Incomplete penetrance and variable expressivity are non‐Mendelian phenomena resulting in the lack of correlation between genotype and phenotype. Not withstanding the diversity in mechanisms, differential expression of homologous alleles within cells manifests as variations in penetrance and expressivity of mutations between individuals of the same genotype. These phenomena are seen most often in dominantly inherited diseases, implying that they are sensitive to concentration of the gene product. In this framework and the advances in understanding the role of microRNA (miRNA) in (...) fine‐tuning gene expression at translational level, we propose miRNA‐mediated regulation as a mechanism for incomplete penetrance and variable expressivity. The presence of miRNA binding sites at 3′ UTR, co‐expression of target gene–miRNA pairs for genes showing incomplete penetrance and variable expressivity derived from available data lend support to our hypothesis. Single nucleotide polymorphisms in the miRNA target site facilitate the implied differential targeting of the transcripts from homologous alleles. (shrink)

I discuss two competing theories of the progressive: the theory proposed in Parsons (1980, 1985, 1989, 1990) and the theory proposed in Landman (1992). These theories differ in more than one way. Landman regards the progressive as an intentional operator, while Parsons doesn't. Moreover, Landman and Parsons disagree on what uninflected predicates denote. For Landman, cross the street has in its denotation complete events of crossing the street; the aspectual contribution of English simple past (perfective aspect) is the identity function. (...) For Parsons, both complete and incomplete events of crossing the street can be in the denotation of the base VP; perfective aspect restricts its denotation to the events that culminate. I present a version of ParsonÕs theory that avoids the problems raised by Landman, in particular the problem posed for Parsons by creation verbs. The repaired version and Landman's theory still differ in the way they analyze uninflected verbs. The repaired version and Landman's theory still differ in the way they analyze uninflected predicates. I present evidence from Slavic languages that both theories are needed. Finally, I discuss some evidence that may favor one or the other approach to the semantics of the English progressive. (shrink)

Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something (...) real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein’s preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein’s attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927. (shrink)

Following Post program, we will propose a linguistic and empirical interpretation of Gödel’s incompleteness theorem and related ones on unsolvability by Church and Turing. All these theorems use the diagonal argument by Cantor in order to find limitations in finitary systems, as human language, which can make “infinite use of finite means”. The linguistic version of the incompleteness theorem says that every Turing complete language is Gödel incomplete. We conclude that the incompleteness and unsolvability theorems find limitations (...) in our finitary tool, which is our complete language. (shrink)

This paper introduces the axiom of Negative Dominance, stating that if a lottery f is strictly preferred to a lottery g, then some outcome in the support of f is strictly preferred to some outcome in the support of g. It is shown that if preferences are incomplete on a sufficiently rich domain, then this plausible axiom, which holds for complete preferences, is incompatible with an array of otherwise plausible axioms for choice under uncertainty. In particular, in this setting, Negative (...) Dominance conflicts with the standard Independence axiom. A novel theory, which includes Negative Dominance, and rejects Independence, is developed and shown to be consistent. (shrink)

The main point of this paper has been to show that the concept of evaluative incompleteness deserves consideration. In addition, I have suggested that it is plausible to accept that certain states of affairs in fact are evaluatively incomplete. But I have not sought to prove that this is so; indeed, I do not know how such proof might be given. Just which states of affairs, if any, are evaluatively incomplete is an extremely vexed question, and it is not (...) one to which I have attempted to supply any systematic answer. My aim has been merely to point out that it is arguable that certain states of affairs are evaluatively incomplete — a fact that ought not to be overlooked due to an unquestioning acceptance of (II) and a fact which, certainly, ought not to be ruled out by fiat due to an adherence to definitions and assumptions which imply that (II) is false. (shrink)

This paper examines the incompleteness of collective preference. We provide a series of Arrovian impossibility theorems without completeness. First, we consider the notion of regularity introduced by Eliaz and Ok (2006, Games and Economic Behavior 56, 61–86); it is an appropriate richness property for strict preference when preference is allowed to be incomplete. We examine the implication of imposing regularity on collective preference. Second, we propose responsiveness, a variation of positive responsiveness. This axiom requires that some changes in individual (...) preferences make an alternative weakly better than another. Third, we consider coherency conditions for collective preferences; this conditionally requires the existence of comparable pairs in a certain manner. We prove an impossibility result for each condition using Arrovian axioms. (shrink)

This paper shows Hilbert system (C+J)-, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system (C+J)- contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system (C+J)- is (...) semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979). (shrink)

A challenge for theories of incomplete descriptions is to capture the consistency of ‘Sobel sequences’ and to account for an asymmetry in the acceptability of utterances of Sobel sequences and ‘reverse Sobel sequences’. David Lewis’s theory of incomplete descriptions answers, unlike many other theories, the challenge from Sobel sequences, but it does not answer the challenge from reverse Sobel sequences. This article presents another asymmetry in the availability of anaphoric readings of Sobel sequences and reverse Sobel sequences, and proposes an (...) explanation of the original asymmetry on its grounds. This provides an answer to the challenge for Lewis’s theory. (shrink)

This paper investigates, in a simple risk-sharing framework, the extent to which the incompleteness of contracts could be attributed to the complexity costs associated with the writing and the implementation of contracts. We show that, given any measure of complexity in a very general class, it is possible to find simple contracting problems such that, when complexity costs are explicitly taken into account, the contracting parties optimally choose an incomplete contract which coincides with the ‘default’ division of surplus. Optimal (...) contracts with complexity costs are constrained efficient in our model. We therefore interpret our results as saying that, in the absence of a strategic role for complexity costs, their effect is entirely determined by their size relative to the size of payoffs. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long ‘search’ for a purely mathematical incompleteness result in first-order Peano arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.

This book concentrates on research done during the last twenty years on the philosophy of quantum mechanics. In particular, the author focuses on three major issues: whether quantum mechanics is an incomplete theory, whether it is non-local, and whether it can be interpreted realistically. Much of the book is concerned with distinguishing various senses in which these questions can be taken, and assessing the bewildering variety of answers philosophers and physicists have given up to now. The book is self-contained in (...) that it presents the necessary parts of the mathematical formalism of quantum mechanics and also covers other interpretative topics, such as the problem of measurement and the uncertainty relations. A considerable portion of the book is based on original arguments presented by the author in lectures and research papers over the past ten years. However, this material is integrated with a broad coverage of most of the recent research in the field, so as to provide a balanced introduction to the whole subject. (shrink)

This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.

The implicit content associated with incomplete definite descriptions is contributed in the form of definite descriptions of situations. A definite description of this kind is contributed by a small structure in the syntax, which is interpreted, in general terms, as ‘the situation that bears R to s’.

There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “$\mathrm {T}$is$\Sigma (...) _n$-ill” is a canonical example of a$\Sigma _{n+1}$formula that is$\Pi _{n+1}$-conservative over$\mathrm {T}$. (shrink)

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

Our purpose is to present some connections between modal incompleteness andmodal logics related to the Gödel-Löb logic GL. One of our goals is to prove that for all m, n, k, l ∈ ℕ the logic K + equation image□i □jp ↔ p) → equation image□ip is incomplete and does not have the fixed point property. As a consequence we shall obtain that the Boolos logic KH does not have the fixed point property.

The purpose of this paper is to discuss two phenomena related to the semantics of definite descriptions: that of incomplete uses of descriptions, and that of the underdetermination of referential uses of descriptions. The Russellian theorist has a way of accounting for incomplete uses of descriptions by appealing to an account of quantifier domain restriction, such as the one proposed in Stanley and Szabó (2000a). But, I argue, the Russellian is not the only one in a position to appeal to (...) such an account of incomplete uses of descriptions. Proponents of other theories, such as the Fregean, which does not treat descriptions as quantifiers, might benefit from this account of domain restriction. In the second part of the paper I discuss referential uses of incomplete definite descriptions. Relative to such uses, Wettstein (1981) and others have argued that the Russellian theory faces a problem of underdetermination of semantic content. Neale (2004) has replied to this objection showing why it does not pose a threat to the Russellian theory. Again, I argue that not only the Russellian, but also the Fregean can subscribe to Neale’s (2004) suggestion. (shrink)

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one (...) of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC. (shrink)

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$ -sound and $\Sigma ^0_1$ -definable do not prove their own $\Pi ^0_1$ -soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$ -sound and $\Sigma ^1_1$ -definable do not prove their own $\Pi ^1_1$ -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of (...) forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) (...) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

How is constitutionalism possible, when people disagree on so many questions about what is good and what is right? The answer lies in two kinds of incompletely theorized agreement - both reached amidst the sharpest disagreements about the fundamental issues in social life. The first consist of agreements on abstract formulations ; these agreements are crucial to constitution-making as a social practice. The second consist of agreements on particular doctrines and practices; these agreements are crucial to life and law under (...) existing constitutions. Incompletely theorized agreements help illuminate an enduring constitutional puzzle: how members of diverse societies can work together on terms of mutual respect amidst intense disagreements about both the right and the good. Such agreements help make constitutions and constitutional law possible, even within nations whose citizens cannot concur on the most fundamental matters. (shrink)

This article discusses the thesis that a subject can have a concept, think thoughts containing it, that she incompletely understands. The central question concerns how to construe the distinction between having a concept and understanding it. Two important versions of the thesis are distinguished: a metasemantic version and an epistemic version. According to the first, the subject may have concept C without being a fully competent user, in virtue of deference to other speakers or to the world. According to the (...) second, the subject may have a concept without being able to provide a proper explication of it. It is argued that whereas the epistemic version is plausible, the metasemantic version faces some challenges. First, it needs to be explained precisely how deference enables a speaker to have C. Second, metasemantic incomplete understanding is in tension with the idea that concepts serve to capture the subject’s cognitive perspective. (shrink)

The incompleteness problem for virtue ethics is inherited by a virtue-based theory of argumentation as developed by Daniel Cohen. A complete normative theory of argumentation should be able to provide reasons for why argumentative virtues such as open-mindedness are worthwhile, along with being able to resolve conflicts of such virtues. Adumbrating virtue-based argumentation theory with a pragmatic utilitarian approach constitutes a more complete theory that can account for why argumentative virtues are worthwhile.

This is an exploration of a cluster of related logical results. Taken together these seem to have something philosophically important to teach us: something about knowledge and truth and something about the logical impossibility of totalities of knowledge and truth. The book includes explorations of new forms of the ancient and venerable paradox of the :Liar, applications and extensions of Kaplan and Montague's paradox of the Knower, generalizations of Godel's work on incompleteness, and new uses of Cantorian diagonalization. Throughout, (...) the concern is with metaphysical and epistemological implications of these results: what such results may have to teach us about knowledge, truth, and possibility. (shrink)

Aiming to unravel the mystery of quantum mechanics, this book is concerned with questions about action-at-a-distance, holism, and whether quantum mechanics gives a complete account of microphysical reality. With rigorous arguments and clear thinking, the author provides an introduction to the philosophy of physics.

We investigate incomplete symbols, i.e. definite descriptions with scope-operators. Russell famously introduced definite descriptions by contextual definitions; in this article definite descriptions are introduced by rules in a specific calculus that is very well suited for proof-theoretic investigations. That is to say, the phrase ‘incomplete symbols’ is formally interpreted as to the existence of an elimination procedure. The last section offers semantical tools for interpreting the phrase ‘no meaning in isolation’ in a formal way.

In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem. We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ and (...) $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds. (shrink)

Tyler Burge’s influential arguments have convinced most philosophers that a thinker can have a thought involving a particular concept without fully grasping or having mastery of that concept. In Burge’s (1979) famous example, a thinker who lacks mastery of the concept of arthritis nonetheless has thoughts involving that concept. It is generally supposed, however, that this phenomenon – incomplete understanding, for short – does not require us to reconsider in a fundamental way what it is for a thought to involve (...) a particular concept. In this paper, I argue that the real significance of incomplete understanding has not been appreciated. To the extent that theorists of content address the phenomenon of thoughts involving incompletely grasped contents at all, they tend to assume that some hand-waving about deference to other thinkers who fully grasp the relevant concepts will take care of the inconvenient cases of incomplete understanding. The main lesson of Burge’s arguments is often taken to be that the content of language and thought is socially determined. On this picture, we do not need to change our basic view about what it is to have a concept; we just need to recognize that some thinkers can manage to have a concept by piggybacking on others. In contrast, on the view I defend, taking incomplete understanding seriously forces us to rethink some of our most basic assumptions about the nature of mental content. Deference is a red herring. The role of society in determining the content of thought is not the main lesson, but at most a useful clue as to the nature of mental and linguistic content. (shrink)

Elizabeth Prior claims that dispositional predicates are incomplete in the sense that they have more than one argument place. To back up this claim, she offers a number of arguments that involve such ordinary dispositional predicates as ‘fragile’, ‘soluble’, and so on. In this paper, I will first demonstrate that one of Prior’s arguments that ‘is fragile’ is an incomplete predicate is mistaken. This, however, does not immediately mean that Prior is wrong that ‘fragile’ is an incomplete predicate. On the (...) contrary, I maintain that she has offered another valid argument that does indeed establish the claim that ‘fragile’ is an incomplete predicate. I will argue further that Prior is right that ‘soluble’ is an incomplete predicate. Then does this mean that all dispositional predicates are incomplete? I don’t think so. I will suggest that there are complete dispositional predicates that have no more than one argument place. Finally, by relying on my discussion of the incompleteness of dispositional predicates, I will attempt to provide a better understanding of the context-dependence and intrinsic nature of dispositional ascriptions. (shrink)

Samuel seeks out Kurt at a pub and initiates a discussion. Soon Kurt becomes engaged. What is it that is incomplete?

How is constitutionalism possible, when people disagree on so many questions about what is good and what is right? The answer lies in two kinds of incompletely theorized agreement - both reached amidst the sharpest disagreements about the fundamental issues in social life. The first consist of agreements on abstract formulations ; these agreements are crucial to constitution-making as a social practice. The second consist of agreements on particular doctrines and practices; these agreements are crucial to life and law under (...) existing constitutions. Incompletely theorized agreements help illuminate an enduring constitutional puzzle: how members of diverse societies can work together on terms of mutual respect amidst intense disagreements about both the right and the good. Such agreements help make constitutions and constitutional law possible, even within nations whose citizens cannot concur on the most fundamental matters. (shrink)

It is shown that there is a Kripke incomplete logic in NExt(KTB ⊕ □2 p → □3 p). Furthermore, it is also shown that there exists a continuum of Kripke incomplete logics in NExt(KTB ⊕ □5 p → □6 p).

What is the best way to make sustained societal progress over time? Non-ideal theory done on its own faces the problem of second best, but ideal theory seems unable to cope with disagreement about how to make progress. If ideal theory gives up its claims to completeness, then we can use the method of incompletely theorized agreements to make progress over time.

I explain and motivate the shutdown problem: the problem of creating artificial agents that (1) shut down when a shutdown button is pressed, (2) don’t try to prevent or cause the pressing of the shutdown button, and (3) otherwise pursue goals competently. I then propose a solution: train agents to have incomplete preferences. Specifically, I propose that we train agents to lack a preference between every pair of different-length trajectories. I suggest a way to train such agents using reinforcement learning: (...) we give the agent lower reward for repeatedly choosing same-length trajectories. (shrink)

Jean-Paul Sartre argues that human beings are fundamentally incomplete. Self-consciousness brings with it a presence-to-self. Human beings consequently seek two things at the same time: to possess a secure and stable identity, and to preserve the freedom and distance that come with self-consciousness. This is an impossible ideal, since we are always beyond what we are and we never quite reach what we could be. The possibility of completion haunts us and we continue to search for it even when we (...) are convinced it can never be achieved. Sartre suggests that we have to continue seeking this ideal in the practical sphere, even when our philosophical reflection shows it to be an impossibility. Sartre puts this existential dilemma in explicitly theological terms. 'God' represents an ideal synthesis of being and consciousness which remains a self-contradictory goal. This dilemma remains unresolved in his thinking. (shrink)

In the early seventies, several logicians developed a semantics for propositional systems of relevance logic. The essential ingredients of this semantics were a privileged point o, an ‘accessibility’ relation R and a special operator * for evaluating negation. Under the truth- conditions of the semantics, each formula A(Pl,…,Pn) could be seen as expressing a first order condition A+(pl,…,pn, o, R,*) on sets p1,…,pn and o, R, *, while each formula-scheme could be regarded as expressing the second-order condition ∀p1,…,∀pn A+(p1,…,pn, o, (...) R, *). It could then be shown that many standard systems of propositional relevance logic were complete in the sense that their theorems were just those formulas true in all models whose components o, R and * conformed to the second-order conditions expressed by the axioms of the system. (shrink)

By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We (...) shall show also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)

In this paper, we argue that the systems Basic Logic of Evidence (BLE) and Logic of Evidence and Truth (LETJ) suffer a kind of semantic incompleteness with respect to the informal notion of evidence. More especifically, we argue that the connective o of the logic LETJ fails to validate intuitive principles about conclusive evidence.

Incomplete Archaeologies takes a familiar archaeological concept--assemblages--and reconsiders such groupings, collections and sets of things from the perspective of the work required to assemble them. The discussions presented here engage with the practices of collection, construction, performance and creation in the past (and present) which constitute the things and groups of things studied by archaeologists--and examine as well how these things and thing-groups are dismantled, rearranged, and even destroyed, only to be rebuilt and recreated. The ultimate aim is to reassert (...) an awareness of the incompleteness of assemblage, and thus the importance of practices of assembling (whether they seem at first creative or destructive) for understanding social life in the past as well as the present. The individual chapters represent critical engagements with this aim by archaeologists presenting a broad scope of case studies from Eurasia and the Mediterranean. Case studies include discussions of mortuary practice from numerous angles, the sociopolitics of metallurgy, human-animal relationships, landscape and memory, the assembly of political subjectivity and the curation of sovereignty. These studies emphasise the incomplete and ongoing nature of social action in the past, and stress the critical significance of a deeper understanding of formation processes as well as contextual archaeologies to practices of archaeology, museology, art history, and other related disciplines. Contributors challenge archaeologists and others to think past the objects in the assemblage to the practices of assembling, enabling us to consider not only plural modes of interacting with and perceiving things, spaces, human bodies and temporalities in the past, but also to perhaps discover alternate modes of framing these interactions and relationships in our analyses. Ultimately then, Incomplete Archaeologies takes aim at the perceived totality not only of assemblages of artefacts on shelves and desks, but also that of some of archaeology's seeming-seamless epistemological objects"--From publisher's website. (shrink)

Incomplete Archaeologies takes a familiar archaeological concept--assemblages--and reconsiders such groupings, collections and sets of things from the perspective of the work required to assemble them. The discussions presented here engage with the practices of collection, construction, performance and creation in the past (and present) which constitute the things and groups of things studied by archaeologists--and examine as well how these things and thing-groups are dismantled, rearranged, and even destroyed, only to be rebuilt and recreated. The ultimate aim is to reassert (...) an awareness of the incompleteness of assemblage, and thus the importance of practices of assembling (whether they seem at first creative or destructive) for understanding social life in the past as well as the present. The individual chapters represent critical engagements with this aim by archaeologists presenting a broad scope of case studies from Eurasia and the Mediterranean. Case studies include discussions of mortuary practice from numerous angles, the sociopolitics of metallurgy, human-animal relationships, landscape and memory, the assembly of political subjectivity and the curation of sovereignty. These studies emphasise the incomplete and ongoing nature of social action in the past, and stress the critical significance of a deeper understanding of formation processes as well as contextual archaeologies to practices of archaeology, museology, art history, and other related disciplines. Contributors challenge archaeologists and others to think past the objects in the assemblage to the practices of assembling, enabling us to consider not only plural modes of interacting with and perceiving things, spaces, human bodies and temporalities in the past, but also to perhaps discover alternate modes of framing these interactions and relationships in our analyses. Ultimately then, Incomplete Archaeologies takes aim at the perceived totality not only of assemblages of artefacts on shelves and desks, but also that of some of archaeology's seeming-seamless epistemological objects"--From publisher's website. (shrink)