A mahāvidyā inference is used for establishing another inference. Its Reason is normally an omnipresent property. Its Target is defined in terms of a general feature that is satisfied by different properties in different cases. It assumes that there is no case that has the absence of its Target. The main defect of a mahāvidyā inference μ is a counterbalancing inference that can be formed by a little modification of μ. The discovery of its counterbalancing inference can invalidate such an (...) inference. This paper will argue that Cantor’s diagonal argument too shares some features of the mahāvidyā inference. A diagonal argument has a counterbalanced statement. Its main defect is its counterbalancing inference. Apart from presenting an epistemological perspective that explains the disquiet over Cantor’s proof, this paper would show that both the mahāvidyā and diagonal argument formally contain their own invalidators. (shrink)
In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there (...) exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)
The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation which claim to be able to solve the halting problem for Turing machines. We show that such arguments are flawed—a contradiction only occurs if a type of machine can compute its own diagonal function. We then demonstrate why such (...) a situation does not occur for the methods of hypercomputation under attack, and why it is unlikely to occur for any other serious methods. Introduction Issues with specific hypermachines Conclusions for hypercomputation. (shrink)
Since the pioneering work of [Aglioti, S., DeSouza, J. F., & Goodale, M. A. . Size-contrast illusions deceive the eye but not the hand. Current Biology, 5, 679–685] visual illusions have been used to provide evidence for the functional division of labour within the visual system—one system for conscious perception and the other system for unconscious guidance of action. However, these studies were criticised for attentional mismatch between action and perception conditions and for the fact that grip size is not (...) determined by the size of an object but also by surrounding obstacles. Stoettinger and Perner [Stoettinger, E., & Perner, J., . Dissociating size representations for action and for conscious judgment: Grasping visual illusions without apparent obstacles. Consciousness and Cognition, 15, 269–284] used the diagonal illusion controlling for the influence of surrounding features on grip size and bimanual grasping to rule out attentional mismatch. Unfortunately, the latter objective was not fully achieved. In the present study, attentional mismatch was avoided by using only the dominant hand for action and for indicating perceived size. Results support the division of labour: Grip aperture follows actual size independent of illusory effects, while finger-thumb span indications of perceived length are clearly influenced by the illusion. (shrink)
Peter Slezak and William Boos have independently advanced a novel interpretation of Descartes's "cogito". The interpretation portrays the "cogito" as a diagonal deduction and emphasizes its resemblance to Godel's theorem and the Liar. I object that this approach is flawed by the fact that it assigns 'Buridan sentences' a legitimate role in Descartes's philosophy. The paradoxical nature of these sentences would have the peculiar result of undermining Descartes's "cogito" while enabling him to "disprove" God's existence.
The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincaré systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not (...) the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed. (shrink)
Environmental rights are diagonal if they are held by individuals or groups against the governments of states other than their own. The potential importance of such rights is obvious: governments' actions often affect the environment beyond their jurisdiction, and those who live in and rely upon the environment affected would like to be able to exercise rights against the governments causing them harm. Although international law has not adopted a comprehensive, uniform approach to such rights, human rights law and (...) international environmental law have begun to develop some possible bases for diagonal environmental rights. Human rights law operates primarily along a vertical axis, setting out individuals' rights against their governments and the corresponding duties owed by the governments, but it may also be diagonal, giving rise to duties on the part of states that extend beyond their own territory. The scope and extent of diagonal human rights are often controversial, and environmental rights face additional difficulties, because the environmental protection required by human rights is clarifying only gradually, on a case-by-case basis. To the extent that human rights require such protection when aligned vertically, it would be logical to conclude that they require the same degree of protection whenever they may be aligned diagonally. Human rights law provides few precedents to support that conclusion, however. Compared to human rights law, international environmental law (IEL) provides a clearer and more specific set of duties with respect to environmental protection. Moreover, most IEL is extraterritorial, in that it requires states to regulate actions within their control that could harm the environment beyond their territory. The problem with grounding diagonal environmental rights in IEL is that, in contrast to human rights law, most IEL operates along a horizontal axis: its duties are owed by states to other states, not to private actors. If the challenge for human rights law is to extend rights from the vertical axis to the diagonal, the challenge for IEL is to derive diagonal rights from horizontal ones. (shrink)
We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the (...) Borel reducbility hierarchy determined. (shrink)
The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by the (...) Curry combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. (shrink)
In the propositional modal treatment of two-variable first-order logic equality is modelled by a ‘diagonal’ constant, interpreted in square products of universal frames as the identity relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is (...) similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first-order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them. (shrink)
This paper deals with textual aspects of the geometric diagonal linear ornamentation that appears on traditional woven Lithuanian bands. Taking into consideration diachronic, local as well as universal perspectives, it aims to determine and classify the basic elements of the ornament that relate to the development of textuality. Previous investigations of Baltic and Lithuanian textile ornaments have been based on a purely geometric analysis of ornamental form, or on creating linguistic inventories of folk pattern denominations. This paper describes a (...) unique, elaborated, interdisciplinary method for studying such ornaments based on historical-typological comparative analyses, the classification of patterns with regard to their form and meaning, and the semiotic interpretation of mythopoetic images of patterns names. Further, the paper discusses whether an authentic folk classification and a tradition of typology based on the forms of patterns and names can be detected. From the traditional point of view the main meaning-carrying element of this ornamentation is the type of pattern. Therefore, reconstructions and interpretations of the semantic field of patterns’ signification may be based on the mythopoetic context of folk culture. (shrink)
Reliabilism about epistemic justification - the thesis that what makes a belief epistemically justified is that it was produced by a reliable process of belief-formation - must face two problems. First, what has been called "the new evil demon problem", which arises from the idea that the beliefs of victims of an evil demon are as justified as our own beliefs, although they are not - the objector claims - reliably produced. And second, the problem of diagnosing why skepticism is (...) so appealing despite being false. I present a special version of reliabilism, "indexical reliabilism", based on two-dimensional semantics, and show how it can solve both problems. (shrink)
We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)
I OFFER AN ANALYSIS OF DESCARTES'S COGITO WHICH IS RADICALLY NOVEL WHILE INCORPORATING MUCH AVAILABLE INSIGHT. BY ENLARGING FOCUS FROM THE DICTUM ITSELF TO THE REASONING OF DOUBT, DREAMING AND DEMON, I DEMONSTRATE A CLOSE PARALLEL TO THE LOGIC OF THE LIAR PARADOX. THIS HELPS TO EXPLAIN FAMILIAR PARADOXICAL FEATURES OF DESCARTES'S ARGUMENT. THE ACCOUNT PROVES TO BE TEXTUALLY ELEGANT AND, MOREOVER, HAS CONSIDERABLE INDEPENDENT PHILOSOPHICAL PLAUSIBILITY AS AN ACCOUNT OF MIND AND SELF.
Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that canno be a computation.
Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that cannot be a computation.
Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. In particular, (...) we show that the analogue of a theorem of Martin fails for these classes. (shrink)
This book is about one of the most baffling of all paradoxes – the famous Liar paradox. Suppose we say: 'We are lying now'. Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers (...) his own solutions and in the process assesses other attempts to solve the paradox. Unlike such attempts, Simmons' 'singularity' solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality – the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (shrink)
We claim that a recent article of P. Cotogno () in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ?logical incomputability? in their being computed over a wider class. Hence this ?logical incomputability? regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. This seems (...) to lead him into a further error in his analysis of the supposed conventional status of the infinite time Turing machines of Hamkins and Lewis (). Theorem 1 refutes this directly. The diagonalisation misunderstanding Infinite computation Conclusion. (shrink)
The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic paradoxes based (...) on diagonal construction are caused by superficial understanding of what a name is. (shrink)
Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.
Robert Stalnaker contrasts two interpretations, semantic and metasemantic, of the two-dimensionalist framework. On the semantic interpretation, the primary intension or diagonal proposition associated with an utterance is a semantic value that the utterance has in virtue of the actual linguistic meaning of the corresponding sentence, and that primary intension is both what a competent speaker grasps and what determines different secondary intensions or horizontal propositions relative to different possible worlds considered as actual. The metasemantic interpretation reverses the order of (...) explanation: an utterance has the primary intension it has because it yields the secondary intensions it yields relative to different possible worlds considered as actual. In these possible worlds, the semantic facts can be different: the metasemantic interpretation is metasemantic in the sense that the secondary intensions are determined relative to possible worlds considered as actual given the meanings the expressions have there. Stalnaker holds a causal picture of the reference of names, according to which names have no meaning over and above their unique referent, and therefore maintains that the semantic interpretation is not an option. He thus endorses the metasemantic interpretation, while insisting that this interpretation does not, contrary to what he originally thought, yield any account of a priori truth and knowledge. My double aim in this paper is to show (i) that the metasemantic interpretation, as sketched by Stalnaker, is not compatible with one natural understanding of the causal picture of reference, on which names are rigid because they have their original bearers essentially , and (ii) that a third kind of interpretation of the framework is available, the metasyntactic interpretation, which grants that names have their bearers essentially and yields some account of a priori knowledge. (shrink)