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Revision Forever!
"... Abstract. Revision is a method to deal with nonmonotonic processes. It has been used in theory of truth an an answer to semantic paradoxes as the liar, but the idea is universal and resurfaces in many areas of logic and applications of logic. In this survey, we describe the general idea in the fram ..."
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Abstract. Revision is a method to deal with nonmonotonic processes. It has been used in theory of truth an an answer to semantic paradoxes as the liar, but the idea is universal and resurfaces in many areas of logic and applications of logic. In this survey, we describe the general idea in the framework of pointer semantics and point out that beyond the formal semantics given by Gupta and Belnap, the process of revision itself and its behaviour may be the central features that allow us to model our intuitions about truth, and is applicable to a lot of other areas like belief, rationality, and many more. 1
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.
Yablo’s paradox and beginningless time
"... The structure of Yablo’s paradox is analysed and generalised in order to show that beginningless stepbystep determination processes can be used to provoke antinomies, more concretely, to make our logical and our ontological intuitions clash. The flow of time and the flow of causality are usually c ..."
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The structure of Yablo’s paradox is analysed and generalised in order to show that beginningless stepbystep determination processes can be used to provoke antinomies, more concretely, to make our logical and our ontological intuitions clash. The flow of time and the flow of causality are usually conceived of as intimately intertwined, so that temporal causation is the very paradigm of a stepbystep determination process. As a consequence, the paradoxical nature of beginningless stepbystep determination processes concerns time and causality as usually conceived. Keywords Antinomy; circularity; ungroundedness; determination structure, recursion. I. Yablo 1993 presents an infinite sequence of sentences s 1, s 2, s 3,... s n,... each of them saying that all the sentences posterior in the sequence are untrue: s n: for all natural numbers m>n, s m is untrue This structure is paradoxical because there is no way to consistently assign a truthvalue to any sentence in the sequence. So far, the situation is the same as in the Liar and Liarlike sentences: we are also incapable of consistently assigning truthvalues to Liarlike sentences. In the case of Liarlike sentences the most widely accepted diagnose is that the kind of selfreference present in such sentences induces circularity in the process of truthvalue determination. But none of Yablo’s sentences seems to be selfreferential, not even indirectly, or involve circularity. In spite of this, Priest 1997 tries to find circularity in Yablo’s sequence. On the other hand, Goldstein 2006 blames underspecification due to ungroundedness but not to circularity. I shall argue that Priest is
Yablo’s Paradox and the Omitting Types Theorem for Propositional Languages
, 2011
"... We start by recapitulating Yablo’s paradox from [1]. We have infinitely many assertions {pi: ∈ IN} and each pi is equivalent to the assertion that all subsequent pj are false. A contradiction follows. There is a wealth of literature on this delightful puzzle, and I have been guilty of a minor contr ..."
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We start by recapitulating Yablo’s paradox from [1]. We have infinitely many assertions {pi: ∈ IN} and each pi is equivalent to the assertion that all subsequent pj are false. A contradiction follows. There is a wealth of literature on this delightful puzzle, and I have been guilty of a minor contribution to it myself. This literature places Yablo’s paradox in the semantical column of Ramsey’s division of the paradoxes into semantical versus logical paradoxes. However—as I hope to show below—there is merit to be gained by regarding it as a purely logical puzzle. Yablo’s Paradox in Propositional Logic If we are to treat Yablo’s paradox as a purely logical puzzle we should try to capture it entirely within a firstorder language with no special predicates. In fact we can even make progress while using nothing more than a propositional language; the obvious language L to use has infinitely many propositional letters {pi: i ∈ IN}. Next we want a propositional theory with axioms pi ↔ ∧