Once more to dissolve the ravens

W. E. Johnson argued that by taking into account both the epistemic and constitutive conditions for using arguments in inferences one could dissolve the paradoxes of material implication. This essay argues that the same sort of consideration can be used to dissolve the paradox of ravens in confirmation theory. It is argued in particular, and in agreement with certain points raised by the Popperians, that those instances of a generalization which are verifying but apparently not confirming cannot raise the posterior of that generalization.     

What the Liar Taught Achilles

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.     

Minimalism and Paradoxes

This paper argues against minimalism about truth. It does so by way of acomparison of the theory of truth with the theory of sets, and considerationof where paradoxes may arise in each. The paper proceeds by asking twoseemingly unrelated questions. First, what is the theory of truth about?Answering this question shows that minimalism bears important similaritiesto naive set theory. Second, why is there no strengthened version ofRussell's paradox, as there is a strengthened Liar paradox? Answering thisquestion shows that like naive set theory, minimalism is unable to makeadequate progress in resolving the paradoxes, and must be replaced by adrastically different sort of theory. Such a theory, it is shown, must befundamentally non-minimalist.     

Vier Philosophen über semantische Paradoxien

In his treatise on sophisms, the medieval logician and philosopher J. Buridan expounded a theory on what we have come to call semantic paradoxes. His theory has not yet been fully understood. The present paper aims at showing that Barwise’s and Etchemendy’s considerations on paradoxes (founded upon Aczel’s non-well-founded sets) provide the framework for an improved understanding. Barwise’s and Etchemendy’s account is contrasted with Kripke’s. Finally, a recent analysis of Buridan’s position by Epstein is criticized     

Three paradoxes of medical diagnosis

Sadegh-zadeh [23] has proposed a theory of the relativity of medical diagnosis in terms of the time at which a diagnosis is accepted, the patient to whom the diagnosis applies, the physician who renders the diagnosis, the medical knowledge used, the diagnostic method applied, and the set of patient observations. Use of classical formal logic as the ‘diagnostic method’ may result in three paradoxes: the paradoxes of consistency, completeness, and justifiable ignorance. These paradoxes may be resolved by the addition of two non-classical operators, the ‘certainty’ and ‘effort’ operators, akin to the non-classical operators of modal logic.     

The paradoxes

Conclusion The paradoxes act as restraints on undisciplined or erroneous reasoning and so perform a valuable role. The fact that they have resisted solution for so long suggests that the current systems of logic are defective. The paradoxes of set theory, in my opinion, completely condemn all the current forms of set theory and their associated definitions of the natural numbers. Nothing short of a complete review of the whole area seems capable of remedying the situation.     

Scharp on replacing truth

Kevin Scharp’s ‘Replacing Truth’ is an ambitious and far reaching account of the semantic paradoxes. In this critical discussion we examine one the books central claims: to have provided a theory of truth that avoids the revenge paradoxes. In the first part we assess this claim, and in the second part we investigate some features of Scharp’s preferred theory of truth, ADT, and compare it with existing theories such as the Kripke–Feferman theory. In the appendix a simple model of Scharp’s theory is presented, and some potential consistent ways to strengthen the theory are suggested.     

Logical, Semantic and Cultural Paradoxes

The property common to three kinds of paradoxes (logical, semantic, and cultural) is the underlying presence of an exclusive disjunction: even when it is put to a check by the paradox, it is still invoked at the level of implicit discourse. Hence the argumentative strength of paradoxical propositions is derived. Logical paradoxes (insolubilia) always involve two contradictory, mutually exclusive, truths. One truth is always perceived to the detriment of the other, in accordance with a succession which is endlessly repetitive. A check is put on the principle of the excluded middle by the logical paradoxes, because self-reference leads to an endlessly repeating circle, out of which no resolution is conceivable. Logical paradoxes are to be compared with the `objective ambiguity' prevalent in oracles (Gallet, 1990). Semantic paradoxes are contextually-determined occurrences, whose resolution at the metalinguistic level is made possible by the discovery of a middle term. They express a wilful ambiguity, in which the interlocutor is invited to take an active part in the construction of sense, since what must be found is the unexpected sense thanks to which A and not-A can be asserted simultaneously. Cultural paradoxes play about doxa (`common sense') and openly challenge common opinion because of their character as inopinata (`unexpected'). My aim is to show that even cultural paradoxes hide sometimes a flaw of argumentation similar to logical or semantic paradox; they too imply an exclusive disjunction leading to the disappearance of the middle terms. Finally, basing myself on the theory of topoi (Anscombre and Ducrot, 1983), a tentative resolution of the cultural paradoxes will be suggested.     

Axiomatic Theories of Partial Ground II

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to ?? ₀ and thus is consistent. We argue that this theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.     

Indefinite Extensibility—Dialetheic Style

In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how.     

The Consistency of The Naïve Theory of Properties

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics.     

Inconsistency Theories of Semantic Paradox

It is argued that a certain form of the view that the semantic paradoxes show that natural languages are "inconsistent" provides the best response to the semantic paradoxes. After extended discussions of the views of Kirk Ludwig and Matti Eklund, it is argued that in its strongest formulation the view maintains that understanding a natural language is sharing cognition of an inconsistent semantic theory for that language with other speakers. A number of aspects of this approach are discussed and a few objections are entertained.     

WINNICOTT AND HELPLESSNESS: DEVELOPMENTAL THEORY,RELIGION, AND PERSONAL LIFE

The author examines Winnicott's theory of development from the perspective of existential helplessness, arguing that (a) his views illuminate healthy (and unhealthy) aspects of religion, and (b) express his stance toward the helplessness of dying and death. The author contends that Winnicott understood the infant's psychic growth in relation to the reality of existential helplessness and absolute dependency. Four interrelated, dynamic paradoxes embedded in Winnicott's developmental perspective are discussed, and these paradoxes are seen as frameworks to depict his notions of ego, transitional objects, and true/false selves. The author posits that religion, which Winnicott included under the rubric of transitional phenomena, can be understood in relation to existential helplessness and can be assessed in terms of the degree to which these paradoxes are dynamic.     

Six Groups of Paradoxes in Ancient China From the Perspective of Comparative Philosophy

This paper divides the sophisms and paradoxes put forth by Chinese thinkers of the pre-Qin period of China (before 221 BCE) into six groups: paradoxes of motion and infinity, paradoxes of class membership, semantic paradoxes, epistemic paradoxes, paradoxes of relativization, other logical contradictions. It focuses on the comparison between the Chinese items and the counterparts of ancient Greek and even of contemporary Western philosophy, and concludes that there turn out to be many similar elements of philosophy and logic at the beginnings of Chinese and Greek civilizations.     

The Simple Consistency of Naive Set Theory using Metavaluations

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches.     

Term Models for Abstraction Principles

Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is poorly behaved unless the background second-order logic is predicative.     

A FailedCassatio: Goldstein on the Liar

The purpose of this note is to express some doubts about Goldstein's cassationist solution to the Liar Paradox. After sketching his theory (§I), we argue that the notions he introduces in order to solve the Strengthened Liar give rise to paradoxes the theory cannot deal with (§II).     

Reference, paradoxes and truth

We introduce a variant of pointer structures with denotational semantics and show its equivalence to systems of boolean equations: both have the same solutions. Taking paradoxes to be statements represented by systems of equations (or pointer structures) having no solutions, we thus obtain two alternative means of deciding paradoxical character of statements, one of which is the standard theory of solving boolean equations. To analyze more adequately statements involving semantic predicates, we extend propositional logic with the assertion operator and give its complete axiomatization. This logic is a sub-logic of statements in which the semantic predicates become internalized (for instance, counterparts of Tarski’s definitions and T-schemata become tautologies). Examples of analysis of self-referential paradoxes are given and the approach is compared to the alternative ones.