Published: Dietrich, E. (2008). The Bishop and Priest: Toward a point-of-view based epistemology of true contradictions. Logos Architekton, v. 2, n. 2, pp. 35-58. THE BISHOP AND PRIEST: TOWARD A POINT-OF-VIEW BASED EPISTEMOLOGY OF TRUE CONTRADICTIONS Eric Dietrich Philosophy Department Binghamton University Binghamton, New York I was in no sense a hypocrite; both sides of me were in dead earnest. Dr. Jekyll Abstract True contradictions are taken increasingly seriously by philosophers and logicians. Yet, the belief that contradictions are always false remains deeply intuitive. This paper confronts this belief head-on by explaining in detail how one specific contradiction is true. The contradiction in question derives from Priest's reworking of Berkeley's argument for idealism. However, technical aspects of the explanation offered here differ considerably from Priest's derivation. The explanation uses novel formal and epistemological tools to guide the reader through a valid argument with, not just true, but eminently acceptable premises, to an admittedly unusual conclusion: a true contradiction. The novel formal and epistemological tools concern points of view and changes in points of view. The result is an understanding of why the contradiction is true. 2 1. Introduction Bishop George Berkeley famously argued that reality consists only of Mind and mental objects; a view called idealism. In his Treatise Concerning the Principles of Human Knowledge (1710), he said ". . . all the choir of heaven and furniture of the earth, in a word all those bodies which compose the mighty frame of the world, have not any subsistence without a mind." Graham Priest, in his book Beyond the Limits of Thought (2002) has turned Berkeley's argument for idealism into an argument for a true contradiction. Priest's Berkeleian contradiction belongs to a distinct class of logical constructs characterized by conceptual limits, such as the limit of what can be expressed, known, or conceived. A contradiction from this class is true (according to Priest), and is called a dialetheia; dialetheism is the view that this class is not empty (Priest, 2006). However, that a contradiction could be true is so shocking that convincing someone that such a thing exists is likely to require more than just producing one; such a mere production is apt to seem like a magic trick akin to pulling a cannon ball out of an empty hat. Rather, what needs to be done is to explain how a contradiction could be true. This is done here. Specifically, I explain how it is that Priest's Berkeleian contradiction is true – that is, what the facts about the world and us as epistemic agents have to be in order for the contradiction to come out true. My explanation requires some new technical machinery which implements a crucial epistemic capacity – the capacity to adopt and then change points of view. This capacity is the key to seeing how dialetheias are true. Also, there is a problem with Priest's reworked Berkeleian argument that is fixed by introducing points of view and their changes. Finally, we end with a re-construction of Priest's argument that both works (concludes in a dialetheia) and makes sense, epistemologically. 3 Priest's reworked argument, like Berkeley's original, concerns conceiving unconceived things.1 The conclusion of Priest's reworked version is that something that is not conceived is also conceived. The mechanism Priest uses is to invoke the notion of unconceived (better: unconceivable) entities (like numbers from the uncountable reals – numbers for which there aren't any names or descriptions, whatsoever). But there is a solid objection to this using such entities, which will be discussed below. Instead of using unconceivable entities, a better strategy is to invoke a special point of view – one related to Nagel's view from nowhere (1979, 1986). Then, using other properties of points of view, we get the flexibility needed to see that things unconceived from the special point of view are nevertheless conceived from another point of view. This doesn't introduce relativism, however, because of the nature of the special point of view. All of this is discussed below. In section 2, I present Priest's reworked Berkeleian argument. In Section 3, the objections to this argument are discussed. Points of view are introduced to handle the central objection. Section 4 presents points of view as they relate to the Priest/Berkeley argument. Section 5, introduces some formalism due to Hales (1997) for manipulating points of view. Section 6 is the actual explanation. Section 7 deals with an objection to my explanation. Section 8 concludes. 1 There is an associated modal argument concerning conceiving the inconceivable. Though the modal argument won't be used here, modal conceiving – conceivability -will be. Also, Berkeley claimed to be primarily concerned with perceiving, but in his master argument, as elsewhere, he runs conceiving and perceiving together. Priest focuses on conceiving, and for the most part, we will too. But as we will see, when it comes to the kinds of points of view we need, there isn't a big distinction between conceiving and perceiving, at least perceiving-with-the-mind's-eye. 4 2. The Priest-Berkeley Argument I begin with Berkeley's original argument from his Three Dialogues Between Hylas and Philonous (1713).2 Then I will turn to Priest's formalization of it. (This part is not an exercise in Berkeley or Priest exegesis.) Phil. . . . But (to pass by all that hath been hitherto said, and reckon it for nothing, if you will have it so) I am content to put the whole upon this issue. If you can conceive it possible for any mixture or combination of qualities, or any sensible object whatever, to exist without the mind, then I will grant it actually to be so. Hyl. If it comes to that the point will soon be decided. What more easy than to conceive a tree or house existing by itself, independent of, and unperceived by, any mind whatsoever? I do at this present time conceive them existing after that manner. Phil. How say you, Hylas, can you see a thing which is at the same time unseen? Hyl. No, that were a contradiction. Phil. Is it not as great a contradiction to talk of conceiving a thing which is unconceived? Hyl. It is. Phil. The, tree or house therefore which you think of is conceived by you? Hyl. How should it be otherwise? Phil. And what is conceived is surely in the mind? Hyl. Without question, that which is conceived is in the mind. Phil. How then came you to say, you conceived a house or tree existing independent and out of all minds whatsoever? Hyl. That was I own an oversight; but stay, let me consider what led me into it.-It is a pleasant mistake enough. As I was thinking of a tree in a solitary place, where no one was present to see it, methought that was to conceive a tree as existing unperceived or unthought of; not considering that I myself conceived it all the while. But now I plainly see that all I can do is to frame ideas in my own mind. I 2 Of course, Berkeley's master argument as been roundly attacked, and defended (see Pappas, 2000 for such a defense; see Flage, 2005, for a good discussion of Berkeley). The attacks are not nearly as successful as is commonly believed. 5 may indeed conceive in my own thoughts the idea of a tree, or a house, or a mountain, but that is all. And this is far from proving that I can conceive them existing out of the minds of all Spirits. Here, now, are the technical definitions required to understand Priest's reworked version of Berkeley's argument (2002, pp. 60 – 64). (A caveat is required before continuing. Many readers get sidetracked at this early stage by taking issue with various aspects of the formal machinery Priest uses in his reworked argument. So, discussion is over before it can start. All of the logical machinery presented in this section is taken from Priest; I've added nothing new. A defense of the axioms and principles used below can be found in Priest, 2002, pp. 60 – 64. But in truth, none of the machinery here is philosophically or logically suspect. So, I ask for the reader's indulgence: play along here, and see if we don't get to somewhere interesting by the end of the paper.) Let Τ be a propositional operator meaning "It is conceived that." Τ(p) thus means "it is conceived that p," where p can be any proposition whatsoever. Conceived, for now, just means thought of, or brought before one's mind. Let τ be a predicate meaning "is conceived." Thus τx means that x is conceived. The relation between Τ and τ will be discussed below. Let c be an arbitrary unconceived thing. We are not stipulating that there are any. The reworked argument is supposed show that c exists and is also conceived. It is important that c be arbitrary (or indefinite). Toward that end, Priest uses an axiom that allows him to pick out an arbitrary thing satisfying a certain predicate, if there are any -an axiom which uses an indefinite description operator, ε (Priest, p. 63).3 We need this 3 There are several ways to select arbitrary or indefinite objects. Two good treatments of the subject are in Shapiro's logic of arbitary objects (Shapiro, 2004), and Fine's book, 1985. Since we are following Priest's argument, we will use the method for picking out arbitrary objects that he uses. 6 so we can be certain of getting an unconceived thing that doesn't presuppose any question-begging properties. Priest's axiom is derived from Hilbert's "transfinite axiom" (as is the operator) which, in turn, is from Hilbert's epsilon calculus.4 Hilbert's axiom uses epsilon terms, for example, εx.A, which selects any (or some) object for which A is true, if there is one. Epsilon terms are governed by the "transfinite axiom," often written A(x) → A(εx.A). This says that if something is A, then something satisfying A is A. Following Priest, our version will be "if something satisfies φ(x), then an arbitrarily selected thing that is φ is one of them." Using εxφ(x) to capture "an arbitrarily selected x that is φ," our version of the axiom will be: ∃xφ(x) → φ(εxφ(x)). Call this the ε axiom. If we let φ be ¬τ, we get: ∃x ¬τ(x) → ¬τ(εx¬τ(x)), which says, "if there are any unconceived things, an arbitrary one of them is indeed unconceived," which seems unobjectionable. Since c is an arbitrary unconceived thing, if there are any (i.e., "c" is just the name given to an arbitrary unconceived thing, if such exists), we can identify c with εx¬τ(x) (p. 63), which gives us: "if there are any unconceived things, then an arbitrarily chosen one of them, c, is not conceived." Or, shorter, "if there are any unconceived things, then c is one of them." Thus, we have Priest's final version of the epsilon axiom: 4 For a good, quick introduction, see http://plato.stanford.edu/entries/epsilon-calculus/ 7 ∃x ¬τ(x) → ¬τ(c).5 This is all needed so we can pick out and get ahold of an unconceived thing without begging any questions. Following Priest, Berkeley's thesis will be "everything is conceived" (p. 60). Assuming this is false, we get the denial of Berkeley's thesis: something is not conceived: ∃x¬τx. Berkeley's thesis is not a summary of Berkeley's idealism. Berkeley held that there could be no mind-independent things. The thesis states only that there aren't any such things. Only the denial of this latter claim is needed, namely, that something is not conceived. The denial of Berkeley's thesis is a premise. In Priest's reconstructed argument, the truth of this premise rests on the fact that, intuitively, Berkeley's thesis seems false – not everything is conceived. Which seems right (below, this is discussed further). Next, Priest notes that Τ satisfies a prefixing principle. Thus, if α entails β then conceiving that α (that α is true) entails conceiving that β (that β is true) (p. 64). Of course, this principle requires idealizing the conceiving agent: the agent must conceive of all the logical consequences of what it conceives. Priest claims, quite 5 Some have asked: "Do you mean: 'We can treat the proper name "c" as intersubstitutable with a epsilon term' or 'we can abbreviate the epsilon term by a single letter'?" Neither. The epsilon term picks out (or allows us to pick out) an arbitrary unconceived thing, assuming such exist. We dub that thing "c". α → β Τα → Τβ. If then 8 plausibly, that in the context of reworking Berkeley's argument this is permissible because Hylas does conceive of the relevant consequences of what he conceives. Finally, Priest needs an axiom called the conception schema (p. 62). It relates Τ and τ, and is quite powerful. The conception schema says that if it is conceived that x is φ then x is conceived. The conception schema thus allows us to move from conceiving a proposition (conceiving that p) to conceiving the denotation or referent of the proposition (conceiving of x). The general conception schema is: Τφ(x) → τx. At the general level, this axiom seems plausible: if I conceive that Tolkien was a writer, I conceive of Tolkien. For now, let's accept it. If we do accept this axiom, then are there any restrictions to instantiating it? Not according to Priest. So, we are free to use an instantiation of the conception schema with x bound to c and φ bound to ¬τ. The final version will thus be: Τ¬τc → τc. This instantiation of the conception schema allows moving from conceiving that an unconceived thing is unconceived to conceiving an unconceived thing. Its role in the reworked Berkeleian argument is, therefore, central. Now, with all of this in place, here is Priest's reworked, dialetheic Berkeleian argument. 9 1. ∃x¬τx → ¬τc An application of the ε axiom 2. Τ∃x¬τx → Τ¬τc 1, Prefixing Principle 3. Τ∃x¬τx Premise (we do conceive of the existential proposition) 4. Τ¬τc 2, 3 M.P. 5. Τ¬τc → τc Conception Schema with φ bound to ¬τ 6. τc 4, 5 M.P. 7. ∃x¬τx Premise (the denial of Berkeley's thesis.) 8. ¬τc 7, 1 M.P. 9. τc /\ ¬τc 6, 8 Conj. Priest sums up this argument nicely: "that there are unconceived things, and that this is itself conceived, together entail a contradiction (p. 65)." Since the premises appear to be true and the axioms unobjectionable, the entailed contradiction seems true . . . it is a dialetheia. 3. The Invariance Question and Other Objections There are, however, a number of questions and objections one can raise about Priest's reworked argument. The most important of these concerns step 5, the conception schema: It is doubtful that this axiom has unrestricted application. Specifically, the schema doesn't seem to apply to c, the very thing the schema was constructed for. If c is an unconceived thing, then once it loses this crucial (essential?) property – which it does in the inference to the consequent in step 5 -hasn't c itself ceased to exist, for it is no longer an unconceived thing? Here's another way to put this question: In step 5, how can we guarantee that the same c is referenced in the move from antecedent to consequent? The two c's referred to in the antecedent and in the consequent of the step are revealed to have completely opposite properties: the former is unconceived and the latter is 10 conceived, so what guarantees that the c in the antecedent is the same c in the consequent? In still other words: what guarantees that εx¬τ(x) picks out the same unconceived thing in both the antecedent and consequent, when in the consequent it is not even unconceived? I call this the Invariance Question -succinctly: How can we guarantee that c remains invariant in the conception schema from antecedent to consequent? In fact, the consequent in step 5 is a contradiction: "an arbitrary unconceived thing is conceived." But without a good answer to the invariance question, one could object that this doesn't result in the desired dialetheia, but results rather in an incoherency. The machinery introduced in section 4 answers the invariance question, as well as provides us a satisfying explanation for how a dialetheia can be true. The central notion of conceiving also turns out to raise some complications. Since Priest wants to avoid the complications of quantifying into intensional contexts as well as other problems associated with de dicto readings of the propositional operator Τ, he is careful to restrict his use of "conceiving" to an extensional one. Extensional conceiving is defined thusly: x = y --> τx ↔ τy. Extensional conceiving allows for a cognitive agent to conceive of something without even believing that that thing exists. For example, extensional conceiving allows us to say that, though Dr. Doom knows nothing about Peter Parker, Doom is conceiving of Peter Parker even when he's only conceiving of Spiderman, because Spiderman is Peter Parker. Extensional conceiving, then, is conceiving under some description or other. But now, doesn't this make step 7, the denial of Berkeley's thesis, false -isn't everything conceived when conceiving is construed extensionally, because, isn't everything conceived, or at least conceivable, under some description or other? The answer to this question is No. To conceive of something extensionally, it is sufficient to bring before one's mind a referring description that picks out that thing (which thing, of course, needn't exist). This referring description can be anything: it can be some purely mental representation, some natural or artificial language description, or even some neuron-based description. No matter what it is, there are only a countably infinite number of such descriptions. There are however, an uncountably infinite number of things: consider the set of real numbers. It follows that there are, in 11 principle, more things (by far) that are not even conceivable, let alone conceived. In short, there are not enough mental descriptions to bring before the mind for singling out each real number. But now suppose someone says, "Yes, but consider the phrase 'Every individual, non-describable real number.' Doesn't this phrase pick out every single real number for which there is no description?" No it doesn't. What it does it is pick out a collection of every single non-describable real number. That collection is of course describable and hence conceivable (as the discussion here proves). But we still don't have every non-describable real number.6 However, the invariance question and the inconceivability of certain entities work together to form a dilemma. Consider again, the conception schema, Τ¬τ (c) → τc, and recall that c is εx¬τ(x). Given this, it appears as if we have no way to conclude that c is conceived – it is, after all, inconceivable. To see this, first, without loss of generality, let's pick an arbitrary inconceivable real number as the entity c is bound to (inconceivable because the real is undescribable in a way that brings it before the mind). Then, the consequent of the conception schema is false; it is false that τc. This just means that c is not a referring description that brings before one's mind, or that allows one to single out, a specific inconceivable real. Rather, c points in the "general direction" of some inconceivable real. Since c is bound to an arbitrary real, we really have no conception of it (the real number). The vagueness of c undoes its ability to function as a descriptor appropriate for conception. But if there were a way to guarantee that c, an arbitrary inconceivable thing, is nevertheless conceived, we then run afoul of the invariance question. We have no reason to be believe, and no way guarantee, that the c's in the antecedent and consequent of the conception schema refer to the same object. 6 It is not clear what the necessary and sufficient conditions for conceiving are (and cognitive science is of no help here: Murphy, 2002). Priest seems to suggest (p. 66) that including perception as well as referring descriptions gives us necessary and sufficient conditions for conceiving. This suggestion (if it is such) might be right. Adding perceptions won't fix the point made here, though, as Priest points out (p. 66), for there are far more things than we can perceive, too. 12 Now it is time to introduce points of view. 4. Points of View The crucial epistemic capacity needed to explain how a contradiction can be true is the ability to adopt various points of view, and to change points of view. The notion of a point of view is common, deriving from the fact that we change points of view. We can walk around Jone's car, for example. Or look inside our computer. We can change points of view without moving. We can imagine our house from above. Though we classify a certain action as morally wrong, we can often understand the point of view of someone who classifies it as morally right – even morally mandatory. In spite of their importance, points of view and their changes are a source of deep philosophical puzzlement (see, e.g., Nagel, 1979, 1986). Yet, points of view and point of view change hold the key to understanding how a contradiction can be true. 4.1. Hylas's and Philonous's points of view When Hylas says "What more easy than to conceive a tree or house existing by itself, independent of, and unperceived by, any mind whatsoever? I do at this present time conceive them existing after that manner," he is picturing in his mind's eye a tree or house unconceived from his point of view. We may suppose that he's imagining a tree that is not perceived or conceived of by anyone. The relevant tree is represented in Hylas's mind as having the property of being unconceived. Then, Philonous says, in effect, "Hylas, the tree which you think of as unconceived is conceived by you, since in thinking of it, you are conceiving it." In saying this, Philonous is pointing out that from his, Philonous's, point of view, which comprehends both Hylas and what Hylas is conceiving as unconceived, the tree is conceived by Hylas. Indeed, Philonous's argument to Hylas is simply getting Hylas to adopt Philonous's point of view. Hylas comes to understand that in conceiving of the 13 unconceived tree, he, Hylas, is conceiving of the tree. As we will see below, Hylas has given up too soon. His point of view is quite special. Call Hylas's point of view that comprehends the tree alone and unconceived, H. Call Philonous's point of view that comprehends the tree and Hylas conceiving of the tree, P. Priest's reworked argument, in formal mode, can now be labeled with these two points of view. (I) 1. ∃x ¬τx → ¬τc An application of the ε axiom 2. Τ∃x ¬τx → Τ¬τc 1, Prefixing Principle 3. Τ∃x ¬τx Premise 4. Τ¬τc 2, 3 M.P. 5. Τ¬τc → τc Conception Schema with φ bound to ¬τ 6. τc 4, 5 M.P. 7. ∃x ¬τx Premise (the denial of Berkeley's thesis.) 8. ¬τc 7, 1 M.P. 9. τc /\ ¬τc 6, 8 Conj. H P 14 (Since both Hylas and Philonous accept premise 1, we can stipulate that it is held from both points of view.) That lines 2 – 6 are from point of view P is due to the use of the propositional operator, Τ. The relevant propositions are conceived, and the Conception Schema (assume it works) allows the conceiving of one of the propositions (¬τc) to transition inside of the proposition so that the denotation of the proposition is itself conceived. The conclusion, τc, is the result, since from point of view P, c is conceived. The same is true, with appropriate changes, for lines 7 – 8. From point of view H, c is not conceived, since there is something not conceived, and if there is, that thing is c. What now of the conclusion, τc /\ ¬τc? One might be tempted to say that it cannot validly be drawn since the differing points of view establish different contexts, so τc and ¬τc are true in different contexts, P and H, and hence cannot be conjoined, at least they cannot be conjoined independently of their contexts, which wouldn't result in the wanted contradiction. To see this, suppose that the predicate, τ, takes points of view as subscripts, giving us the correct conclusion: τPc /\ ¬τHc, This is obviously not a contradiction: c is conceived from Philonous's point of view, and unconceived from Hylas's. This fact seems uninteresting. It looks like introducing points of view dissolves the dialetheism, rather than explains it. One might conclude that with the introduction of points of view, we can see how one might mistakenly conclude that dialetheias are true, even though they aren't. But this is hasty, because it incorrectly describes Hylas's point of view. 4.2. From nowhere to no one. Consider again, the tree unconceived from Hylas's point of view. Philonous's argument is not that there is another point of view from which the tree is conceived, but rather that Hylas has made a mistake there are no points of view from which the tree is unconceived. 15 But Hylas has not made a mistake. When he imagines a tree existing unconceived he is adopting a modified version of the view from nowhere with respect to the tree; he is conceiving of the tree sub specie aeternitatis, and from that perspective, imagining the tree to be unconceived.7 Hylas's viewpoint is actually a variant of the view from nowhere (explained below). I dub this variant the view of no one. The view from nowhere is Nagel's characterization of objectivity, or a path to objectivity. Nagel introduces the view from nowhere thusly: Since a kind of intersubjective agreement characterizes even what is most subjective, the transition to a more objective viewpoint is not accomplished merely through intersubjective agreement. Nor does it proceed by an increase of imaginative scope that provides access to many subjective points of view other than one's own. Its essential character . . . is externality or detachment. The attempt is made to view the world not from a place within it, or from the vantage point of a special type of life and awareness, but from nowhere in particular and no form of life in particular at all. The object is to discount for the features of our pre-reflective outlook that make things appear to us as they do, and thereby to reach an understanding of things as they really are. We flee the subjective under the pressure of an assumption that everything must be something not to any point of view, but in itself. (1979, p. 208) The view from nowhere is achieved via repeated acts of removing oneself from one's subjective world. Eventually, the viewer vanishes altogether. As Nagel explains: 7 For more on the view from nowhere, see Nagel, 1979, the seminal work on the topic. Also, see his 1986. 16 To acquire a more objective understanding of some aspect of life or the world, we step back from our initial view of it and form a new conception which has that view and its relation to the world as its object. In other words, we place ourselves in the world that is to be understood. The old view then comes to be regarded as an appearance, more subjective than the new view, and correctable or confirmable by reference to it. The process can be repeated, yielding a still more objective conception. (1986, p. 4) Hylas's point of view – the view of no one -is not Nagel's view from nowhere. This is because the view from nowhere still includes conceivers. The view of no one does not.8 One can follow Hylas to his view of no one by imagining a universe with no conceivers in it at all. And this imagining, one can clearly do. For example, one can imagine our universe bereft of any intelligent life. A stronger intuition pump can be built by imagining our universe with no life in it whatsoever. This gets rid of the tree; replace it with a rock. Now, it is easy to imagine this rock existing unconceived in the universe, for not only are there no conceivers, there is no life. (According to current theories, life was a chance occurrence in our universe. Perhaps it was likely, even very likely, but it was still due to chance. So, all one has to do is imagine that that chance never materialized.) In Hylas's mind's eye, he has removed himself from the picture, along with everyone else, which is precisely what the view of no one demands. From this austere, centerless, pointless, point of view, the rock (or whatever) is indeed unconceived. 8 There's another reason to avoid the view from nowhere. Nagel's view from nowhere is a rendering of objectivity, so it cannot be used here. This is because the introduction of points of view as foci of different knowledge (or knowledges) renders quite problematic the very notion of objectivity. I am certainly not saying that objectivity is impossible within an epistemology based on points of view, but this paper is not the place to sort out all the complexities. In order to avoid getting entangled in these complicated issues here, I simply introduce the view of no one. This is Hylas's true viewpoint. 17 So, it is Philonous who has made a mistake; it is the mistake of everyone who claims to be in charge of an objective point of view. And he has duped Hylas into making the mistake with him. Hylas, instead of saying "That was I own an oversight," should have said "No so fast, Philo, old boy. From your point of view, the rock is indeed conceived. We need only imagine that you conceive of me conceiving of my lifeless universe and said rock. But it doesn't follow from this that the rock is "in reality" conceived. There is no such unique reality. To assume that there is is to beg the question against me. Your point of view includes mine, but that doesn't make it more real or more correct. My point of view excludes yours, and from my no-one, centerless viewpoint, the rock is indeed unconceived. Viewed by no one, it is somewhat easy to imagine unconceived things. And they are, in fact, unconceived from that point of view." 4.3. More on points of view Points of view are complicated, and they have several important properties. Here, we will only need two of their properties: the nesting dolls structure and the invariance property. I take these in order. Every point of view, V1, admits of an "outer" point of view, V2, from which V1 (the "inner" viewpoint) is perceived as a point of view (I use "perceive" here, but if the perception is with the mind's eye, it is some sort of conceiving-perceiving mix that is at work). Sometimes, from V2, one can perceive what is being perceived from V1. If Jones sees a white dog, I can see that Jones sees the dog. We do this sort of thing all the time. Also, from the outer point of view, things might appear differently. From my point of view, I can see Jones and a black dog, but I can also see that Jones doesn't see the black dog. However, being at a more inclusive, "outer" point of view is no guarantee that one will have any cognitive access to what is being viewed from the "inner" point of view. I can see that Jones is looking around the corner of building, but I cannot see what Jones is looking at. So, the more inclusive, outer point of view, cannot necessarily, simply in virtue of being an encompassing point of view, comprehend what the inner point of view 18 comprehends. This property recurses: points of view, in principle, form a hierarchy. We see, then, that points of view have a nesting doll structure. This will be called the Nesting Dolls Property.9 The nesting dolls principle partly describes what is going on between Hylas and Philonous. From Hylas's point of view, there is something unconceived. But Philonous's point of view is more inclusive and wider than Hylas's point of view. And it is the fact that Philonous's point of view is the outer, comprehending point of view that makes Philonous erronously conclude what he concludes, namely, that Hylas is mistaken when he claims to conceive of something unconceived. The invariance property is this. Points of view form families, where each point of view in a family is a view on one and the same thing. That thing is an invariant among the relevant points of view. Getting different points of view on Jones' car by walking around it only works because Jones' car is an invariant among the relevant points of view. Nothing can count as a point of view unless it belongs to a family of points of view. Hence nothing can count as a point of view unless this invariance property obtains for it. It is this that answers the invariance question raised in section 3. (There is far more to points of view, but this will suffice for here.) Finally, we need one other point of view, or rather one other locus of points of view: the Reader's. The Reader is crucial here. It is the Reader which first inhabits the view of no one and then Philonous's point of view. And it is this sequential inhabiting that gives us the long-sought-after dialetheia. 5. A Modicum Formal Machinery Let's begin by picking some perfectly ordinary thing that is easily conceived – say Mt. Everest. As shown in section3, we can't pick an inconceivable thing, but we can pick 9 There is a question about whether or not this property applies to the view from nowhere. I leave this aside here, since we aren't using the view from nowhere. The property definitely applies to the view of no one. 19 something conceivable, and then "unconceive" it. That is the function of the view of no one. So, let c be Mt. Everest ("c" for its real name: Chomolungma – approximately translated, variously, as Goddess Mother of the Universe, or Goddess Mother of the Snows). As before, the predicate τ will mean "conceived.". Now, I introduce some of the apparatus from Hales's Relativism Logic (RL) (1997). ■Φ will mean that Φ (a proposition) is absolutely true; i.e., true from all points of view. ◆Φ will mean that Φ is relatively true; i.e., true from some point of view or other. In analogy with classical alethic modal logic: ◆Φ ↔ ¬ ■¬Φ, and ¬◆¬Φ ↔ ■Φ. Two theorems will also be required. Theorem 1): ■◆p → ◆p (Theorem 1 is true in almost all modal logics, and is true in RL.) Theorem 2): ◆■p → ■p Theorem 2 is also from Hales, 1997. It is an analogue of a well-known theorem in the modal logic S5.10 10 There is a technical difficulty with this theorem for points of view. Here, briefly, is the difficulty and its resolution. Both possible worlds and points of view have "accessibility" relations. These relations define which worlds or views one can get to given that one is already at some world or view. S5 requires that the accessibility relations be symmetrical and transitive: world W1 is accessible from W2 if and only if W2 is accessible from W1, and if W1 is accessible from W2 and W2 is accessible from W3, then W1 is accessible from W3. Points of view aren't possible worlds. One way to see this is to note that in the logic of points of view there are two different accessibility relations. One accessibility relation is informational, based on what is true at one point of view relative to another (called visibility), and the other is based on actually occupying one point of view by moving to it from another (called occupiability). In general, for visibility, points of view lack symmetry; occupiability, however, is symmetrical. To see the lack of symmetry for visibility, note at Philonous's point of view, there is information about what is going on at Hylas's point of view, but from Hylas's point of view there is no information whatsoever about what is going on at Philonous's. This is because Hylas's view is the view of no one, and from there, there aren't any other points of view at all. So symmetry is violated (so, 20 Now for the long-promised explanation. 6. The Explanation. The explanation occurs in three parts. The first part is a rendition of what transpires from Hylas's special point of view. The second part is what transpires from Philonous's. Unlike what happens at Hylas's viewpoint, what happens at Philonous's involves reasoning. And finally, there is what happens at the Reader's points of view. Recall, that Philonous's point of view contains Hylas's, and one of the Reader's contains Philonous's (and by transitivity contains Hylas's, too.) 6.1. Hylas and Philonous. I. This is what occurs from Hylas's point of view – which is the View of No One: [This is blank because from here, there are no minds; hence no thoughts] II. This is the reasoning from Philonous's point of view: Hylas is asserting that he is not conceiving c In symbols: Assert (Hylas, ¬τc) But if he's doing that, Hylas must be conceiving of c. In symbols: Assert (Hylas, ¬τc) → τc Ergo, c is conceived after all, In symbols: τc by the way, is transitivity). But the second theorem needed for the explanation only requires occupiability, and that is symmetrical here (as well as transitive). Hylas can, and does, leave the view of no one and go to Philonous's point of view. But crucially, it is the Reader who also makes such a trip, occupying first the view of no one, and then Philonous's. So using the S5 analogue theorem is legal. 21 That c is conceived is apodictic to me [Philonous]. So I conclude that everyone will come to the same conclusion as me. Ergo, τc is true from all points of view, including Hylas's view of no one. In symbols: ■τc III. From the Reader's points of view, first Hylas's, then Philonous's, then the Readers own: Clearly, from Hylas's point of view (the view of no one), it is apodictic that ¬τc: anyone who goes to the view of no one will not conceive of c. Hence, from any other point of view (i.e., any other than the view of no one), if I go to the view of no one, it will be true that ¬τc. (This requires only understanding the view of no one.) Hence, it is absolutely true that from Hylas's view (the view of no one), c is not conceived. In symbols: ■H[¬τc] Clearly, from Philonous's point of view, he's right: i.e., from every point of view containing Hylas's, which is all the relevant points of view (including Philonous's), it will be obvious that τc. Hence, everyone will come to the same conclusion as Philonous himself. So, from Philonous's point of view, it is absolutely true that c is conceived. In symbols: P [■τc]. Ergo, ■H[¬τc] ∧ P[■τc] Note: the left conjunct exhibits an important property of the view of no one: everyone who goes there will see the same thing, i.e., it is an absolute truth that from H (or that by going to H), a certain proposition is true (this proposition, of course, is able to be written or represented only from a 22 point of view that contains Hylas's). The right conjunct exhibits the opposite property: that from a certain point of view, P, it is absolutely true that a certain, other, proposition is true. But, if some proposition is true from someone's point of view, then that proposition is relatively true – true from that point of view. That's what we've got here: ¬τc is true from Hylas's point of view, and from Philonous's point of view, τc is absolutely true. Hence, absolutely ¬τc is relatively true, and it is relatively true that τc is absolutely true. So now we have both that it is absolutely true that it is relatively true that ¬τc, and it is relatively true that it is absolutely true that τc. In symbols: ■◆¬τc ∧ ◆■τc. Using the two theorems, we now have: ◆¬τc ∧ ■τc. Which is our needed contradiction. Or, to be explicit, and using the definition of ◆Φ as ¬■¬Φ: ◆¬τc ∧ ¬◆¬τc Hence, from the Reader's point of view it is both a relative truth that c is unconceived, yet it is an absolute truth that c is conceived. This is our true contradiction, our dialetheia. 6.2. Generalizing The explanation above can be generalized. Let Φ be any proposition that is true from the view of no one, let V1 be the view of no one, and let it nest inside of point of view V2. Let the Reader's point of view be R. Then: 23 From point of view V1: [Nothing] From point of view V2: V1[Assert Φ] V1[Assert Φ] → ¬Φ ¬Φ by Modus Ponens From point of view R: ■V1[Φ] ∧ V2[■ ¬Φ] Hence, ■◆Φ ∧ ◆■ ¬Φ Hence, ◆Φ ∧ ■ ¬Φ (by the two theorems). 7. Handling an Objection Here I refute two versions of an objection. Some readers (of this paper) believe that the dialetheia is purchased too cheaply. One form of this is the objection that there is no such view as the view of no one. But, there is clearly a view from nowhere, a view sub specie aeternitatis. All the view of no one is is this view in a world with no conceivers – which we guaranteed by considering a world with no life in it whatsoever. One can get to one instance of the view of no one by considering the possible world where there is no life, but which is otherwise exactly like the actual world, and then moving to the view from nowhere in that world. In that world, Chomolungma (for example) is unconceived. Another form of the too-cheaply objection takes is this: If my explanation worked, the following argument would be sound: The sky is blue; 24 Imagine that the sky is pink. Contradiction! Voila', a dialetheia! Clearly, the Blue/Pink sky argument fails, so my explanation must fail, too. It is very difficult to see how imagining that the sky is pink results in any contradiction at all: the sky is pink in someone's mind, but blue in the external world. No contradiction there. The Blue/Pink sky argument is really just a non sequitur. But the worry behind the argument is legitimate, and it is this: How can simply imagining that something is unconceived make that thing unconceived. This worry, however, misunderstands the nature of the view of no one. The view of no one is not achieved by simple imagining (I am granting here what I think is false: that imagining is itself simple, and that there is such a thing as a simple imagining). If it were, then my explanation would be obvious, which it is not. That my explanation puts some strain on the imagination shows that simple imagining is not sufficient to achieve the view of no one. Rather, getting to the view of no one is difficult, requiring some work and practice. 8. Conclusion This then is the explanation of how a contradiction could be true. From the view of no one, there is something unconceived by everyone. Hylas was right all along. But Philonous is right, too. Philonous, from his point of view, observes Hylas, and notes that he (Hylas) is conceiving of the (to Philonous, alleged) unconceived thing. The reader perceives what is occurring at P and H and thereby comes to the dialetheic conclusion. Point of view P is not unusual in any way. But H, the view of no one, takes some getting used to. Once used to it, however, matters unfold pretty much as Philonous thinks they do; it is just that he's misunderstood where Hylas is coming from.11 11 For comments on previsious versions of this paper, I thank David Chalmers, Thony Gillies, Charles Goodman, Josh Julian, Graham Priest, Julie Rose, Zach Weber, and the 25 participants at several philosophy seminars at Binghamton University and the University of Buffalo. Not surprisingly, their comments were contradictory, so I used all of them. 26 References Berkeley, G. (1713). Three Dialogues Between Hylas and Philonous in Opposition to Skeptics and Atheists. (A supplement to his Treatise Concerning the Principles of Human Knowledge.) Vol. XXXVII, Part 2. The Harvard Classics. New York: P.F. Collier & Son, 1909–14; also published 2001 by Bartleby.com. See also: The Works of George Berkeley, Bishop of Cloyne (1948-1957). A.A. Luce and T.E. Jessop (eds.). London, Thomas Nelson and Sons. 9 vols. For a good, short summary, see http://www.utm.edu/research/iep/b/berkeley.htm. Fine, K. (1985). Reasoning with Arbitrary Objects. New York: Blackwell. Flage, D. (2005). George Berkeley (1685-1753). In Internet Encyclopedia of Philosophy. http://www.iep.utm.edu/b/berkely.htm. Hales, S. (1997). Consistent Relativism, Mind, 106, pp. 33 – 52. Murphy, G. (2002). The Big Book of Concepts. Cambridge, MA: MIT Press. Nagel, T. (1979). Subjective and Objective, in Mortal Questions. Cambridge, UK: Cambridge University Press, pp. 196-213. Nagel, T. (1986). The View From Nowhere. New York: Oxford University Press. Pappas, G. S. (2000). Berkeley's Thought. Ithaca: Cornell University Press. Priest, G. (2002). Beyond the Limits of Thought. Oxford: Oxford University Press. Shapiro, Stuart. (2004). A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: 27 Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004, 565-575.