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8 Circularity and Paradox
, 2004
"... Both the paradoxes Ramsey called semantic and the ones he called settheoretic look to be paradoxes of circularity. What does it mean to say this? I suppose it means that they look to turn essentially on circular notions of the relevant disciplines: semantics and set theory. But what does it mean to ..."
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Both the paradoxes Ramsey called semantic and the ones he called settheoretic look to be paradoxes of circularity. What does it mean to say this? I suppose it means that they look to turn essentially on circular notions of the relevant disciplines: semantics and set theory. But what does it mean to call a notion circular? I suppose that a circular notion (of discipline D) is one of the form selfR, for R a key relation of that discipline. Reference and predication are key semantic relations, so selfreference and selfpredication are circular notions of semantics. Membership is a key settheoretic relation, so selfmembership is a circular notion of set theory. The set and semantic paradoxes look to be paradoxes of circularity because they look to turn essentially on notions like selfmembership and selfreference. This approach to circularity might seem insufficiently discriminating. Do we really want to count selfdeception and selfincrimination in with selfreference and selfmembership? Well, why not? Remember, the target here is not circular notions as such but circularitybased paradox. We get a circularitybased paradox when a circular notion generates absurdities, with the circularity of the notion playing an essential role. I don’t know whether selfdeception and selfincrimination generate absurdities in this way. But if they do, then I for one am happy to speak of circularitybased paradoxes of psychoanalysis or legal theory.
Vagueness and Truth
"... In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and ..."
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In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and consider reasons for subscribing to the principle of uniform solution. 1 Introducing the Principle of Uniform Solution It would be very odd to give different responses to two paradoxes depending on minor, seeminglyirrelevant details of their presentation. For example, it would be unacceptable to deal with the paradox of the heap by invoking a multivalued logic, ̷L∞, say, and yet, when faced with the paradox of the bald man, invoke a supervaluational logic. Clearly these two paradoxes are of a kind—they are both instances of the sorites paradox. And whether the sorites paradox is couched in terms of heaps and grains of sand, or in terms of baldness and the number of hairs on the head, it is essentially the same problem and therefore must be solved by the same means. More generally, we might suggest that similar paradoxes should be resolved by similar means. This advice is sometimes elevated to the status of a principle, which usually goes by the name of the principle of uniform solution. This principle and its motivation will occupy us for much of the discussion in this paper. In particular, I will defend a rather general form of this principle. I will argue that two paradoxes can be thought to be of the same kind because (at a suitable level of abstraction) they share a similar internal structure, or because of external considerations such as the relationships of the paradoxes in question to other paradoxes in the vicinity, or the way they respond to proposed solutions. I will then use this reading of the principle of uniform solution to make a case for the sorites and the liar paradox being of a kind.
The Elimination of SelfReference: Generalized YabloSeries and the Theory of Truth 1,2
"... Abstract: Although it was traditionally thought that selfreference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is selfreferential but which, taken together, are paradoxical. Yablo's parado ..."
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Abstract: Although it was traditionally thought that selfreference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is selfreferential but which, taken together, are paradoxical. Yablo's paradox consists of a countable series of linearly ordered sentences s(0), s(1), s(2),..., where each s(i) says: For each k>i, s(k) is false (or equivalently: For no k>i is s(k) true). We generalize Yablo's results along two dimensions. First, we study the behavior of generalized Yabloseries in which each sentence s(i) has the form: For Q k>i, s(k) is true, where Q is a generalized quantifier (e.g. no, most, every, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves selfreference can be emulated without selfreference. Various translation procedures that eliminate selfreference from a nonquantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo's paradox and the translations we offer do not involve selfreference. It was traditionally thought that selfreference is a crucial ingredient of semantic paradoxes. However Yablo (1993, 2004) showed that this was not so by displaying an infinite