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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 informationprocessing machine that computes beyond the universal Turing machine. To this end, I present a more (...) 







Synchronic norms of theory choice, a traditional concern in scientific methodology, restrict the theories one can choose in light of given information. Diachronic norms of theory change, as studied in belief revision, restrict how one should change one’s current beliefs in light of new information. Learning norms concern how best to arrive at true beliefs. In this paper, we undertake to forge some rigorous logical relations between the three topics. Concerning, we explicate inductive truth conduciveness in terms of optimally direct (...) 

Explaining the connection, if any, between simplicity and truth is among the deepest problems facing the philosophy of science, statistics, and machine learning. Say that an efficient truth finding method minimizes worst case costs en route to converging to the true answer to a theory choice problem. Let the costs considered include the number of times a false answer is selected, the number of times opinion is reversed, and the times at which the reversals occur. It is demonstrated that (1) (...) 

Hilary Putnam on Logic and Mathematics, edited by HellmanGeoffrey and CookRoy T. Cham: Springer, 2018. Pp. x + 274. 

O presente volume se trata de uma coletânea de artigos que reúne alguns dos trabalhos propostos para o evento “III International Colloquium of Analytic Epistemology and VII Conference of Social Epistemology”, realizado entre os dias 27 e 30 de Novembro de 2018, na Universidade Federal de Santa Maria. O “III International Colloquium of Analytic Epistemology and VII Conference of Social Epistemology” é um dos principais eventos de Epistemologia analítica da América Latina e reúne especialistas do Brasil e do exterior para (...) 

Ebbhinghaus, H., J. Flum, and W. Thomas. 1984. Mathematical Logic. New York, NY: SpringerVerlag. Forster, T. Typescript. The significance of Yablo’s paradox without selfreference. Available from http://www.dpmms.cam.ac.uk. Gold, M. 1965. Limiting recursion. Journal of Symbolic Logic 30: 28–47. Karp, C. 1964. Languages with Expressions of Infinite Length. Amsterdam. 





I propose that empirical procedures, like computational procedures, are justified in terms of truthfinding efficiency. I contrast the idea with more standard philosophies of science and illustrate it by deriving Ockham's razor from the aim of minimizing dramatic changes of opinion en route to the truth. 



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 Turingmachine 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 (...) 

The notion of an ideal reasoner has several uses in epistemology. Often, ideal reasoners are used as a parameter of (maximum) rationality for finite reasoners (e.g. humans). However, the notion of an ideal reasoner is normally construed in such a high degree of idealization (e.g. infinite/unbounded memory) that this use is unadvised. In this dissertation, I investigate the conditions under which an ideal reasoner may be used as a parameter of rationality for finite reasoners. In addition, I present and justify (...) 

This paper describes the cornerstones of a meansends approach to the philosophy of inductive inference. I begin with a fallibilist ideal of convergence to the truth in the long run, or in the 'limit of inquiry'. I determine which methods are optimal for attaining additional epistemic aims (notably fast and steady convergence to the truth). Meansends vindications of (a version of) Occam's Razor and the natural generalizations in a Goodmanian Riddle of Induction illustrate the power of this approach. The paper (...) 

Chaitin’s incompleteness result related to random reals and the halting probability has been advertised as the ultimate and the strongest possible version of the incompleteness and undecidability theorems. It is argued that such claims are exaggerations. 

Przedmowa Problematyka związana z zależnościami przyczynowymi, ich modelowaniem i odkrywa¬niem, po długiej nieobecności w filozofii i metodologii nauk, budzi współcześnie duże zainteresowanie. Wiąże się to przede wszystkim z dynamicznym rozwojem, zwłaszcza od lat 1990., technik obli¬czeniowych. Wypracowane w tym czasie sieci bayesowskie uznaje się za matematyczny język przyczynowości. Pozwalają one na daleko idącą auto¬matyzację wnioskowań, co jest także zachętą do podjęcia prób algorytmiza¬cji odkrywania przyczyn. Na potrzeby badań naukowych, które pozwalają na przeprowadzenie eksperymentu z randomizacją, standardowe metody ustalania zależności przyczynowych (...) 

Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's (...) 

A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a diﬀerence between intuitionistic and classical theories is isolated. 

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'' (...) 

Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is (...) 

Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mindchangebounded learnability is equivalent to finite View the MathML source2piecewise computability 2 denotes the difference of two View the MathML sourceΠ10 sets), errorbounded learnability is equivalent to finite View (...) 

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. 



Formal learning theory is the mathematical embodiment of a normative epistemology. It deals with the question of how an agent should use observations about her environment to arrive at correct and informative conclusions. Philosophers such as Putnam, Glymour and Kelly have developed learning theory as a normative framework for scientific reasoning and inductive inference. 

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. 



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 (...) 



The Hilbert–Bernays Theorem establishes that for any satisfiable firstorder quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for (...) 

TrailAndError machines have been proposed by Hintikka and Mutanen as an alternative formulation of the notion of computation. These machines extend the capabilities of the Turing machine and widen the theory of computation. 

Various proposals have suggested that an adequate explanatory theory should reduce the number or the cardinality of the set of logically independent claims that need be accepted in order to entail a body of data. A (and perhaps the only) wellformed proposal of this kind is William Kneale’s: an explanatory theory should be finitely axiomatizable but it’s set of logical consequences in the data language should not be finitely axiomatizable. Craig and Vaught showed that Kneale theories (almost) always exist for (...) 

Trial and error classifiers, corresponding to concepts which change their extensions over time, are introduced and briefly philosophically motivated. A fragment of the language of classical firstorder logic is given a new semantics, using \sequences of classical models, in order to interpret the basic predicates as classifiers of this kind. It turns out that we can use a natural deduction proof system which differs from classical logic only in the conditions for application of existential elimination. Soundness and completeness theorems are (...) 



We investigate and classify the notion of final derivability of two basic inconsistencyadaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...) 



What''s computation? The received answer is that computation is a computer at work, and a computer at work is that which can be modelled as a Turing machine at work. Unfortunately, as John Searle has recently argued, and as others have agreed, the received answer appears to imply that AI and Cog Sci are a royal waste of time. The argument here is alarmingly simple: AI and Cog Sci (of the Strong sort, anyway) are committed to the view that cognition (...) 



This paper pursues a thoroughgoing instrumentalist, or meansends, approach to the theory of inductive inference. I consider three epistemic aims: convergence to a correct theory, fast convergence to a correct theory and steady convergence to a correct theory (avoiding retractions). For each of these, two questions arise: (1) What is the structure of inductive problems in which these aims are feasible? (2) When feasible, what are the inference methods that attain them? Formal learning theory provides the tools for a complete (...) 

Occam’s razor directs us to adopt the simplest hypothesis consistent with the evidence. Learning theory provides a precise definition of the inductive simplicity of a hypothesis for a given learning problem. This definition specifies a learning method that implements an inductive version of Occam’s razor. As a case study, we apply Occam’s inductive razor to causal learning. We consider two causal learning problems: learning a causal graph structure that presents global causal connections among a set of domain variables, and learning (...) 



Synchronic norms of theory choice, a traditional concern in scientific methodology, restrict the theories one can choose in light of given information. Diachronic norms of theory change, as studied in belief revision, restrict how one should change one’s current beliefs in light of new information. Learning norms concern how best to arrive at true beliefs. In this paper, we undertake to forge some rigorous logical relations between the three topics. Concerning, we explicate inductive truth conduciveness in terms of optimally direct (...) 

We study the global properties of [Formula: see text], the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order of [Formula: see text] is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, [Formula: see text] is not a Σ1substructure of [Formula: see text]. 

We analyse the extent of possible computations following Hogarth ([2004]) conducted in Malament–Hogarth (MH) spacetimes, and Etesi and Németi ([2002]) in the special subclass containing rotating Kerr black holes. Hogarth ([1994]) had shown that any arithmetic statement could be resolved in a suitable MH spacetime. Etesi and Németi ([2002]) had shown that some relations on natural numbers that are neither universal nor couniversal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. (...) 

1. Characterizing randomness. Consider a physical process that, if suitably idealized, generates an indefinite sequence of independent random bits. One such process might be radioactive decay of a lump of uranium whose mass is kept at just the level needed to ensure that the probability is onehalf that no alpha particle is emitted in the nth microsecond of the experiment. Let us think of the bits as drawn from {0, 1} and denote the resulting sequence by x with coordinates x0, (...) 

We begin with the history of the discovery of computability in the 1930’s, the roles of Gödel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines . To whom did Gödel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine and Post’s development of it in [1944, §11], [1948], and finally KleenePost [1954] into its present form. A number of topics arose from Turing functionals including continuous functionals on (...) 

Following a science and ontology conference in Barbizon, France, Layla Raïd and Karim Belabas published an article on Peirce and Quine that focuses on truth considered as the convergence of opinions or theories. 2 The article is a productive collaboration between a philosopher and mathematician, identifying two problems that Quine poses: first, the use of numerical analogy in Peirce’s account of truth, and second, the uniqueness of the final opinion, which can presumably be defeated or undermined by arguments from underdetermination (...) 

