Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

My dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientific by further developing recent interpretive efforts to explain Carnap’s mature philosophical work as a form of engineering. It does this by looking in detail at his philosophical practice in his most sustained mature project, his work on pure and applied inductive logic. I, first, specify the sort of engineering Carnap is engaged in as involving an engineering design problem and then draw out the complications of design (...) problems from current work in history of engineering and technology studies. I then model Carnap’s practice based on those lessons and uncover ways in which Carnap’s technical work in inductive logic takes some of these lessons on board. This shows ways in which Carnap’s philosophical project subtly changes right through his late work on induction, providing an important corrective to interpretations that ignore the work on inductive logic. Specifically, I show that paying attention to the historical details of Carnap’s attempt to apply his work in inductive logic to decision theory and theoretical statistics in the 1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophical point of the practical/theoretical distinction in his late work, offering thus a new interpretation of Carnap’s technical work within the broader context of philosophy of science and analytical philosophy in general. (shrink)

Rudolf Carnap's 1930s philosophy of logic, including his adherence to the principle of tolerance, is discussed. What theses did Carnap commit himself to, exactly? I argue that while Carnap did commit himself to a certain multitude thesis—there are different logics of different languages, and the choice between these languages is merely a matter of expediency—there is no evidence that he rejected a language-transcendent notion of fact, contrary to what Warren Goldfarb and Thomas Ricketts have prominently argued. (In fact, it is (...) obscure just what Goldfarb and Ricketts claim about Carnap). Toward the end I critically discuss Michael Friedman's suggestion that Carnap believed in a relative a priori. (shrink)

This paper considers the question whether the notion of the relative apriori, central to Michael Friedman’s transcendentalist programme for philosophy of science, is available also to philosophers who reject appeals to a synthetic a priori. After tracing the rediscovery of the relative a priori and delineating its potential, the question is considered whether Friedman’s arguments against Quinean naturalism and Carnap’s attenuated logicism tell against a conception of philosophy as scientific metatheory that combines logical and empirical inquiries. Finding an opening here (...) it is argued that this conception can also help itself to the notion of a relative a priori which, it is agreed, is central to historically informed theorising about science. (shrink)

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.

A survey of more philosophical applications of Gödel's incompleteness results.

Some versions of empiricism have been accused of being neither empirically confirmable nor analytically true and therefore meaningless or unknowable by their own lights. Carnap, and more recently van Fraassen, have responded to this objection by construing empiricism as a stance containing non-cognitive attitudes. The resulting stance empiricism is not subject to the norms of knowledge, and so does not self-defeat as per the objection. In response to this proposal, several philosophers have argued that if empiricism is a stance, then (...) there can be no distinctively epistemic reasons in favor of adopting it, but only prudential or moral reasons. I defend stance empiricism against this objection by showing that stance empiricism furthers many plausibly epistemic goals, such as false belief avoidance, wisdom, and justification. I respond to three objections to my argument: that I assume a conception of epistemic reason that leads to problematic tradeoffs, that to have epistemic reason is just to be epistemically justified, and that my premise that experience is the only source of information has no empirical content. (shrink)

The classic “self-undermining objection” to the verificationist criterion of meaning states that the criterion does not meet its own standard: since verificationism is not empirically confirmable, analytic, or contradictory, verificationism implies its own meaninglessness. This essay reconstructs and motivates Carnap’s response to this objection. The interpretation presented is contrasted with those of Putnam and Ricketts. I argue that Carnap’s basic move in response to the self-undermining objection is to construe his verificationism as an analytic definition of “meaningfulness” that is meaningful (...) by its own lights. I then discuss possible motivations for this definition. I argue, against Reichenbach, Ayer, and Hempel, that it is not an analysis of the everyday concept of meaning. Instead, I claim, the definition is motivated _pragmatically_: verificationism replaces the ordinary conception of meaning with one that purports to capture all and only the expressions that are pragmatically useful to the scientist. Finally, I consider whether pragmatism faces a self-undermining objection to of its own. I argue that pragmatism is a preference concerning formal languages, and that, since preferences need not apply to themselves, pragmatism is not self- undermining. (shrink)

The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...) objection to Carnap’s version of the invariance conception has been repeated several times in the literature. It is based on Gödel incompleteness, which is puzzling, since Carnap’s Syntax is otherwise quite careful to take account of Gödel. We show here that, in fact, the objection is invalid and is based on a confusion about determinacy. Sorting this out is worthwhile not only for the purpose of better understanding Carnap’s thinking in the Syntax, though. The invariance conception is also related to recent work in the philosophy of logic regarding “logicality”—the characterization of logical concepts—following a proposal of Tarski’s. It is even connected to some very recent developments in the foundations of mathematics. (shrink)

Throughout the late nineteenth and early twentieth centuries, philosophers of various sorts, including Helmholtz, Avenarius, Husserl, Russell, Carnap, Neurath, and Heidegger, were united in promulgating a new, “scientific” philosophy. This article documents some of the varieties of scientific philosophy and argues that the history of scientific philosophy is crucial to the development of analytic philosophy and the division between analytic and continental philosophy. Scientific philosophy defined itself via criticisms of old-fashioned systematic metaphysics and, in the twentieth century, of Lebensphilosophie. It (...) offered a modernist vision of philosophy participating in a progressive, problem-solving, piecemeal, and collaborative scientific ethos. The article argues that the rise of scientific philosophy indicates a change of the conception of science as well as philosophy in the late nineteenth century and notes some tensions in the accounts of science offered by scientific philosophers. The article offers some preliminary lessons for the interpretation of logical empiricism and phenomenology as episodes within a larger history of scientific philosophy. (shrink)

Criteria of empirical significance are supposed to state conditions under which reference to an unobservable object or property is “empirically meaningful”. The intended kind of empirical meaningfulness should be necessary for admissibility into the selective contexts of scientific inquiry. I defend Justus’s recent argument that the reasons generally given for rejecting the project of defining a significance criterion are unpersuasive. However, as I show, this project remains wedded to an overly narrow conception of its subject matter. Even the most cutting (...) edge significance criteria identify empirical significance with predictive power, and thereby rule out vocabulary with legitimate scientific functions. In a nutshell, the problem is that there are terms that reduce the computational burden of extracting predictions from theory, and that may therefore be scientifically useful, but that do not produce any additional predictions, and so are ruled scientifically inadmissibility by existing significance criteria. I spell out this objection by specifying terms of this kind that are ruled inadmissible by Creath’s and Schurz’s criteria. Having objected in this way to extant criteria, and to the equation of empirical significance with predictive power in general, I discuss an approach to defining empirical significance that is capable of avoiding my objection and, more ambitiously, that may break the cycle of “punctures and patches” that has plagued the project from the beginning: I gloss Goldfarb and Ricketts’s idea of “case-by-case” delineations of empirically significant terms as the provision of special rather than general explications of the informal concept of empirical significance. (shrink)

On a widespread reading, the Carnap-Quine debate about ontology concerns the objectivity and non-triviality of ontological claims. I argue that this view mischaracterizes Carnap’s aims in “Empiricism, Semantics, and Ontology” : Carnap’s fundamental goal is to free up decisions about scientific language from constraints deriving from ontological doctrine. The contention, based on his internal/external distinction, that ontological claims are either meaningless or trivial was Carnap’s means to achieving this more fundamental goal. Setting the record straight on this point brings out (...) three important and often overlooked features of Carnap’s views on ontology. First, the target of Carnap’s critique in ESO is not Quine’s mature views on ontology, as laid out in “On What There Is”. Rather, Carnap is responding to arguments for nominalism that were given by Tarski, Goodman, and Quine in the 1940s. Second, a more general rejection of conservatism in theory choice is essential to Carnap’s aims and is implicit in his fundamental views on language. Third, even if it turns out to be tenable, the i/e distinction is not adequate for Carnap’s aim in ESO. Drawing on his basic conception of scientific language, I will suggest an alternative approach on his behalf. (shrink)

Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture. The author is grateful to Michael Friedman for valuable comments. Part of the research (...) towards this paper was carried out while the author was a Visiting Fellow at the Center for Philosophy of Science at the University of Pittsburgh. The paper was presented to the Center's Fellowship Reunion Conference in Athens in 1992. It was committed for publication in the Proceedings of that conference, but those Proceedings never appeared. By the time it became evident that they would never appear, both the hard copy and the source file had been mislaid. The hard copy re-surfaced in 2007. The literature on this topic since 1992 appears to leave some space for the ideas and arguments presented here. Although the paper has been updated in light of the more recent literature, its basic thesis, presented in 1992, remains the same. Only 3 is new, questioning a basic presumption made by more recent commentators in their presentation of Gödel's criticism of Carnap in his Gibbs Lecture. For helpful comments on the current version, the author is indebted to Robert Kraut, Stewart Shapiro, and Adam Podlaskowski. CiteULike Connotea Del.icio.us What's this? (shrink)

In this paper, the connection between logicism and the principle of tolerance in Carnap’s philosophy of logic and mathematics is to be presented in terms of the history of its development. Such development is conditioned by two lines of criticism to Carnap’s attempt to combine Logicism and Conventionalism, the first of which comes from Gödel, the second from Alfred Tarski. The presentation will take place in three steps. First, the Logicism of Carnap before the publication of The Logical Syntax of (...) Language will be analyzed in its main features. Second, the relationship between Logicism and the Principle of Tolerance at the time of the publication of Logical Syntax, and thirdly, in the so-called semantical phase of Carnap’s thought will be examined. On the basis of the two preceding points, we will then critically analyze two interpretations of the possibility to maintain a philosophical position like that of Carnap nowadays. (shrink)

This dissertation makes two primary contributions. The first three chapters develop an interpretation of Carnap's Meta-Philosophical Program which places stress upon his methodological analysis of the sciences over and above the Principle of Tolerance. Most importantly, I suggest, is that Carnap sees philosophy as contiguous with science—as a part of the scientific enterprise—so utilizing the very same methods and subject to the same limitations. I argue that the methodological reforms he suggests for philosophy amount to philosophy as the explication of (...) the concepts of science through the construction and use of suitably robust meta-logical languages. My primary interpretive claim is that Carnap's understanding of logic and mathematics as a set of formal auxiliaries is premised upon this prior analysis of the character of logico-mathematical knowledge, his understanding of its role in the language of science, and the methods used by practicing mathematicians. Thus the Principle of Tolerance, and so Carnap's logical pluralism, is licensed and justified by these methodological insights. This interpretation of Carnap's program contrasts with the popular Deflationary reading as proposed in Goldfarb & Ricketts. The leading idea they attribute to Carnap is a Logocentrism: That philosophical assertions are always made relative to some particular language, and that our choice of syntactical rules for a language are constitutive of its inferential structure and methods of possible justification. Consequently Tolerance is considered the foundation of Carnap's entire program. My third chapter argues that this reading makes Carnap's program philosophically inert, and I present significant evidence that such a reading is misguided. The final chapter attempts to extend the methodological ideals of Carnap's program to the analysis of the ongoing debate between category- and set-theoretic foundations for mathematics. Recent criticism of category theory as a foundation charges that it is neither autonomous from set theory, nor offers a suitable ontological grounding for mathematics. I argue that an analysis of concepts can be foundationally informative without requiring the construction of those concepts from first principles, and that ontological worries can be seen as methodologically unfruitful. (shrink)

We compare Carnap's and Kuhn's views on science. Although there are important differences between them, the similarities are striking. The basis for the latter is a pragmatically oriented semantic conventionalist picture of science, which suggests that the view that post-positivist philosophy of science constitutes a radical revolution which has no interesting affinities with logical positivism must be seriously mistaken.

This volumes aim is to provide an introduction to Carnaps book from a historical and philosophical perspective, each chapter focusing on one specific issue. The book will be of interest not only to Carnap scholars but to all those interested in the history of analytical philosophy.

In “Empiricism, Semantics, and Ontology” Carnap drew his famous distinction between ‘internal’ and ‘external’ questions of existence, pronouncing the former meaningful and the latter meaningless. In The Significance of Philosophical Scepticism, Barry Stroud understands Carnap to be applying the verification criterion of meaningfulness in order to refute Cartesian skepticism. I suggest that Stroud misrepresents both Carnap’s aim and method. Carnap was responding to critics who suggested that his willingness to quantify over abstract entities in his work in semantics violated his (...) commitment to empiricism. He rejected that criticism as presupposing a super-scientific standpoint from which constraints on the admissible domain of entities of science could be delivered. Carnap wanted to insulate science from the imposition of frrst-philosophical metaphysical prejudice, not to defuse scepticism by appeal to verificationism. (shrink)

del's appeal to mathematical intuition to ground our grasp of the axioms of set theory, is notorious. I extract from his writings an account of this form of intuition which distinguishes it from the metaphorical platonism of which Gödel is sometimes accused and brings out the similarities between Gödel's views and Dummett's.

The waning popularity of logical empiricism and the supposed discovery of insurmountable technical difficulties led most philosophers to abandon the project to formulate a formal criterion of empirical significance. Such a criterion would delineate claims that observation can confirm or disconfirm from those it cannot. Although early criteria were clearly inadequate, criticisms made of later, more sophisticated criteria were often indefensible or easily answered. Most importantly, Carnap’s last criterion was seriously misinterpreted and an amended version of it remains tenable.

Michael Friedman has recently argued that Carnap'sLogical Syntax of Language is fundamentally flawed in a way that reveals the ultimate failure of logical positivism. Friedman's argument depends crucially on two claims: (1) that Carnap was committed to the view that there is a universal metalanguage and (2) that given what Carnap wanted from a metalanguage, in particular given that he wanted a definition of analytic for an object language, he was in fact committed to a hierarchy of stronger and stronger (...) metalanguages. We argue that neither of these claims need be accepted. We show that there is no textual evidence for (1) and that if metalanguages are to be used for merely descriptive and not also justificatory purposes, Carnap does not need to define analyticity sufficiently for proving consistency, and so could have given a definition that does not entail a hierarchy of metalanguages. (shrink)

We discuss in this paper the question of the scope of the principle of tolerance about languages promoted in Carnap's The Logical Syntax of Language and the nature of the analogy between it and the rudimentary conventionalism purportedly exhibited in the work of Poincaré and Hilbert. We take it more or less for granted that Poincaré and Hilbert do argue for conventionalism. We begin by sketching Coffa's historical account, which suggests that tolerance be interpreted as a conventionalism that allows us (...) complete freedom to select whatever language we wish—an interpretation that generalizes the conventionalism promoted by Poincaré and Hilbert which allows us complete freedom to select whatever axiom system we wish for geometry. We argue that such an interpretation saddles Carnap with a theory of meaning that has unhappy consequences, a theory we believe he did not hold. We suggest that the principle of linguistic tolerance in fact has a more limited scope; but within that scope the analogy between tolerance and geometric conventionalism is quite tight. (shrink)

The relationship between Carnap’s Logical Syntax of Language and Wittgenstein’s Tractatus Logico-Philosophicus has been interpreted in several ways during past decades. One of the interpretations has gained keen advocates among Carnap scholars. It was originally provoked by what Caranp said in LSL, and it consists of two parts. First, it indicates that in TLP the possibility of speaking about the logical form of a language within the same language had been foresworn by Wittgenstein, but Carnap proved him wrong by producing (...) a book written exactly in the manner which had been proscribed by Wittgenstein. This is the debate about the possibility of speaking about logical form. Second, Wittgenstein’s dogmatism with regard to the existence of a unique correct grammar at the foundation of the language has been contrasted with Carnap’s open-mindedness in conceiving a boundless ocean of possibilities for constructing logical systems. Interestingly enough, Wittgenstein rambled with rage in reaction to Carnap’s view about the LSL-TLP relationship. But unlike Carnap’s view, which led to a dominant interpretation of the relationship, Wittgenstein’s testimony about the case has been strangely ignored in the history of analytic philosophy. In this paper, I try to make an inquiry about the grounds for Wittgenstein’s dissatisfaction with the Carnapian reading of the LSL-TLP relationship. I will show that Wittgenstein was not totally unfair in his judgment, and that some salient aspects of LSL could be best understood in the light, or rather the gloom, of TLP, and bear a significant resemblance to it. This, however, does not need to diminish the logical and historical significance of LSL. (shrink)

Criteria of empirical significance are supposed to state conditions under which reference to an unobservable object or property is “empirically meaningful.” The intended kind of empirical meaningfulness should be necessary for admissibility into the selective contexts of scientific inquiry. I defend Justus’s recent argument that the reasons generally given for rejecting the project of defining a significance criterion are unpersuasive. However, as I show, this project remains wedded to an overly narrow conception of its subject matter. Even the most cutting (...) edge significance criteria identify empirical significance with predictive power, and thereby rule out vocabulary with legitimate scientific functions. In a nutshell, the problem is that there are terms that reduce the computational burden of extracting predictions from theory, and that may therefore be scientifically useful, but that do not add to the theory’s observational consequences, and so are ruled scientifically inadmissibility by existing significance criteria. I spell out this objection by specifying shortcut terms that are ruled inadmissible by Creath’s and Schurz’s criteria. Having objected in this way to extant criteria, and to the equation of empirical significance with predictive power in general, I discuss an approach to defining empirical significance that is capable of avoiding my objection and, more ambitiously, that may break the cycle of “punctures and patches” that has plagued the project from the beginning. I gloss Goldfarb and Ricketts’s idea of “case-by-case” delineations of empirically significant terms as the provision of special rather than general explications of the informal concept of empirical significance. (shrink)