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.

The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that all modal systems are characterizable by classes of frames defined by first-order conditions on a binary relation. However, the discovery of certain incomplete modal systems has undermined the (...) all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all. ;In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic. ;Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory. (shrink)

It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal (...) logic of linear discrete time with gaps follows. (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.

ABSTRACT The question of axiomatizability of first-order temporal logics is studied w.r.t. different semantics and several restrictions on the language. The validity problem for logics admitting flexible interpretations of the predicate symbols or allowing at least binary predicate symbols is shown to be ?1 1-complete. In contrast, it is decidable for temporal logics with rigid monadic predicate symbols but without function symbols and identity.

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.

Reference [12] introduced a novel formula to formula translation tool that enables syntactic metatheoretical investigations of first-order modallogics, bypassing a need to convert them first into Gentzen style logics in order torely on cut elimination and the subformula property. In fact, the formulator tool,as was already demonstrated in loc. cit., is applicable even to the metatheoreticalstudy of logics such as QGL, where cut elimination is unavailable. This paper applies the formulator approach to show the independence of the axiom schema ☐A (...) → ☐∀ A of the logics M3and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics. (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)

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)

Abstract:Russellian incomplete symbols are usually conceived as an analytical residue—as what remains of the would-be entities when properly analyzed. This article aims to reverse the approach in raising another question: what, if any, does the incomplete symbol contribute to the completely analyzed language? I will first show that, from a technical point of view, there is no difference between the way Russell defines his denoting phrases in “On Denoting” and the way Frege defines his second-order concepts. But I (...) will secondly support that the two notions have two widely different conceptual meanings: the same logical procedures which are used by Frege to increase the deductive power of his system allow Russell to logically relate quantificational notation to ordinary language. Focusing on Russell’s treatment of OD’s puzzles, I will thirdly argue that this shift constitutes the source of a deep transformation in the way logic and language are related. (shrink)

A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...) logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence. (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)

Let n be a positive integer. By a $\beta_{n}-model$ we mean an $\omega-model$ which is elementary with respect to $\sigma_{n}^{1}$ formulas. We prove the following $\beta_{n}-model$ version of $G\ddot{o}del's$ Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a $\beta_{n}-model$ of S, then there exists a $\beta_{n}-model$ of S + "there is no countable $\beta_{n}-model$ of S". We also prove a $\beta_{n}-model$ version of $L\ddot{o}b's$ Theorem. As a corollary, we obtain (...) a $\beta_{n}-model$ which is not a $\beta_{n+1}-model$. (shrink)

We define a property R(A 0 , A 1 ) in the partial order E of computably enumerable sets under inclusion, and prove that R implies that A 0 is noncomputable and incomplete. Moreover, the property is nonvacuous, and the A 0 and A 1 which we build satisfying R form a Friedberg splitting of their union A, with A 1 prompt and A promptly simple. We conclude that A 0 and A 1 lie in distinct orbits under automorphisms (...) of E, yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit. (shrink)

This paper addresses the phenomenon of incomplete preferences in disaster risk management. If an agent finds two options to be incomparable and thus has an incomplete preference ordering, i.e., neither prefers one option over the other nor finds them equally as good, it is not possible for the agent to perform a value tradeoff, necessary for an informed decision, between these two options. In this paper we suggest a way to model incomplete preference orderings by means (...) of probabilistic preferences, and how to reveal an agent’s incomplete preference ordering within a behaviorist framework. (shrink)

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however (...) complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π. (shrink)

We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question (...) in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness. (shrink)

The search for moral objectivity has been constant throughout the history of philosophy, although interpretations of the nature and scope of objectivity have varied. One aim of the pursuit of moral objectivity has been the demonstration of what may be termed its epistemological thesis, that is, the claim that the truth of assertions of the goodness or rightness of moral acts is as legitimate, reliable, or valid as the truth of assertions involving other forms of human knowledge, such as common (...) sense, practical expertise, science, or mathematics. Another aim of the quest for moral objectivity may be termed its pragmatic formulation ; this refers to the development of a method or procedure that will mediate among conflicting moral views in order to realize a convergence or justified agreement about warranted or true moral conclusions. In the ethical theories of Aristotle, David Hume, and John Dewey, theories that represent three of the four variants of ethical naturalism that are surveyed in this essay, the epistemological thesis and the pragmatic formulation are integrated or combined. The distinction between these two elements is significant for the present essay, however, since I want to show that linguistic naturalism, the fourth variant I shall examine, has provided a demonstration of the epistemological thesis about moral knowledge, even if the pragmatic formulation has not been successfully realized. (shrink)

The search for moral objectivity has been constant throughout the history of philosophy, although interpretations of the nature and scope of objectivity have varied. One aim of the pursuit of moral objectivity has been the demonstration of what may be termed its epistemological thesis, that is, the claim that the truth of assertions of the goodness or rightness of moral acts is as legitimate, reliable, or valid as the truth of assertions involving other forms of human knowledge, such as common (...) sense, practical expertise, science, or mathematics. Another aim of the quest for moral objectivity may be termed its pragmatic formulation ; this refers to the development of a method or procedure that will mediate among conflicting moral views in order to realize a convergence or justified agreement about warranted or true moral conclusions. In the ethical theories of Aristotle, David Hume, and John Dewey, theories that represent three of the four variants of ethical naturalism that are surveyed in this essay, the epistemological thesis and the pragmatic formulation are integrated or combined. The distinction between these two elements is significant for the present essay, however, since I want to show that linguistic naturalism, the fourth variant I shall examine, has provided a demonstration of the epistemological thesis about moral knowledge, even if the pragmatic formulation has not been successfully realized. (shrink)

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation (...) of set theory and arithmetic are demonstrated. (shrink)

According to the displacement model of secularization, religious-theological concepts, themes, and values have been reinterpreted in non-religious contexts without fully dispensing with the religious content. Secularization is thus incomplete. The incomplete secularization argument can be used as a lens through which to read Ethan Kleinberg’s deconstructive approach to the past. In his narrative, as reconstructed here, deconstruction promises to bring us closer to a secular relationship to the past than the ontological realism Kleinberg says still dominates contemporary historical (...) theory. By contrasting Kleinberg’s analysis with Hayden White’s, whose oeuvre can be read as structured by the idea of incomplete secularization and a wish to liberate history from religious themes in order to enable a direct confrontation with meaninglessness, I argue that Kleinberg’s deconstructive approach does not fulfill its promise. Rather, it opens up a post-secular historiography in which religious themes might find a place at the very heart of historical reasoning. (shrink)

Bayesian psychology, in what is perhaps its most familiar version, is incomplete: Jeffrey conditionalization is not a complete account of rational belief change. Jeffrey conditionalization is sensitive to the order in which the evidence arrives. This order effect can be so pronounced as to call for a belief adjustment that cannot be understood as an assimilation of incoming evidence by Jeffrey's rule. Hartry Field's reparameterization of Jeffrey's rule avoids the order effect but fails as an account of how new (...) evidence should be assimilated. (shrink)

Bayesian psychology, in what is perhaps its most familiar version, is incomplete: Jeffrey conditionalization is not a complete account of rational belief change. Jeffrey conditionalization is sensitive to the order in which the evidence arrives. This order effect can be so pronounced as to call for a belief adjustment that cannot be understood as an assimilation of incoming evidence by Jeffrey's rule. Hartry Field's reparameterization of Jeffrey's rule avoids the order effect but fails as an account of how new (...) evidence should be assimilated. (shrink)

An ordered field is said to be Scott complete iff it is complete with respect to its uniform structure. Zakon has asked whether nonstandard real lines are Scott complete. We prove in ZFC that for any complete Boolean algebra B which is not (ω, 2)-distributive there is an ultrafilter U of B such that the Boolean ultrapower of the real line modulo U is not Scott complete. We also show how forcing in set theory gives rise to examples of Boolean (...) ultrapowers of the real line which are not Scott complete. (shrink)

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic (...) regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski. Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle. (shrink)

We study preferences over lotteries which do not necessarily satisfy completeness. We provide a characterization which generalizes Expected Utility theory. We show in particular that various sure-thing axioms are needed to guaranteee the representability in terms of utility intervals rather than numbers, and to provide a linear interval order representation which is very much in the spirit of Expected Utility theory.

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like (...) FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of (...) the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt [12], and later extended to quantified modal relevant logics by Ferenz [6]. Modular soundness and completeness are proved for constant domain semantics, (...) using non-general frames with Mares–Goldblatt truth conditions. I further detail the relation between the demand that classical possible worlds have Tarskian truth conditions and incompleteness results in quantified relevant logics. (shrink)

This article examines some aspects of the natural philosophy of Juan Gallego de la Serna, royal physician to the Spanish kings Philip III and Philip IV. In his account of animal generation, Gallego criticizes widely accepted views: (1) the view that animal seeds are animated, and (2) the alternative view that animal seeds, even if not animated, possess active potencies sufficient for the development of animal souls. According to his view, animal seeds are purely material beings. This, of course, raises (...) the question of how living beings can arise from inanimate matter. Gallego is aware that two other thinkers who understood animal seeds as purely material beings, Duns Scotus and the Louvain-based physician Thomas Feyens, did not solve this problem. Gallego’s solution makes use of the notion of incomplete entities developed by the Spanish Jesuit Francisco Suarez. While Suarez applies this notion to soul and body in order to explain why souls have a natural tendency towards organic bodies and organic bodies have a natural tendency towards souls, Gallego applies this notion to the natural tendency of animal seeds towards each other and towards further substances in their respective environment. In his view, this natural tendency of animal seeds to incorporate further substances explains that origin of material structures complex enough to constitute an animal soul. (shrink)

This book covers completeness of first-order logic, some model theory, Gödel's incompleteness theorems and related results, and a smattering of computability theory. The text is self-contained and provides full proofs of the main facts. Ideally, the reader of this work has already taken at least one introductory logic course; however, everything needed to understand the syntax and semantics of first-order logic is presented herein. Students from philosophy, linguistics, computer science, physics, and other related subjects will find this work useful and (...) enlightening. (shrink)

This article argues that values that apply to health care allocation entail the possibility of “spectrum arguments,” and that it is plausible that they often fail to determine a best alternative. In order to deal with this problem, a two-step process is suggested. First, we should identify the Strongly Uncovered Set that excludes all alternatives that are worse than some alternatives and not better in any relevant dimension from the set of eligible alternatives. Because the remaining set of alternatives often (...) contain more than one element, we need some complementary method of selecting a unique alternative. In order to address this issue, I suggest that we must invoke caps on the values that are used to evaluate alternatives, and that these caps must be grounded in collective commitments. (shrink)

Peirce did not achieve a final systematization of his work. Beyond the difficulties in explaining so many philosophical tools that he introduced—suffice it to mention semiotic, abductive logic, a heuristic based on continuity, scholastic realism—, there is a theoretical reason for this incompletion. All those new philosophical tools indicated a conception of synthesis very different from the one he received from Kant. Peirce did not realize the profound direction of his enquiry so that he did not directly question neither Kant’s (...) legacy on this issue nor the idea of necessity that presides over it. Starting from Peirce’s conception of continuity and change, this paper will give a new definition of synthetic and analytic judgments and reasoning, completing the picture with a third “vague” judgment and reasoning. In this new definition a synthetic judgment is a judgment that recognizes identity through changes. An analytical judgment is a judgment that loses identity through changes. A vague judgment is a judgment that it is blind to identity through changes. How do we perform synthetic reasoning? Following Peirce’s semiotic study of elements of Gamma Graphs as the sheet of assertion and the line of identity, the paper will first individuate the semiotic characteristics necessary for the recognition of identity. These characteristics lead us to discover “complete gesture” as the tool that we use in our every-day reasoning in order to acquire new knowledge synthetically. “Complete gestures” are actions through which we carry and recognize significant meanings. This new paradigm should provide an improved account of common-sense knowledge as well as of particular creative and hypothetic stages of conception in both scientific and humanistic thought. (shrink)

How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and time, (...) such as symmetries and order, or on the generality of Herbrand's notion of ‘prototype proof’. (shrink)

Kurt Gödel’s Incompleteness theorem is well known in Mathematics/Logic/Philosophy circles. Gödel was able to find a way for any given P (UTM), (read as, “P of UTM” for “Program of Universal Truth Machine”), actually to write down a complicated polynomial that has a solution iff (=if and only if), G is true, where G stands for a Gödel-sentence. So, if G’s truth is a necessary condition for the truth of a given polynomial, then P (UTM) has to answer first that (...) G is true in order to secure the truth of the said polynomial. But, interestingly, P (UTM) could never answer that G was true. This necessarily implies that there is at least one truth a P (UTM), however large it may be, cannot speak out. Daya Krishna and Karl Potter’s controversy regarding the construal of India’s Philosophies dates back to the time of Potter’s publication of “Presuppositions of India’s Philosophies” (1963, Englewood Cliffs Prentice-Hall Inc.) In attacking many of India’s philosophies, Daya Krishna appears to have unwittingly touched a crucial point: how can there be the knowledge of a ‘non-cognitive’ mokṣa? [‘mokṣa’ is the final state of existence of an individual away from Social Contract]—See this author’s Indian Social Contract and its Dissolution (2008) mokṣa does not permit the knowledge of one’s own self in the ordinary way with threefold distinction, i.e., subject–knowledge-object or knower–knowledge–known. But what is important is to demonstrate whether such ‘knowledge’ of non-cognitive mokṣa state can be logically shown, in a language, to be possible to attain, and that there is no contradiction involved in such demonstration, because, no one can possibly express the ‘experience-itself’ in language. Hence, if such ‘knowledge’ can be shown to be logically not impossible in language, then, not only Daya Krishna’s arguments against ‘non-cognitive mokṣa’ get refuted but also it would show the possibility of achieving ‘completeness’ in its truest sense, as opposed to Gödel’s ‘Incompleteness’. In such circumstances, man would himself become a Universal Truth Machine. This is because the final state of mokṣa is construed as the state of complete knowledge in Advaita. This possibility of ‘completeness’ is set in this paper in the backdrop of Śrī Śaṅkarācārya’s Advaitic (Non-dualistic) claim involved in the mahāvākyas (extra-ordinary propositions). (Mahāvākyas that Śaṅkara refers to are basically taken from different Upaniṣads. For example, “Aham Brahmāsmi” is from Bṛhadāraṇyaka Upanisad, and “Tattvamasi” is from Chāndogya Upaniṣad. Śrī Śaṅkarācārya has written extensively. His main works include his Commentary on Brahma-Sūtras, on major Upaniṣads, and on ŚrīmadBhagavadGītā, called Bhāṣyas of them, respectively. Almost all these works are available in English translation published by Advaita Ashrama, 5 Dehi Entally Road, Calcutta, 700014.) On the other hand, the ‘Incompleteness’ of Gödel is due to the intervening G-sentence, which has an adverse self-referential element. Gödel’s incompleteness theorem in its mathematical form with an elaborate introduction by R.W. Braithwaite can be found in Meltzer (Kurt Gödel: on formally undecidable propositions of principia mathematica and related systems. Oliver & Boyd, Edinburgh, 1962). The present author believes first that semantic content cannot be substituted by any amount of arithmoquining, (Arithmoquining or arithmatization means, as Braithwaite says,—“Gödel’s novel metamathematical method is that of attaching numbers to the signs, to the series of signs (formulae) and to the series of series of signs (“proof-schemata”) which occur in his formal system…Gödel invented what might be called co-ordinate metamathematics…”) Meltzer (1962 p. 7). In Antone (2006) it is said “The problem is that he (Gödel) tries to replace an abstract version of the number (which can exist) with the concept of a real number version of that abstract notion. We can state the abstraction of what the number needs to be, [the arithmoquining of a number cannot be a proof-pair and an arithmoquine] but that is a concept that cannot be turned into a specific number, because by definition no such number can exist.”.), especially so where first-hand personal experience is called for. Therefore, what ultimately rules is the semanticity as in a first-hand experience. Similar points are voiced, albeit implicitly, in Antone (Who understands Gödel’s incompleteness theorem, 2006). (“…it is so important to understand that Gödel’s theorem only is true with respect to formal systems—which is the exact opposite of the analogous UTM (Antone (2006) webpage 2. And galatomic says in the same discussion chain that “saying” that it ((is)) only true for formal systems is more significant… We only know the world through “formal” categories of understanding… If the world as it is in itself has no incompleteness problem, which I am sure is true, it does not mean much, because that is not the world of time and space that we experience. So it is more significant that formal systems are incomplete than the inexperiencable ‘World in Itself’ has no such problem.—galatomic”) Antone (2006) webpage 2. Nevertheless galatomic certainly, but unwittingly succeeds in highlighting the possibility of experiencing the ‘completeness’ Second, even if any formal system including the system of Advaita of Śaṅkara is to be subsumed or interpreted under Gödel’s theorem, or Tarski’s semantic unprovability theses, the ultimate appeal would lie to the point of human involvement in realizing completeness since any formal system is ‘Incomplete’ always by its very nature as ‘objectual’, and fails to comprehend the ‘subject’ within its fold. (shrink)

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.

It is well known that a degree-of-belief function P is coherent if and only if it satisfies the probability calculus. In this paper, we show that the notion of coherence can be extended to higher order probabilities such as P(P(h)=p)=q, and that a higher order degree-of-belief function P is coherent if and only if it satisfies the probability calculus plus the following axiom: P(h)=p iff P(P(h)=p)=1. Also, a number of lemmata which extend an incomplete probability function to a complete (...) one are established. (shrink)

The paper examines Husserl’s interactions with logicians in the 1930s in order to assess Husserl’s awareness of Gödel’s incompleteness theorems. While there is no mention about the results in Husserl’s known exchanges with Hilbert, Weyl, or Zermelo, the most likely source about them for Husserl is Felix Kaufmann (1895–1949). Husserl’s interactions with Kaufmann show that Husserl may have learned about the results from him, but not necessarily so. Ultimately Husserl’s reading marks on Friedrich Waismann’s Einführung in das mathematische Denken: die (...) Begriffsbildung der modernen Mathematik, 1936, show that he knew about them before his death in 1938. (shrink)

Let S be a deductive system such that S-derivability (⊦s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and ⊦s, it follows constructively that the K-completeness of ⊦s implies MP(S), a form of Markov's Principle. If ⊦s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if ⊦s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when ⊦s is many-one (...) complete, MP(S) implies the usual Markov's Principle MP. An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implies MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP. These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Godel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article. (shrink)

This paper investigates the claim that the second-order consequence relation is intractable because of the incompleteness result for SOL. The opponents’ claim is that SOL cannot be proper logic since it does not have a complete deductive system. I argue that the lack of a completeness theorem, despite being an interesting result, cannot be held against the status of SOL as a proper logic.

This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are (...) neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Gödel used is applicable in a wide variety of circumstances. Gödel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations. (shrink)

This book offers a thorough reanalysis of Charles Darwin's Origin of Species, which for many people represents the work that alone gave rise to evolutionism. Of course, scholars today know better than that. Yet, few resist the temptation of turning to the Origin in order to support it or reject it in light of their own work. Apparently, Darwin fills the mythical role of a founding figure that must either be invoked or repudiated. The book is an invitation to move (...) beyond what is currently expected of Darwin's magnum opus. Once the rhetorical varnish of Darwin's discourses is removed, one discovers a work of remarkably indecisive conclusions. The book comprises two main theses: The Origin of Species never remotely achieved the theoretical unity to which it is often credited. Rather, Darwin was overwhelmed by a host of phenomena that could not fit into his narrow conceptual framework. In the Origin of Species, Darwin failed at completing the full conversion to evolutionism. Carrying many ill-designed intellectual tools of the 17th and 18th centuries, Darwin merely promoted a special brand of evolutionism, one that prevented him from taking the decisive steps toward an open and modern evolutionism. It makes an interesting read for biologists, historians and philosophers alike. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like (...) FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

The history of building automated theorem provers for higher-order logic is almost as old as the field of deduction systems itself. The first successful attempts to mechanize and implement higher-order logic were those of Huet [13] and Jensen and Pietrzykowski [17]. They combine the resolution principle for higher-order logic (first studied in [1]) with higher-order unification. The unification problem in typed λ-calculi is much more complex than that for first-order terms, since it has to take the theory of αβη-equality into (...) account. As a consequence, the higher-order unification problem is undecidable and sets of solutions need not even always have most general elements that represent them. Thus the mentioned calculi for higher-order logic have take special measures to circumvent the problems posed by the theoretical complexity of higher-order unification. In this paper, we will exemplify the methods and proof- and model-theoretic tools needed for extending first-order automated theorem proving to higherorder logic. For the sake of simplicity take the tableau method as a basis (for a general introduction to first-order tableaux see part I.1) and discuss the higherorder tableau calculi HT and HTE first presented in [19]. The methods in this paper also apply to higher-order resolution calculi [1, 13, 6] or the higher-order matings method of Peter [3], which extend their first-order counterparts in much the same way. Since higher-order calculi cannot be complete for the standard semantics by Gödel’s incompleteness theorem [11], only the weaker notion of Henkin models [12] leads to a meaningful notion of completeness in higher-order logic. It turns out that the calculi in [1, 13, 3, 19] are not Henkin-complete, since they fail to capture the extensionality principles of higher-order logic. We will characterize the deductive power of our calculus HT (which is roughly equivalent to these calculi) by the semantics of functional Σ-models. To arrive at a calculus that is complete with respect to Henkin models, we build on ideas from [6] and augment HT with tableau construction rules that use the extensionality principles in a goal-oriented way.. (shrink)

Within this paper we consider a model of Nash bargaining with incomplete information. In particular, we focus on fee games, which are a natural generalization of side payment games in the context of incomplete information. For a specific class of fee games we provide two axiomatic approaches in order to establish the Expected Contract Value, which is a version of the Nash bargaining solution.

Assume that the problem P0 is not solvable in polynomial time. Let T be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{ConT} as the minimal extension of T proving for some algorithm that it decides P0 as fast as any algorithm B with the property that T proves that B decides P0. Here, ConT claims the consistency of T. As a byproduct, we obtain a version of Gödelʼs Second Incompleteness Theorem. Moreover, we characterize problems (...) with an optimal algorithm in terms of arithmetical theories. (shrink)