Are theories 'underdetermined by the evidence' in any way that should worry the scientific realist? I argue that no convincing reason has been given for thinking so. A crucial distinction is drawn between data equivalence and empirical equivalence. Duhem showed that it is always possible to produce a data equivalent rival to any accepted scientific theory. But there is no reason to regard such a rival as equally well empirically supported and hence no threat to realism. Two theories (...) are empirically equivalent if they share all consequences expressed in purely observational vocabulary. This is a much stronger requirement than has hitherto been recognised—two such 'rival' theories must in fact agree on many claims that are clearly theoretical in nature. Given this, it is unclear how much of an impact on realism a demonstration that there is always an empirically equivalent 'rival' to any accepted theory would have—even if such a demonstration could be produced. Certainly in the case of the version of realism that I defend—structural realism—such a demonstration would have precisely no impact: two empirically equivalent theories are, according to structural realism, cognitively indistinguishable. (shrink)
Does quantum mechanics clash with the equivalence principle—and does it matter? Content Type Journal Article Pages 133-145 DOI 10.1007/s13194-010-0009-z Authors Elias Okon, Philosophy Department, UC San Diego, 9500 Gilman Dr., La Jolla CA, 92093, USA Craig Callender, Philosophy Department, UC San Diego, 9500 Gilman Dr., La Jolla CA, 92093, USA Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 1.
In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of (...) observational equivalence are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)
Contrary to popular belief, I argue that Leibniz is not hopelessly confused about motion: Leibniz is indeed both a relativist and an absolutist about motion, as suggested by the textual evidence, but, appearances to the contrary, this is not a problem; Leibniz’s infamous doctrine of the equivalence of hypotheses is well-supported and well-integrated within Leibniz’s physical theory; Leibniz’s assertion that the simplest hypothesis of several equivalent hypotheses can be held to be true can be explicated in such a way (...) that it makes good sense; the mere Galilean invariance of Leibniz’s conservation law does not compromise Leibniz’s relativism about motion; and Leibniz has a straightforward response to Newton’s challenge that the observable effects of the inertial forces of rotational motions empirically distinguish absolute from relative motions. (shrink)
On the observational equivalence of continuous-time deterministic and indeterministic descriptions Content Type Journal Article Pages 193-225 DOI 10.1007/s13194-010-0011-5 Authors Charlotte Werndl, Department of Philosophy, Logic and Scientific Method, London School of Economics, Houghton Street, London, WC2A 2AE UK Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 2.
We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time (...) generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely $\Delta^1_2$ reductions, and hence to the infinite time computable reductions. (shrink)
Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in  that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in . Other authors have extended this result to the (...) cases of k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature. (shrink)
Let E be a Σ1 1 equivalence relation for which there does not exist a perfect set of inequivalent reals. If 0# exists or if V is a forcing extension of L, then there is a good ▵1 2 well-ordering of the equivalence classes.
This paper deals with the concept of simultaneity in classical and relativistic physics as construed in terms of group-invariant equivalence relations. A full examination of Newton, Galilei and Poincaré invariant equivalence relations in ℝ4 is presented, which provides alternative proofs, additions and occasionally corrections of results in the literature, including Malament’s theorem and some of its variants. It is argued that the interpretation of simultaneity as an invariant equivalence relation, although interesting for its own sake, does not (...) cut in the debate concerning the conventionality of simultaneity in special relativity. (shrink)
The concept of substantial equivalence,introduced for the risk assessment of geneticallymodified (GM) food, is a reducing concept because itignores the context in which these products have beenproduced and brought to the consumer at the end of thefood chain. Food quality cannot be restricted to meresubstance and food acts on human beings not only atthe level of nutrition but also through theirrelationship to environment and society. To make thiscontext explicit, I will introduce an ``equivalencescale'' for the evaluation of food chains (...) (GM or notGM). By contrast with substantial equivalence, whichinvolves mainly quantitative, analytical methods ofevaluation, ``qualitative equivalence'' refers to ``less''or non-substantial factors that require new methodsof evaluation based on qualitative principles.``Ethical equivalence'' refers to factors that show themoral value contained in food products. To analyze thedifferent levels at which ethics is needed in foodchains, I will use the French principles: ``Liberty,Equality, Fraternity,'' or freedom, equality,solidarity, and add a fourth principle:sustainability. Sustainability, solidarity, andfreedom can be applied to the evaluation ofenvironmental, socio-economic, and socio-culturalethical equivalence, respectively. Equality refers tojustice and should operate so as to guarantee thatsustainability, solidarity, and freedom are satisfied.I suggest that ethics can provide a basis for arenewal of the food chain concept. Besides QualityAssurance, it is now essential to develop an ``EthicalAssurance'' and this equivalence scale could provide abasis to set up ``Ethical Assurance Standards'' (EAS)for food chains. (shrink)
Substantial equivalence (SE) has beenintroduced to assess novel foods, includinggenetically modified (GM) food, by means ofcomparison with traditional food. Besides anumber of objections concerning its scientificvalidity for risk assessment, the maindifficulty with SE is that it implies that foodcan be qualified on a purely substantial basis.SE embodies the assumption that only reductivescientific arguments are legitimate fordecision-making in public policy due to theemphasis on legal issues. However, the surge ofthe food debate clearly shows that thistechnocratic model is (...) not accepted anymore.Food is more than physico-chemical substanceand encompasses values such as quality andethics. These values are legitimate in theirown right and require that new democraticprocesses are set up for transverse,transdisciplinary assessment in partnershipwith society. The notion of equivalence canprovide a reference scale in which to examinethe various legitimate factors involved:substance (SE), quality (QualitativeEquivalence: QE), and ethics (EthicalEquivalence: EE). QE requires that newqualitative methods of evaluation that are notbased on reductive principles are developed. EEcan provide a basis for the development of anEthical Assurance as a counterpart of QualityAssurance in the food sector. In France, asecond circle of expertise is being set up toaddress the social issues in food public policybeside classical risk assessment by the firstcircle of expertise. Since ethics is likely tobecome an organizing principle of the secondcircle, the equivalence ethical framework canprove instrumental in this context. (shrink)
A precise fomulation of the strong Equivalence Principle is essential to the understanding of the relationship between gravitation and quantum mechanics. The relevant aspects are reviewed in a context including General Relativity but allowing for the presence of torsion. For the sake of brevity, a concise statement is proposed for the Principle: An ideal observer immersed in a gravitational field can choose a reference frame in which gravitation goes unnoticed. This statement is given a clear mathematical meaning through an (...) accurate discussion of its terms. It holds for ideal observers (time-like smooth non-intersecting curves), but not for real, spatially extended observers. Analogous results hold for gauge fields. The difference between gravitation and the other fundamental interactions comes from their distinct roles in the equation of force. (shrink)
Let E be a coanalytic equivalence relation on a Polish space X and (A n ) n∈ω a sequence of analytic subsets of X. We prove that if lim sup n∈K A n meets uncountably many E-equivalence classes for every K ∈ [ω] ω , then there exists a K ∈ [ω] ω such that ⋂ n∈K A n contains a perfect set of pairwise E-inequivalent elements.
Einstein’s equivalence principle has a number of problems, and it is often applied incorrectly. Clocks on the earth do not seem to be affected by the sun’s gravitational potential. The most commonly accepted reason given is a faulty application of the equivalence principle. While no valid reason is available within either the special or general theories of relativity, ether theories can provide a valid explanation. A clock bias of the correct magnitude and position dependence can convert the Selleri (...) transformation of ether theories into an apparent Lorentz transformation, which gives rise to an apparent equivalence of inertial frames. The results indicate that the special theory is invalid and that only an apparent relativity exists. (shrink)
We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦ δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □ δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.
We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.
It is known that if $\kappa < \lambda$ are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness (...) is indestructible under κ-directed closed forcing. (shrink)
Louveau and Rosendal  have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ℒω1ω) is far from complete (see [5, 2]). In this article (...) we strengthen the results of  by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in  about the comparison between isomorphism and bi-embeddability. Finally, as in  our results apply not only to classes of countable structures defined by sentences of ℒω1ω, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids. (shrink)
Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is (...) a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion. (shrink)
In this note we develop a method for constructing finite totally-ordered m-zeroids and prove that there exists a categorical equivalence between the category of finite, totally-ordered m-zeroids and the category of pseudo Łukasiewicz-like implicators.
We asked younger and older human participants to perform computer-based configural discriminations that were designed to detect acquired equivalence. Both groups solved the discriminations but only the younger participants demonstrated acquired equivalence. The discriminations involved learning the preferences (‘like’ [+] or ‘dislike’ [-]) for sports (e.g., tennis [t] and hockey [h]) of four fictitious people (e.g., Alice [A], Beth [B], Charlotte [C] & Dorothy [D]). In one experiment, the discrimination had the form: At+, Bt-, Ct+, Dt-, Ah-, Bh+, (...) Ch-, Dh+. Notice that, e.g., Alice and Charlotte are ‘equivalent’ in liking tennis but disliking hockey. Acquired equivalence was assessed in ancillary components of the discrimination (e.g., by looking at the subsequent rate of ‘whole’ versus ‘partial’ reversal learning). Acquired equivalence is anticipated by a network whose hidden units are shared when inputs (e.g., A and C) signal the same outcome (e.g., +) when accompanied by the same input (t). One interpretation of these results is that there are age-related differences in the mechanisms of configural acquired equivalence. (shrink)
Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets.
Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see ) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally equivalent but not (...) bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones. (shrink)
We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R n to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition (...) such as continuity. (shrink)
The aim of this work is to develop a declarative semantics for N-Prolog with negation as failure. N-Prolog is an extension of Prolog proposed by Gabbay and Reyle (1984, 1985), which allows for occurrences of nested implications in both goals and clauses. Our starting point is an operational semantics of the language defined by means of top-down derivation trees. Negation as finite failure can be naturally introduced in this context. A goal-G (...) may be inferred from a database if every top-down derivation of G from the database finitely fails, i.e., contains a failure node at finite height.Our purpose is to give a logical interpretation of the underlying operational semantics. In the present work (Part 1) we take into consideration only the basic problems of determining such an interpretation, so that our analysis will concentrate on the propositional case. Nevertheless we give an intuitive account of how to extend our results to a first order language. A full treatment of N-Prolog with quantifiers will be deferred to the second part of this work. (shrink)
We argue that purely local experiments can distinguish a stationary charged particle in a static gravitational field from an accelerated particle in (gravity-free) Minkowski space. Some common arguments to the contrary are analyzed and found to rest on a misidentification of “energy.”.