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)

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)

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’.

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)

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)

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.

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.

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)

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)

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.

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)

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)

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP ≠ coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be (...) special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences. -/- We have introduced the most fundamental conjectures in our earlier works. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before. (shrink)

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)

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)

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 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)

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)

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.

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.

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.

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)

John Rawls’s use of the “fully cooperating assumption” has been criticized for hindering attempts to address the needs of disabled individuals, or non-cooperators. In response, philosophers sympathetic to Rawls’s project have extended his theory. I assess one such extension by Cynthia Stark, that proposes dropping Rawls’s assumption in the constitutional stage (of his four-stage sequence), and address the needs of non-cooperators via the social minimum. I defend Stark’s proposal against criticisms by Sophia Wong, Christie Hartley, and Elizabeth Edenberg and Marilyn (...) Friedman. Nevertheless, I argue that Stark’s proposal is crucially incomplete. Her formulation of the social minimum lacks accompanying criteria with which the adequacy of the provisions for non-cooperators may be assessed. Despite initial appearances, Stark’s proposal does not fully address the needs of non-cooperators. I conclude by considering two payoffs of identifying this lack of criteria. (shrink)

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or (...) Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow. (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.

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

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)

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.

Although G¨odel proved the first incompleteness theorem by intuitionistically respectable means, G¨odel’s formula, true although undecidable,seems to offer a counter-example to the general constructivist or anti-realistclaim that truth may not transcend recognizability in principle. It is arguedhere that our understanding of the formula consists in a knowledge of itstruth-conditions, that it is true in a minimal sense and, finally, that it is recognized as such given the consistencyand ω-consistency of P. The philosophical lesson to be drawn from G¨odel’sproof is that (...) our capacities for justification in favour of minimal truth exceedwhat is strictly speaking formally provable in P by means of an algorithm. (shrink)

§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)

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.

Using video recordings of draft meetings conducted as part of an intramural basketball program as data, this conversation analytic study examines the use of an incomplete utterance in a joint evaluative activity. In particular, we focus on how the participants, volunteer coaches, who meet to draft players for their respective teams, produce a syntactically incomplete utterance as a means to critically assess a player, a non-present third party to the interaction. Analysis reveals that the participants use an incomplete utterance as (...) part of dispreferred design; it allows them to withhold articulating overt criticism of others. By trailing off where the criticism is due, the participants display reluctance to verbalize what is to be said and treat its articulation as delicate. The syntactic structure of the utterance that includes a contrastive conjunction and accompanying embodied actions such as head shakes help them convey a critical stance. We examine the use of incomplete utterances in both agreement and disagreement sequences; the recipients display their unproblematic understanding of the critical assessment and respond by providing their own assessments that either affiliate or disaffiliate with the conveyed critical stance. (shrink)

In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for T (...) . We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A . We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA . The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing. (shrink)

Cappelen and Lepore (2005) maintain that Incompleteness Arguments for context sensitivity are fallacious. In their view, Incompleteness Arguments are non sequitur fallacies whose conclusions are not logically related to premises. They affirm that the conclusions of Incompleteness Arguments are metaphysical claims about the existence of entities that might be constituents of propositions, while their premises concern psychological data about speakers' dispositions to truth evaluate sentences in contexts of utterance. Cappelen and Lepore reject Incompleteness Arguments because psychological data have no bearing (...) on metaphysical issues. I do not dispute Cappelen and Lepore's claim that psychological data have no relevance to metaphysics. Nonetheless, I argue that Cappelen and Lepore's criticism is vitiated by a misunderstanding of the nature of Incompleteness Arguments that makes them overlook the link between semantics and linguistic competence. I try to shed light on the real nature of Incompleteness Arguments by pointing at the failure of Cappelen and Lepore's criticism. My aim is not to defend Semantic Contextualism, but to bring to attention the link between semantics and linguistic competence that should never be overlooked in debates in the field of semantics. (shrink)

Standard attempts to defend Russell's Theory of Descriptions against the problem posed by incomplete descriptions, are discussed and dismissed as inadequate. It is then suggested that one such attempt, one which exploits the notion of a contextually delimited domain of quantification, may be applicable to incomplete quantifier expressions which are typically treated as quantificational: expressions of the form AllF's, NoF's, SomeF's, Exactly eightF's, etc. In this way, one is able to retain the plausible claim that such expressions ought to receive (...) their usual quantificational analyses. The conclusion tentatively drawn is that perhaps definite descriptions arenot amenable to a (Russellian) quantificational analysis. (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 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 is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, (...) we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$. (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)

Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...) prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial. The present entry surveys the two incompleteness theorems and various issues surrounding them. (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)

Bayesian models typically assume that agents are rational, logically omniscient and opinionated. The last of these has little descriptive or normative appeal, however, and limits our ability to describe how agents make up their minds (as opposed to changing them) or how they can suspend or withdraw their opinions. To address these limitations this paper represents the attitudinal states of non-opinionated agents by sets of (permissible) probability and desirability functions. Several basic ways in which such states of mind can be (...) changed are then characterised and compared with those found in AGM style models of attitude revision. Finally these models are employed to describe how agents make up their mind when deliberating. (shrink)

Many philosophers have discussed the ability of thinkers to think thoughts that the thinker cannot justify because the thoughts involve concepts that the thinker incompletely understands. A standard example of this phenomenon involves the concept of the derivative in the early days of the calculus: Newton and Leibniz incompletely understood the derivative concept and, hence, as Berkeley noted, they could not justify their thoughts involving it. Later, Weierstrass justified their thoughts by giving a correct explication of the derivative concept. This (...) paper discusses various accounts of how a thinker manages to think with a concept that they incompletely understand and finds them wanting in the case at hand. Part of the overlooked complexity is that there are many derivative concepts, and it is unclear in virtue of what a thinker would be thinking with one of them rather than another. After critical evaluation of standard accounts, this paper suggests a novel account of how one should think about the derivative concepts with which Newton and Leibniz thought and how Weierstrass could have managed to justify their thoughts even if their thoughts did not involve the same derivative concept as Weierstrass’s. (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).

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.

By demostrating how Kapital is incomplete, The Incomplete Marx provides the basis for a re-interpretation of Marx that looks beyond the Marx of Kapital.