In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how to deduce Gödel's results (...) by alternative means, ones arguably more elementary than those of Kreisel. We avoid DC and Herbrand's Theorem by marshalling simple facts about negative translations and Markov's Rule. Second, the theorems of Gödel and Kreisel are commonly interpreted as demonstrating the unprovability of completeness for HPL, if means of proof are confined within strictly intuitionistic metamathematics. In the closing section, we assay some doubts about such interpretations. (shrink)

We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with atruth value defined classically, (...) and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic. (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.

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

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.

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)

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

Digital physics claims that the entire universe is, at the very bottom, made out of bits; as a result, all physical processes are intrinsically computational. For that reason, many digital physicists go further and affirm that the universe is indeed a giant computer. The aim of this article is to make explicit the ontological assumptions underlying such a view. Our main concern is to clarify what kind of properties the universe must instantiate in order to perform computations. We analyse the (...) logical form of the two models of computation traditionally adopted in digital physics, namely, cellular automata and Turing machines. These models are computationally equivalent, but we show that they support different ontological commitments about the fundamental properties of the universe. In fact, cellular automata are compatible with a rather traditional form of physicalism, whereas Turing machines support a dualistic ontology, which could be understood as a realism about the laws of nature or, alternatively, as a kind of panpsychism. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)

Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to Frege/Tarski, of a certain sort of semantics, and the seeming absence of intensional contexts from mathematical discourse. The philosophical import of these facts is at best murky. Extensionalists will emphasize the success and clarity of the model theoretic semantics; others will emphasize the relative poverty of the mathematical idiom; still others will question the aptness (...) of the standard extensional semantics for mathematics. In this paper I investigate some implications of the Gödel Second Incompleteness Theorem for these positions. I argue that the realm of mathematics, proof theory in particular, has been a breeding ground for intensionality and that satisfactory intensional semantic theories are implicit in certain rigorous technical accounts. (shrink)

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

Ranging from Alan Turing’s seminal 1936 paper to the latest work on Kolmogorov complexity and linear logic, this comprehensive new work clarifies the relationship between computability on the one hand and constructivity on the other. The authors argue that even though constructivists have largely shed Brouwer’s solipsistic attitude to logic, there remain points of disagreement to this day. Focusing on the growing pains computability experienced as it was forced to address the demands of rapidly expanding applications, the content maps the (...) developments following Turing’s ground-breaking linkage of computation and the machine, the resulting birth of complexity theory, the innovations of Kolmogorov complexity and resolving the dissonances between proof theoretical semantics and canonical proof feasibility. Finally, it explores one of the most fundamental questions concerning the interface between constructivity and computability: whether the theory of recursive functions is needed for a rigorous development of constructive mathematics. This volume contributes to the unity of science by overcoming disunities rather than offering an overarching framework. It posits that computability’s adoption of a classical, ontological point of view kept these imperatives separated. In studying the relationship between the two, it is a vital step forward in overcoming the disagreements and misunderstandings which stand in the way of a unifying view of logic. (shrink)

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term (...) itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Semantics connected to some information based metaphor are well-known in logic literature: a paradigmatic example is Kripke semantic for Intuitionistic Logic. In this paper we start from the concrete problem of providing suitable logic-algebraic models for the calculus of attribute dependencies in Formal Contexts with information gaps and we obtain an intuitive model based on the notion of passage of information showing that Kleene algebras, semi-simple Nelson algebras, three-valued ukasiewicz algebras and Post algebras of order three are, in a sense, (...) naturally and directly connected to partially defined information systems. In this way wecan provide for these logic-algebraic structures a raison dêetre different from the original motivations concerning, for instance, computability theory. (shrink)

The epsilon substitution method was proposed by D. Hilbert as a tool for consistency proofs. A version for first order predicate logic had been described and proved to terminate in the monograph “Grundlagen der Mathematik”. As far as the author knows, there have been no attempts to extend this approach to the second order case. We discuss possible directions for and obstacles to such extensions.

Embora algumas posições filosóficas de Gödel sejam bem conhecidas, como o platonismo, a sua teoria do conhecimento é, em comparação, menos divulgada. A partir do «Problema da Evidência» de Hilbert-Bernays, I, pg. 20 seq., apresento a seguir os traços essenciais da posição de Gödel sobre a caracterização epistemológica da evidência finitista, com especial relevo para a história dos conceitos utilizados.

We propose a rudimentary formal framework for reasoning about recursion equations over inductively generated data. Our formalism admits all equational programs , and yet singles out none. While being simple, this framework has numerous extensions and applications. Here we lay out the basic concepts and definitions; show that the deductive power of our formalism is similar to that of Peano's Arithmetic; prove a strong normalization theorem; and exhibit a mapping from natural deduction derivations to an applied λ -calculus, à la (...) Curry–Howard, which provides a transparent proof of Gödel's result that the provably recursive functions of PA are definable by primitive recursion in finite types. Also of interest is the extent, uncommon among similar constructions, to which our mapping is oblivious to first order terms, including first order quantification and equality. Additional applications and extensions of the method will be presented in sequel papers. In particular, the methodology introduced here lends itself to systematic links between natural sub-formalisms and computational complexity classes. Since these sub-formalisms are largely transparent, allowing one to reason formally in the full system while leaving to automatic software tools the task of verifying that restrictions are complied with, the framework is particularly attractive for the development of Feasible Mathematics. (shrink)

A substitutional account of logical validity for formal first‐order languages is developed and defended against competing accounts such as the model‐theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its substitution (...) instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is taken to be a primitive notion and axiomatized.For every set‐theoretic model in the sense of model theory there exists a corresponding substitutional interpretation in a sense to be specified. Conversely, however, there are substitutional interpretations – in particular the ‘intended’ interpretation – that lack a model‐theoretic counterpart. The substitutional definition of logical validity overcomes the weaknesses of more restrictive accounts of substitutional validity; unlike model‐theoretic logical consequence, the substitutional notion is trivially and provably truth preserving. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of intuitive validity. (shrink)

Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. (...) It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched. (shrink)

This paper examines the question of the extensional correctness of Tarskian definitions of logical truth and logical consequence. I identify a few different informal properties which are necessary for a sentence to be an informal logical truth and look at whether they are necessary properties of Tarskian logical truths. I examine arguments by John Etchemendy and Vann McGee to the effect that some of those properties are not necessary properties of some Tarskian logical truths, and find them unconvincing. I stress (...) the point that since the hypothesis that Tarski's definitions are extensionally correct is deeply entrenched, the burden of proof is still on the shoulders of Tarski's critics, who have not lifted the burden. (shrink)

I attribute an 'intensional reading' of the second incompleteness theorem to its author, Kurt G del. My argument builds partially on an analysis of intensional and extensional conceptions of meta-mathematics and partially on the context in which G del drew two familiar inferences from his theorem. Those inferences, and in particular the way that they appear in G del's writing, are so dubious on the extensional conception that one must doubt that G del could have understood his theorem extensionally. However, (...) on the intensional conception, the inferences are straightforward. For that reason I conclude that G del had an intensional understanding of his theorem. Since this conclusion is in tension with the generally accepted view of G del's understanding of mathematical truth, I explain how to reconcile that view with the intensional reading of the theorem that I attribute to G del. The result is a more detailed account of G del's conception of meta-mathematics than is currently available. (shrink)

There are two different ways to introduce the notion of truthin constructive mathematics. The first one is to use a Tarskian definition of truth in aconstructive (meta)language. According to some authors, (Kreisel, van Dalen, Troelstra ... ),this definition is entirely similar to the Tarskian definition of classical truth (thesis A).The second one, due essentially to Heyting and Kolmogorov, and known as theBrouwer–Heyting–Kolmogorov interpretation, is to explain informally what it means fora mathematical proposition to be constructively proved. According to other authors (...) (Martin-Löfand Shapiro), this interpretation and the Tarskian definition of truth amount to thesame (thesis B). My aim in this paper is to show that thesis A is only reasonable, that thesis Bis false and to answer the following question: what is defined by the Tarskian definition ofconstructive truth? (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

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)

The recent debate on hypercomputation has raised new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics.We propose here the idea of “effective physical process” as the essentially physical notion of computation. By using the Bohm and Hiley active information concept we analyze the differences between the standard form (quantum gates) and the non-standard one (adiabatic and morphogenetic) of Quantum Computing, and we point out how its Super-Turing potentialities derive from an incomputable information (...) source in accordance with Bell’s constraints. On condition that we give up the formal concept of “universality”, the possibility to realize quantum oracles is reachable. In this way computation is led back to the logic of physical world. (shrink)

In this paper I discuss some issues concerning the semantics of set theory. The set-theoretical nature of the semantics of axiomatic set theory raises a problem of circularity. It is well-known that when we adopt the model-theoretic point of view in the study of mathematical theories we decide to consider primarily structures in their relationship with languages. But for the fundamental structure adopted in a set-theoretic setting, namely the collection of all sets, together with the relation of membership, we would (...) have in that case a structure whose universe would be a set, and at the same time it would have the power of the collection of all sets. This is precisely the origin of the fundamental problem in the semantics of set theory. Apparently, the recourse to an intuitive or pre-theoretic semantics in terms of the iterative conception of sets is unavoidable. However, this strategy doesn´t seem to solve some basic philosophical topics involved and linked to the central semantic problem of set theory, namely, how to understand quantification over the totality of sets. Finally, I put forward some meta-philosophical considerations concerning the semantic problem I deal with. (shrink)

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

A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach (...) may be used under (iii)for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way. (shrink)