularity is a necessary feature of liarlike paradoxes. It might be thought we could go further and offer an analysis of paradoxicality in the context of infinite sets of the sentences with just a bit more development of the notion of RP families. But I think we will see that it would take significantly more work to come up with necessary and sufficient conditions for such paradoxicality. To see this, consider an infinite list of sentences indicated by the following schema: 43. sn: (3x > n)(Vy > x) sy is true. Another rendition of this sentence might be "sn: 'infinitely many sentences of index greater than n are untrue.'" This list of sentences forms a paradox: if so is true then for every m > 0 Sm is true because there are always an infinite number of sentences with index greater than m which are untrue, but in this case so must be untrue, there being no untrue sentences that follow it. On the other hand if so is untrue then there are only finitely many sentences with higher index that are untrue, but this situation leads to contradiction because there must be some > 0 such that sp is true, but this is the case only if infinitely many of the sentences that follow sp are untrue, in which case so would have been true after all. Similar reasoning holds for sentences with arbitrary index. This set of sentences is paradoxical but is not captured by the RP family characterization. What has gone wrong? Even though this set of sentences is predicatively circular, the sentences do not recursively describe a single set of truth values for the subsequent sentences (as did each member of an RP family formally as a function from a subset of the natural numbers to {0,1 }), rather the sentences describe sets of truthvalues for all subsequent sentences. To illustrate, (43) describes sets of truth values for all subsequent sentences in which there are infinitely many untrue sentences. Part of the "bite" comes from the fact that for any natural numbers n and m, the intersection of the sets described by nth and mth sentences of such a set is nonempty and so ifn = 0, and the first sentence is true then for any natural number the mth must be true. All the subsequent sentences must be true, but each claims that infinitely many of the following are untrue. A general characterization of this sort of phenomenon might be available something, for example, that made more explicit use of set theory but that project exceeds the scope of what we would set out to do here. Finally, a few last points about the possibility of formal results available from paradoxes comprising infinite sets of liarlike sentences. If the demonstrations offered in the appendices are successful, the result is that using the notion of RP families we can prove both an undefinability theorem similar to Theorem 1 of Mostowski, Robinson and Tarski (1953)7 and an incompleteness theorem similar to that given by Smullyan (1992).8 Theorem 1 of the former work shows that if formal system Tis to be consistent then the set V of all and only those numbers which are godel codes of valid sentences in T and diagonal function D9 cannot both be definable. Appendix B shows that there is a set Y defined with the help of predicate that simulates an RP family that cannot be definable along with Vif the formal system in which they're defined is to be consistent. Appendix C shows that for certain systems in which an RPlist can be defined, there is a godel numbering for the expressions of that language relative to which there are infinitely many undecidable sentences for that system. It is interesting to note that both of these results to nothing to lower the standard that's required for the existence of an undefinable set within a system or an undecidable sentence. In each case, undefinable sets and undecidable sentences would already be features of systems which were powerful enough to define RP families, i.e. systems that were capable of defining recursive sets. Another point of interest is that the nonrecursive RP families we labored so long to demonstrate cannot be used in either formal result. Both appendices require that the RP family in question be recursive, otherwise in the case of Appendix B we could not speak of " and in the case of Appendix C we could not recover the sequence of godel codes of so, si, s2, .... So it is unclear whether any additional technical result is available from the existence of nonrecursive P families SFor all references to Mostowski, Robinson and Tarski see (Mostowski, Robinson and Tarski 1953) 8 For all references to Smullyan see (Smullyan 1992). 9 "Dn is the number correlated with the expression which is obtained from En by replacing everywhere the variable u by the term An." (46) For example, if, for one place predicate H, APPENDIX A USING DIGRAPHS TO DETERMINE WHEN A SET OF SENTENCES FORMS A PARADOX We give a method for determining a sufficient condition for whether a set of referentially circular sentences forms a paradox using A, an algorithm which colors in stages (nondeterministically) the vertices of G with color1 and color2. A { Let x, y be a variables ranging over the natural numbers. Let Tbe a variable ranging over sets. Set T 0. Choose i to be the least index of the vertices of G. Set x < i. Assign Vx either color1 or color2. While there is some vertex Vj such that j T { If Vx is color1, consider which assignment of truth values to all and only the sentences whose corresponding vertices share edgesfrom Vx will make sentence i true (these sentences must be all and only those sentences to which sentence i directly refers by construction of G), color (nondeterministically) vertices of this set with color1 whose corresponding sentences must be true in order for sentence i to be true, color vertices of this set with color2 whose corresponding sentences must be untrue in order for sentence i to be true. All the vertices that share edges from Vx may not be colored. If Vx is color2, consider which assignment of truth values to all and only the sentences whose corresponding vertices share edgesfrom Vx will make sentence i untrue, color (nondeterministically) the vertices of this set with color1 whose corresponding sentences must be true in order for sentence i to be untrue, color the vertices of this set with color2 whose corresponding sentences must be untrue in order for sentence i to be untrue. All the vertices that share edges from Vx may not be colored. Choose one of the vertices, say Vy, just colored. If x y, set T< Tu {x} (the color of Vx is "fixed" iffx e T). Set x y. } } Claim A. 1: A graphically circular set of sentences is paradoxical if the corresponding graph has no "stable coloring" in the following sense. For any execution of A on the corresponding previously uncolored graph G, either there are colored vertices Vi and Vj such that i e Tbutj V T(that is Vi has been assigned color1 colorr) at some previous stage and Vj has just been assigned a color and that color hasn't been fixed), there is an edge from Vj to Vi and A assigns Vi color2 colorsn) or the color of a vertex which cannot be fixed alternates endlessly on the execution of A. Proof of Claim A. 1: Assume otherwise for contradiction: that there is a set, R, of sentences which is not paradoxical but whose corresponding graph G contains a cycle, yet has no stable coloring. If a set of sentences each of which only makes truth claims of the others is not paradoxical then there is an assignment of "true" and "false" to those sentences that does not lead to contradiction. For each x, if vertex Vx is a vertex of G, and if Vx were colored with color1 just in case the xth sentence of R were true and with color2 just in case the xth sentence of R were untrue, then this coloring would be stable in terms of A, in contradiction to the original assumption. 0 1 A vertex cannot be fixed in case the corresponding sentence is directly selfreferential. APPENDIX B AN UNDEFINABILITY THEOREM In our development, we're far from that which is usually considered a semantic paradox. I want to suggest that what we're doing is not so strange in light of how traditional liar sentences can be used in the study of formal systems. Mostowski, Robinson and Tarski's (Mostowski, Robinson and Tarski 1953) Theorem 1 shows that any formalized theory (call it T) in which both V (the predicate applying to codes of valid sentences) and function D (a diagonal function) are definable must be inconsistent. We will demonstrate here that there is a formula with one free variable which defines a set of natural numbers Y such that V and Y are not both definable if the theory T of the formal system in which they're to be defined is consistent. Let y, be a formula of the language of theory Twith one free variable. Now, let Ybe a predicate of natural numbers such that e Yjust in case for all p > m, <\a(p)> e Y ifd,m(p) = 1 and <\a(p)> v Yif da,m(p) = 0. Claim A.2: If Tis consistent, then Yand Vare not both definable. Proof of Claim A.2: Suppose for contradiction that there is some predicate Y so defined by j, and predicate V such that n e Viff the sentence of T whose godel code is n is valid. If construction of Da, that for some m > 0, we can infer that ... are valid, then since there is some m such that d,,m is nullish and <(Vx)( Y(x))> is in V, we have warrant to infer V, and so we can infer <(Vx)( Y(x))> is not in V, in contradiction to our assumption. Y and V cannot both be definable. 0 APPENDIX C AN SMULLYAN STYLE INCOMPLETENESS THEOREM Using an RP family Da, we can generate an incompleteness result. Let L be a language similar to the one developed by Smullyan (1992, chp. 14). Following Smullyan, if there is an arithmetically definable proof procedure based on some axiom schemas and rule of inference, then we can speak of a one place predicate Px which defines proofs is satisfied by all and only the godel numbers (for any suitable numbering) of the sentences of L which are provable according to this procedure. If the negation of sentence is provable then the negatum is refutable. If the system determined by the language and the proof procedure is consistent, then no sentence is provable and refutable. IfPx satisfies the following two additional conditions then we can get a kind of incompleteness result: (1) if P( ), then P(x), and (2) for metalinguistic variables ), and so P'(k) for an odd k greater than 2j. But 