In this paper I argue that Turing’s responses to the mathematical objection are straightforward, despite recent claims to the contrary. I then go on to show that by understanding the importance of learning machines for Turing as related not to the mathematical objection, but to Lady Lovelace’s objection, we can better understand Turing’s response to Lady Lovelace’s objection. Finally, I argue that by understanding Turing’s responses to these objections more clearly, we discover a hitherto unrecognized, substantive thesis in his philosophical (...) thinking about the nature of mind. (shrink)

We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy's thesis.

Alan Turing anticipated many areas of current research incomputer and cognitive science. This article outlines his contributionsto Artificial Intelligence, connectionism, hypercomputation, andArtificial Life, and also describes Turing's pioneering role in thedevelopment of electronic stored-program digital computers. It locatesthe origins of Artificial Intelligence in postwar Britain. It examinesthe intellectual connections between the work of Turing and ofWittgenstein in respect of their views on cognition, on machineintelligence, and on the relation between provability and truth. Wecriticise widespread and influential misunderstandings of theChurch–Turing thesis (...) and of the halting theorem. We also explore theidea of hypercomputation, outlining a number of notional machines thatcompute the uncomputable. (shrink)

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary (...) to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

A survey of the field of hypercomputation, including discussion of a variety of objections.

Since the mid-twentieth century, the concept of the Turing machine has dominated thought about effective procedures. This paper presents an alternative to Turing's analysis; it unifies, refines, and extends my earlier work on this topic. I show that Turing machines cannot live up to their billing as paragons of effective procedure; at best, they may be said to provide us with mere procedure schemas. I argue that the concept of an effective procedure crucially depends upon distinguishing procedures as definite courses (...) of action(- types) from the particular courses of action(-tokens) that actually instantiate them and the causal processes and/or interpretations that ultimately make them effective. On my analysis, effectiveness is not just a matter of logical form; `content' matters. The analysis I provide has the advantage of applying to ordinary, everyday procedures such as recipes and methods, as well as the more refined procedures of mathematics and computer science. It also has the virtue of making better sense of the physical possibilities for hypercomputation than the received view and its extensions, e.g. Turing's o-machines, accelerating machines. (shrink)

Does what guides a pastry chef stand on par, from the standpoint of contemporary computer science, with what guides a supercomputer? Did Betty Crocker, when telling us how to bake a cake, provide an effective procedure, in the sense of `effective' used in computer science? According to Cleland, the answer in both cases is ``Yes''. One consequence of Cleland's affirmative answer is supposed to be that hypercomputation is, to use her phrase, ``theoretically viable''. Unfortunately, though we applaud Cleland's ``gadfly philosophizing'' (...) (as, in fact, seminal), we believe that unless such a modus operandi is married to formal philosophy, nothing conclusive will be produced (as evidenced by the problems plaguing Cleland's work that we uncover). Herein, we attempt to pull off not the complete marriage for hypercomputation, but perhaps at least the beginning of a courtship that others can subsequently help along. (shrink)

This paper offers an account of what it is for a physical system to be a computing mechanism—a system that performs computations. A computing mechanism is a mechanism whose function is to generate output strings from input strings and (possibly) internal states, in accordance with a general rule that applies to all relevant strings and depends on the input strings and (possibly) internal states for its application. This account is motivated by reasons endogenous to the philosophy of computing, namely, doing (...) justice to the practices of computer scientists and computability theorists. It is also an application of recent literature on mechanisms, because it assimilates computational explanation to mechanistic explanation. The account can be used to individuate computing mechanisms and the functions they compute and to taxonomize computing mechanisms based on their computing power. (shrink)

In the technical literature of computer science, the concept of an effective procedure is closely associated with the notion of an instruction that precisely specifies an action. Turing machine instructions are held up as providing paragons of instructions that "precisely describe" or "well define" the actions they prescribe. Numerical algorithms and computer programs are judged effective just insofar as they are thought to be translatable into Turing machine programs. Nontechnical procedures (e.g., recipes, methods) are summarily dismissed as ineffective on the (...) grounds that their instructions lack the requisite precision. But despite the pivotal role played by the notion of a precisely specified instruction in classifying procedures as effective and ineffective, little attention has been paid to the manner in which instructions "precisely specify" the actions they prescribe. It is the purpose of this paper to remedy this defect. The results are startling. The reputed exemplary precision of Turing machine instructions turns out to be a myth. Indeed, the most precise specifications of action are provided not by the procedures of theoretical computer science and mathematics (algorithms) but rather by the nontechnical procedures of everyday life. I close with a discussion of some of the rumifications of these conclusions for understanding and designing concrete computers and their programming languages. (shrink)

According to the conventional wisdom, Turing said that computing machines can be intelligent. I don't believe it. I think that what Turing really said was that computing machines –- computers limited to computing –- can only fake intelligence. If we want computers to become genuinelyintelligent, we will have to give them enough “initiative” to do more than compute. In this paper, I want to try to develop this idea. I want to explain how giving computers more ``initiative'' can allow them (...) to do more than compute. And I want to say why I believe that they will have to go beyond computation before they can become genuinely intelligent. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...) provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one’s implicit commitments, and sheds light on why this is so. We argue that the theory has significant epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent analyses of implicit commitment based on truth or epistemic notions. (shrink)

Wittgenstein’s “machines-as-symbols” are considered with respect to their historical sources and their symbolic and logical nature. Among these sources and precursors, along with Leonardo’s drawings of machines, there are illustrated “machine books”, a kind of book published in the period from the 16th to the 18th centuries which consist of pictures and descriptions of a variety of mechanical devices. Most probably, these books were one of Wittgenstein’s inspirations for his view of machines as components of language-games. The picture of homo (...) volans in Vrančić’s machine book possessed by Wittgenstein is taken as an example. In particular, homo volans is shown to contain patterns of logical laws and rules and to be abstractly interpretable as a logical symbol. A machine, taken as a symbol, is shown to be a precondition of a meaningful “overview” of a mechanical work that exceeds the limits of decidability, and to possess causal features if causality is understood teleologically and in a deeper sense of a “binding” life. (shrink)

Georg Kreisel suggested various ways out of the Gödel incompleteness theorems. His remarks on ways out were somewhat parenthetical, and suggestive. He did not develop them in subsequent papers. One aim of this paper is not to develop those remarks, but to show how the basic idea that they express can be used to reason about the Lucas-Penrose-Putnam arguments that human minds are not finitary computational machines. Another aim is to show how one of Putnam’s two anti-functionalist arguments avoids the (...) logical error in the Lucas-Penrose arguments, extends those arguments, but succumbs to an absurdity. A third aim is to provide a categorization of the Lucas-Penrose-Putnam anti-functionalist arguments. (shrink)

Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...) implement the idea of "infinite calculation" is the main subject. (shrink)

What Bar-On and Simmons call 'Conceptual Deflationism' is the thesis that truth is a 'thin' concept in the sense that it is not suited to play any explanatory role in our scientific theorizing. One obvious place it might play such a role is in semantics, so disquotationalists have been widely concerned to argued that 'compositional principles', such as -/- (C) A conjunction is true iff its conjuncts are true -/- are ultimately quite trivial and, more generally, that semantic theorists have (...) misconceived the relation between truth, meaning, and logic. This paper argues, to the contrary, that even such simple compositional principles as (C) have substantial content that cannot be captured by deflationist 'proofs' of them. The key thought is that (C) is supposed, among other things, to affirm the truth-functionality of conjunction and that disquotationalists cannot, ultimately, make sense of truth-functionality. -/- This paper is something of a companion to "The Logical Strength of Compositional Principles". (shrink)

Turing was an exceptional mathematician with a peculiar and fascinating personality and yet he remains largely unknown. In fact, he might be considered the father of the von Neumann architecture computer and the pioneer of Artificial Intelligence. And all thanks to his machines; both those that Church called “Turing machines” and the a-, c-, o-, unorganized- and p-machines, which gave rise to evolutionary computations and genetic programming as well as connectionism and learning. This paper looks at all of these and (...) at why he is such an often overlooked and misunderstood figure. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

A critical survey of some attempts to define ‘computer’, beginning with some informal ones, then critically evaluating those of three philosophers, and concluding with an examination of whether the brain and the universe are computers.

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both (...) Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake. (shrink)

This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...) remarks on _Lebensformen_ is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the _Entscheidungsproblem_“, as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of _Lebensform_ in the PI is then given in terms of a logical “regression” to _Lebensform_ as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?”. (shrink)

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I (...) argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. (shrink)

The behavior/structure methodological dichotomy as locus of scientific inquiry is closely related to the issue of modeling and theory change in scientific explanation. Given that the traditional tension between structure and behavior in scientific modeling is likely here to stay, considering the relevant precedents in the history of ideas could help us better understand this theoretical struggle. This better understanding might open up unforeseen possibilities and new instantiations, particularly in what concerns the proposed technological modification of the human condition. The (...) sequential structure of this paper is twofold. The contribution of three philosophers better known in the humanities than in the study of science proper are laid out. The key theoretical notions interweaving the whole narrative are those of mechanization, constructability and simulation. They shall provide the conceptual bridge between these classical thinkers and the following section. Here, a panoramic view of three significant experimental approaches in contemporary scientific research is displayed, suggesting that their undisclosed ontological premises have deep roots in the Western tradition of the humanities. This ontological lock between core humanist ideals and late research in biology and nanoscience is ultimately suggested as responsible for pervasively altering what is canonically understood as “human”. (shrink)

Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...) and principles of mathematical and transfinite induction as well as the status of the latter with respect to various foundational characterizations of number theory. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, (...) for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π1-induction schema. This result, however, cannot be much improved: by adapting a (...) theorem of D. Zambella and G. Mints we show that the logic of Π1-conservativity of primitive recursive arithmetic properly extends ILM.In the third part of the paper we give an ordinal classification of -consequences of the standard fragments of Peano arithmetic in terms of reflection principles. This is interesting in view of the general program of ordinal analysis of theories, which in the most standard cases classifies Π-classes of sentences. (shrink)

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...) T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (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)

Bynum (Putting information first: Luciano Floridi and the philosophy of information. NY: Wiley-Blackwell, 2010) identifies Floridi’s focus in the philosophy of information (PI) on entities both as data structures and as information objects. One suggestion for examining the association between the former and the latter stems from Floridi’s Herbert A. Simon Lecture in Computing and Philosophy given at Carnegie Mellon University in 2001, open problems in the PI: the transduction or transception, and how we gain knowledge about the world as (...) a complex, living, information environment. This paper addresses PI across a model of interoperating levels: perception (P)—intuition (N)—computation (C)—information (I), as factored by cognitive continuity (1), temporality (2), and constitution (3). How might we begin to characterize our experience of an abstract information object across such a matrix? Chudnoff’s rationalist distinctions between perception and intuition serve as a first rung of the ladder. Turing’s brief references to the utility of intuition, in an allied, rationalist-Cartesian sense, provide the next step up to computation. Floridi provides the final link from computation to information. (shrink)

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more (...) mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

Our limited a priori-reasoning skills open a gap between our finding a proposition conceivable and its metaphysical possibility. A prominent strategy for closing this gap is the postulation of ideal conceivers, who suffer from no such limitations. In this paper I argue that, under many, maybe all, plausible unpackings of the notion of ideal conceiver, it is false that ideal negative conceivability entails possibility.

This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...) a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. 2005 Elsevier Inc. All rights reserved.

We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

A myth has arisen concerning Turing's paper of 1936, namely that Turing set forth a fundamental principle concerning the limits of what can be computed by machine - a myth that has passed into cognitive science and the philosophy of mind, to wide and pernicious effect. This supposed principle, sometimes incorrectly termed the 'Church-Turing thesis', is the claim that the class of functions that can be computed by machines is identical to the class of functions that can be computed by (...) Turing machines. In point of fact Turing himself nowhere endorses, nor even states, this claim (nor does Church). I describe a number of notional machines, both analogue and digital, that can compute more than a universal Turing machine. These machines are exemplars of the class of _nonclassical_ computing machines. Nothing known at present rules out the possibility that machines in this class will one day be built, nor that the brain itself is such a machine. These theoretical considerations undercut a number of foundational arguments that are commonly rehearsed in cognitive science, and gesture towards a new class of cognitive models. (shrink)

In his PhD thesis (1938) Turing introduced what he described as 'a new kind of machine'. He called these 'O-machines'. The present paper employs Turing's concept against a number of currently fashionable positions in the philosophy of mind.

The aim of this paper is twofold: first, I provide a cluster of theories of truth in classical logic that is (internally) consistent with global reflection principles: the theories of positive truth (and falsity). After that, I analyse the _epistemic value_ of such theories. I do so employing the framework of cognitive projects introduced by Wright (Proc Aristot Soc 78:167–245, 2004), and employed—in the context of theories of truth—by Fischer et al. (Noûs 2019. https://doi.org/10.1111/nous.12292 ). In particular, I will argue (...) that theories of positive truth are _trustworthy_, analogously to the theories of full disquotational truth. Moreover, I argue that, for a given cognitive project, _if_ the acceptance of trustworthy theories is taken to be an _epistemic norm_ of cognitive project, then one has good reasons to accept theories of positive truth over other rival theories of truth in classical logic. On the other hand, the latter theories are deemed epistemically unacceptable. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains so far an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still (...) lacking. We provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one's implicit commitments, and sheds light on the reason why this is so. We argue that the theory has interesting epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent proposals for the analysis of implicit commitment based on truth or epistemic notions. (shrink)

Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper (...) is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)