We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces . We prove that any classical probability space has a causally closed extension; any finite classical probability space with positive rational probabilities on the atoms of the event algebra can be extended to a causally up-to-three-closed finite space; and any classical probability space can be extended to a space in which all correlations between events that are logically independent modulo measure zero event (...) have a countably infinite common-cause system. Collectively, these results show that it is surprisingly easy to find Reichenbach-style ‘explanations' for correlations, underlining doubts as to whether this approach can yield a philosophically relevant account of causality. 1 Introduction2 Basic Definitions and Results in the Literature3 Causal Completability the Easy Way: ‘Splitting the Atom’4 Causal Completability of Classical Probability Spaces: The General Case5 Infinite Statistical Common-Cause Systems for Arbitrary Pairs6 ConclusionAppendix A. (shrink)

Maps and territory suggest problems which have to do with the opening of pathways in some poorly explored domain.

The different versions of the Kochen-Specker Theorem show that quantum mechanics cannot be supplemented by hidden variables given two constraints. These results are generally interpreted as showing that QM is complete in the following way. A QM system S has only those values of observables for which the state yields probability 1. For all other probabilities S adopts a value during the measurement interaction. But this interpretation is fundamentally problematic. In fact it cannot yield a general and coherent interpretation of (...) the trace formula. I present an argument to this effect, discuss a possible objection and sketch implications. (shrink)

The old Bohr–Einstein debate about the completeness of quantum mechanics (QM) was held on an ontological ground. The completeness problem becomes more tractable, however, if it is preliminarily discussed from a semantic viewpoint. Indeed every physical theory adopts, explicitly or not, a truth theory for its observative language, in terms of which the notions of semantic objectivity and semantic completeness of the physical theory can be introduced and inquired. In particular, standard QM adopts a verificationist theory (...) of truth that implies its semantic nonobjectivity; moreover, we show in this paper that standard QM is semantically complete, which matches Bohr's thesis. On the other hand, one of the authors has provided a Semantic Realism (or SR) interpretation of QM that adopts a Tarskian theory of truth as correspondence for the observative language of QM (which was previously mantained to be impossible); according to this interpretation QM is semantically objective, yet incomplete, which matches EPR's thesis. Thus, standard QM and the SR interpretation of QM come to opposite conclusions. These can be reconciled within an integrationist perspective that interpretes non-Tarskian theories of truth as theories of metalinguistic concepts different from truth. (shrink)

Eukaryotic DNA replication initiates at multiple origins. In early fly and frog embryos, chromosomal replication is very rapid and initiates without sequence specificity. Despite this apparent randomness, the spacing of these numerous initiation sites must be sufficiently regular for the genome to be completely replicated on time. Studies in various eukaryotes have revealed that there is a strict temporal separation of origin “licensing” prior to S phase and origin activation during S phase. This may suggest that replicon size must be (...) already established at the licensing stage. However, recent experiments suggest that a large excess of potential origins are assembled along chromatin during licensing. Thus, a regular replicon size may result from the selection of origins during S phase. We review single molecule analyses of origin activation and other experiments addressing this issue and their general significance for eukaryotic DNA replication. BioEssays 25:116–125, 2003. © 2003 Wiley Periodicals, Inc. (shrink)

This paper investigates a problem related to quantifiers which has some analogies to that of propositional completeness I give a definition of quantifier in many-valued logics generalizing the cases which already occur in first order many- valued logics. Though other definitions are possible, this particular one, which I call distribution quantifiers, generalizes the classical quantifiers in a very natural way, and occurs in finite numbers in every m-valued logic. We then call the problem of quantificationa2 completeness (...) in m-valued logic the problem of characterizing which are the quantifiers in a given language which can generate all other quantifiers in this language, using the connectives, as is the case, for example, of the universal and exis- tential quantifiers in classical logic, using negation. We are interested, in particular, in those many-valued quantifiers which closely mimic the behavior of existential an universal quantifiers in generating all other quantifiers using negation: these I call perfect quantifiers, as defined below. The main result of this paper is the characterization of all perfect quantifiers in 3-valued logics, which are complete if the logic is functionally complete. As a byproduct, we obtain the same result for the classical logic, which we include mainly for motivation. (shrink)

As an important corollary of this interpretation of absolute knowledge, the dissertation concludes with the suggestion that Hegelian philosophy need not be regarded merely as an interesting curiosity in the history of ideas, but rather that it can serve as a vital and potentially rewarding source of fresh theoretical insights. ;Instead, the concrete completeness of speculative philosophy can only consist in the activity of a dynamical, ceaselessly self-examining and self-regulating intellectual community. In one sense, of course, no finite system (...) can ever be complete, insofar as it will inevitably be guilty of errors and misconceptions. Yet at the same time, the infinite System is perfect and complete, insofar as it contains within itself the means whereby its own unavoidable flaws will be discovered, corrected, and "forgiven"--i.e. aufgehoben into vanishing moments of a universal totality. ;In opposition to most traditional interpretations of Hegel, the position is maintained that Hegel himself deliberately faced this problem, and that a true understanding of his proposed solution requires a radical Aufhebung of the concept of cognition as such. Relying primarily on the penultimate chapters of the Phenomenology of Spirit, the dissertation argues that absolute knowledge can neither be a finalized system of fully explicit content, nor an endless movement toward such an ideal state. ;This dissertation undertakes a critical examination of one central problem in Hegelian philosophy: viz., whether the final realization of "absolute knowledge" is logically consistent with significant epistemic progress in the system's continuing development. Serious consideration of the concept of systematic completeness, as interpreted on Hegel's terms, uncovers the existence of a profound paradox. On the one hand, if the Truth is the Whole, then the truth of any finite part or aspect of that Whole depends upon its place within the system as a totality. In order to grasp the part and comprehend it correctly, one must already have a systematically ordered knowledge of the universe in its entirety. Yet the methodological open-endedness of dialectical thinking precludes the acceptance of any particular theoretical formulation as final or complete in itself. Hegel interprets the history of philosophy as a progression of successive formulations of the Truth, each of which improves on its predecessors by reconciling their internal contradictions, but no one of which is adequate in itself. Hence, no theoretical world view, including Hegel's own, can rest content within its limits, for the inevitable advance of thought must in the end throw it down. (shrink)

This paper introduces the notions of perfect quantifiers in general many-valued logics and investigates the problem of quantificational completeness for such logics as well as the problem of characterizing all perfect quantifiers in 3-valued logics using techniques of combinatorial group theory.

In what sense, and to what extent, do rules of inference determine the meaning of logical constants? Motivated by the principle of charity, a natural constraint on the interpretation of logical constants is to make the rules of inference come out sound. But, as Carnap observed, although this constraint does rule out some non-standard interpretations, it does not rule them all out. This is known as Carnap’s problem. I suggest that a charitable interpretation of the logical constants should, as (...) far as possible, make the rules of inference both sound and complete, and I show how this idea can be brought to bear on a successful solution to Carnap’s problem in the case of classical propositional logic, as well as classical first-order logic. In fact, the solution generalizes to any logic whose rules of inference are sound and complete with respect to a bivalent semantics that is classical with respect to negation. (shrink)

We consider a deterministic rewrite system for combinatory logic over combinators , and . Terms will be represented by graphs so that reduction of a duplicator will cause the duplicated expression to be "shared" rather than copied. To each normalizing term we assign a weighting which is the number of reduction steps necessary to reduce the expression to normal form. A lambda-expression may be represented by several distinct expressions in combinatory logic, and two combinatory logic expressions are considered equivalent if (...) they represent the same lambda-expression (up to --equivalence). The problem of minimizing the number of reduction steps over equivalent combinator expressions (i.e., the problem of finding the "fastest running" combinator representation for a specific lambda-expression) is proved to be NP-complete by reduction from the "Hitting Set" problem. (shrink)

While we endorse Heidegger’s effort to reclaim Kant’s Critique of Pure Reason as a work concerned with the possibility of metaphysics, we hold, first, that his reading is less original than is often assumed and, second, that it unduly marginalizes the critical impetus of Kant’s philosophy. This article seeks to shed new light on Kant and the Problem of Metaphysics and related texts by relating Heidegger’s interpretation of Kant to, on the one hand, the epistemological approach represented by Cohen’s (...) Kant’s Theory of Experience and, on the other, the metaphysical readings put forward by Heimsoeth, Wundt and others in the 1920s. On this basis, we argue that Heidegger’s interpretation of Kant remains indebted to the methodological distinction between ground and grounded that informed Cohen’s reading and was transferred to the problem of metaphysics by Wundt. Even if Heidegger resists a ‘foundationalist’ mode of this distinction, we argue that his focus on the notions of ground and grounding does not... (shrink)

The Luck Argument seems to show that libertarianism is false, since indeterministic free will is impossible. We should be wary of this argument, however, since a very similar argument shows that indeterministic causation1 is impossible. Further, since chancy events require causes, but are not determined, it would also follow that chancy events do not exist. If we are to conclude that free actions are all deterministic (or nonexistent), then the same reasoning should also persuade us that events with physical chances (...) do not exist. The Luck Argument, in its various formulations, assumes that a human being, like any physical system, has a set of complete (or exact, or precise) possible states. The same assumption drives the similar argument against indeterministic causation. This spells disaster for both free actions and chancy events, as these require causes. The assumption that physical systems have precise states should therefore be subjected to the closest scrutiny, which is not usually the case. On the contrary, it enjoys a wide and uncritical acceptance. (shrink)

Perceptual completion fills the gap for discrete perception to become continuous. Similarly, dynamic perceptual completion provides an experience of dynamic continuity. Our recent discovery of the ‘happening’ element of DPC completes the total experience for dynamism in the flow of time. However, a phenomenological explanation for these experiences is essential. The Snapshot Hypotheses especially the Dynamic Snapshot View provides the most comprehensive explanation. From that understanding the ‘two times’ problem can be addressed. The static time of spacetime cosmologies has (...) been irreconcilable with the dynamic FOT. Dismissing the FOT as an illusion is unsatisfactory. Therefore, we provide four hypotheses for the TTP.1) Since cosmological static time demands that all events are discrete, DPC elements for dynamism should likewise be expected to be discrete and accounted for by a snapshot phenomenology such as the DSV. 2) If temporality can be demonstrated to be similar to apparent motion by being a snapshot phenomenon and not demanding temporal extension it would confirm the DSV and permit reconciliation with static time. 3) If the ‘present moment’ is subjective as static time theories suggest, it should be possible experimentally for an observer to choose his own ‘present’ by moving to various points in the past with the aid of virtual reality. 4) If dynamism e.g. motion can be precluded without significant information loss or violating physics principles it is a cognitive add-on, thereby contradicting non-static time theories which suggest that time is ‘real.’ We confirm those hypotheses. (shrink)

By probability theory the probability space to underlie the set of statistical data described by the squared modulus of a coherent superposition of microscopically distinct (sub)states (CSMDS) is non-Kolmogorovian and, thus, such data are mutually incompatible. For us this fact means that the squared modulus of a CSMDS cannot be unambiguously interpreted as the probability density and quantum mechanics itself, with its current approach to CSMDSs, does not allow a correct statistical interpretation. By the example of a 1D completed scattering (...) and double slit diffraction we develop a new quantum-mechanical approach to CSMDSs, which requires the decomposition of the non-Kolmogorovian probability space associated with the squared modulus of a CSMDS into the sum of Kolmogorovian ones. We adapt to CSMDSs the presented by Khrennikov (Found. Phys. 35(10):1655, 2005) concept of real contexts (complexes of physical conditions) to determine uniquely the properties of quantum ensembles. Namely we treat the context to create a time-dependent CSMDS as a complex one consisting of elementary (sub)contexts to create alternative subprocesses. For example, in the two-slit experiment each slit generates its own elementary context and corresponding subprocess. We show that quantum mechanics, with a new approach to CSMDSs, allows a correct statistical interpretation and becomes compatible with classical physics. (shrink)

A number of philosophers have argued that causation is not an objective feature of the microphysical world but rather is a perspectival phenomenon that holds only between “coarse-grained” entities such as those that figure in the special sciences. This view seems to pose a problem for arguments for physicalism that rely on the alleged causal completeness of physics. In this paper, I address this problem by arguing that the completeness of physics has two components, only one (...) of which is causal. These two components of completeness can be used in an argument for physicalism that is supported by strong inductive evidence even in the absence of microphysical causation. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

From the author: This dissertation undertakes a critical examination of one central problem in Hegelian philosophy: viz., whether the final realization of “absolute knowledge” is logically consistent with significant epistemic progress in the system’s continuing development. Serious consideration of the concept of systematic completeness, as interpreted on Hegel’s terms, uncovers the existence of a profound paradox. On the one hand, if the Truth is the Whole, then the truth of any finite part or aspect of that Whole depends (...) upon its place within the system as a totality. In order to grasp the part and comprehend it correctly, one must already have a systematically ordered knowledge of the universe in its entirety. Yet the methodological open-endedness of dialectical thinking precludes the acceptance of any particular theoretical formulation as final or complete in itself. Hegel interprets the history of philosophy as a progression of successive formulations of the Truth, each of which improves on its predecessors by reconciling their internal contradictions, but no one of which is adequate in itself. Hence, no theoretical world view, including Hegel’s own, can rest content within its limits, for the inevitable advance of thought must in the end throw it down. (shrink)

This paper aims to solve a potential problem for the methodology of constitutive inference offered by Harbecke. The methodology is ultimately based on Mill’s “method of difference”, which requires a complete variation of factors in a given frame. In constitutive contexts, such a complete variation is often impossible. The offered solution utilizes the notion of a “mechanism slice”. In a first step, an example of a currently accepted explanation in neuroscience is reconstructed, which serves as a reference point of (...) the subsequent discussion. The regularity theory of mechanistic constitution and the corresponding methodology of constitutive inference are then introduced. Eventually, it is argued that the proposed solution accommodates well all schematic situations in which the impossibility of varying all test factors could be expected either to lead to false inferences or to preclude the establishment of correct constitutive claims. (shrink)

The causal compatibility question asks whether a given causal structure graph — possibly involving latent variables — constitutes a genuinely plausible causal explanation for a given probability distribution over the graph’s observed categorical variables. Algorithms predicated on merely necessary constraints for causal compatibility typically suffer from false negatives, i.e. they admit incompatible distributions as apparently compatible with the given graph. In 10.1515/jci-2017-0020, one of us introduced the inflation technique for formulating useful relaxations of the causal compatibility problem in terms (...) of linear programming. In this work, we develop a formal hierarchy of such causal compatibility relaxations. We prove that inflation is asymptotically tight, i.e., that the hierarchy converges to a zero-error test for causal compatibility. In this sense, the inflation technique fulfills a longstanding desideratum in the field of causal inference. We quantify the rate of convergence by showing that any distribution which passes the nth-order inflation test must be On−1/2$\begin{array}{} \displaystyle {O}{\left(n^{{{-}{1}}/{2}}\right)} \end{array}$-close in Euclidean norm to some distribution genuinely compatible with the given causal structure. Furthermore, we show that for many causal structures, the (unrelaxed) causal compatibility problem is faithfully formulated already by either the first or second order inflation test. (shrink)

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total (...) Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics (...) define the same consequence relations for a given spi-logic. (shrink)

The fundamental problem of Christology is the apparent contradiction of Christ as recorded at Chalcedon. Christ is human and Christ is divine. Being divine entails being immutable. Being human entails being mutable. Were Christ two different persons there’d be no apparent contradiction. But Chalcedon rules as much out. Were Christ only partly human or only partly divine there’d be no apparent contradiction. But Chalcedon rules as much out. Were the very meaning of ‘mutable’ and/or ‘immutable’ other than what they (...) are, there’d be no apparent contradiction. But the meaning is what it is, and changing the meaning of our terms to avoid the apparent contradiction of Christ is an apparent flight from reality. What, in the end, is the explanation of the apparent contradiction of Christ? Theologians and philosophers have long advanced many consistency-seeking answers, all of which increase the metaphysical or semantical complexity of the otherwise strikingly simple but radical core of Christianity’s GodMan. In this paper, I put the simplest explanation on the theological table: namely, Christ appears to be contradictory because Christ is contradictory. This explanation may sound complicated to the many who are steeped in the mainstream account of logic according to which logic precludes the possibility of true contradictions. But the mainstream account of logic can and should be rejected. Ridding theology of the dogma of mainstream logic illuminates the simple though striking explanation of the apparent contradiction of Christ — namely, that Christ is a contradictory being. Just as the simplest explanation to the apparent roundness of the earth has earned due acceptance, so too should the simplest explanation of the apparent contradiction of Christ. (shrink)

Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.

Computational complexity is a discipline of computer science and mathematics which classifies computational problems depending on their inherent difficulty, i.e. categorizes algorithms according to their performance, and relates these classes to each other. P problems are a class of computational problems that can be solved in polynomial time using a deterministic Turing machine while solutions to NP problems can be verified in polynomial time, but we still do not know whether they can be solved in polynomial time as well. A (...) solution for the so-called NP-complete problems will also be a solution for any other such problems. Its artificial-intelligence analogue is the class of AI-complete problems, for which a complete mathematical formalization still does not exist. In this chapter we will focus on analysing computational classes to better understand possible formalizations of AI-complete problems, and to see whether a universal algorithm, such as a Turing test, could exist for all AI-complete problems. In order to better observe how modern computer science tries to deal with computational complexity issues, we present several different deep-learning strategies involving optimization methods to see that the inability to exactly solve a problem from a higher order computational class does not mean there is not a satisfactory solution using state-of-the-art machine-learning techniques. Such methods are compared to philosophical issues and psychological research regarding human abilities of solving analogous NP-complete problems, to fortify the claim that we do not need to have an exact and correct way of solving AI-complete problems to nevertheless possibly achieve the notion of strong AI. (shrink)