In recent years, Reichenbach's 1920 conception of the principles of coordination has attracted increased attention after Michael Friedman's attempt to revive Reichenbach's idea of a "relativized a priori". This paper follows the origin and development of this idea in the framework of Reichenbach's distinction between the axioms of coordination and the axioms of connection. It suggests a further differentiation among the coordinating axioms and accordingly proposes a different account of Reichenbach's "relativized a priori".

Relativized Metaphysical Modality (RMM: Murray and Wilson, 'Relativized metaphysical modality', Oxford Studies in Metaphysics, 2012; Murray, Perspectives on Modal Metaphysics, 2017) exploits 'two-dimensionalist' resources to metaphysical, rather than epistemological, ends: the second dimension offers perspective-dependence without contingency, diverting attacks on 'Classical' analyses of modals (in effect, analyses validating S5 and the Barcan Formulae). Here, we extend the RMM program in two directions. First, we harvest resources for RMM from Lewis's 1980 'Context--Index' (CI) framework: (a) the ban in CI on binding (...) into context-arguments (akin to Kaplan's 'monstrosity' ban) projects a bright line between perspective-dependence and contingency; and (b) CI-postulated connections among meaning, content, truth, argument-structure, context, and modality collectively generate a 'Generalized Humphrey Problem' for any non-Classical analysis (examples covered include appeals to accessibility, contingent domains, and counterpart relations). Second, we sharpen the tools of RMM-based metaphysical analysis, and extend their domain of coverage across familiar anomalies for Classical modals: we revisit earlier RMM-based bulwarks for S5 (against 'Chisholm's Paradox' for moderate flexibility of essence, and nomological necessitarianism); and we now similarly shore up the Barcan Formulae (against the apparent contingency of existence and nonexistence). (shrink)

According to the sensitivity account of knowledge, S knows that p only if S's belief in p is sensitive in the sense that S would not believe that p if p were false. It is widely accepted that the sensitivity condition should be relativized to belief-formation methods to avoid putative counterexamples. A remaining issue for the account is how belief-formation methods should be individuated. In this paper, I argue that while a coarse-grained individuation is still susceptible to counterexamples, a fine-grained (...) individuation makes the target belief trivially insensitive. Therefore, there is no principled way of individuating belief-formation methods that helps the sensitivity account to accommodate different cases. (shrink)

This paper answer the question how propositions whose truth is relativized to times, places, asserters or assessers can, despite their relativity, be used to represent the world.

An influential strand in philosophy of science claims that scientific paradigms can be understood as relativized a priori frameworks. Here, Kant’s constitutive a priori principles are no longer held to establish conditions of possibility for knowledge which are unchanging and universally true, but are restricted only to a given scientific domain. Yet it is unclear how exactly a relativized a priori can be construed as both stable and dynamical, establishing foundations for current scientific claims while simultaneously making intelligible the transition (...) to a subsequent framework. In this article, I show that important resources for this problem have been overlooked in Kant’s theory of reflective judgement in the third Critique. I argue that Kant accorded the task of formulating new scientific laws to reflective judgement, which is charged with forming new ‘universals’ that guide the experience of nature. I show that this is the very task attributed to the relativized a priori: the constitution of a given conceptual framework, not of the conditions for object-reference as such. I conclude that Kant’s considered conception of science encompasses the operations of both reflective and determining judgement. Relativizations of the a priori should follow Kant’s lead. (shrink)

We introduce a way of relativizing operational set theory that also takes care of application. After presenting the basic approach and proving some essential properties of this new form of relativization we turn to the notion of relativized regularity and to the system OST that extends OST by a limit axiom claiming that any set is element of a relativized regular set. Finally we show that OST is proof-theoretically equivalent to the well-known theory KPi for a recursively inaccessible universe.

We consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACENP and PTIMENP. For these classes, characterisations are known in terms of NP computable Lindström quantifiers which hold on ordered structures. We show that these characterisations are unlikely to extend to arbitrary structures, since this would imply the collapse of certain exponential complexity hierarchies. We also observe, however, that PTIMENP can be characterised in terms of Lindström quantifers , though it remains (...) open whether this can be done for LOGSPACENP. (shrink)

In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.

_Wh_-questions with the modal verb _can_ admit both mention-some (MS) and mention-all (MA) answers. This paper argues that we should treat MS as a grammatical phenomenon, primarily determined by the grammar of the _wh_-interrogative. I assume that MS and MA answers can be modeled using the same definition of answerhood (Fox in Mention-some interpretations, MIT seminar, 2013 ) and attribute the MS/MA ambiguity to structural variations within the question nucleus. The variations are: (i) the scope ambiguity of the higher-order _wh_-trace (...) and (ii) the absence/presence of an anti-exhaustification operator. However, treating MS answers as complete answers in this way contradicts the widely adopted analysis of uniqueness effects in questions of Dayal (Locality in _wh_ quantification: Questions and relative clauses in Hindi, 1996 ), according to which the uniqueness effects of singular _which_-phrases arise from an exhaustivity presupposition, namely that a question must have a unique exhaustive true answer. To solve this dilemma, I propose that question interpretations presuppose Relativized Exhaustivity: roughly, the exhaustivity in questions is evaluated relative to the accessible worlds as opposed to the anchor/utterance world. Relativized Exhaustivity preserves the merits of Dayal’s exhaustivity presupposition while permitting MS; moreover, it explains the local-uniqueness effects in modalized singular _wh_-questions. (shrink)

Relativizing Newton" is a first step towards a simple and beautiful theory of everything. The theory, termed "Information Relativity" (IR) takes a novel approach to physics that overlooks all post-Newtonian physics. It stands on the shoulders of Newtonian dynamics, but modifies it by accounting for the time-travel of information from one reference-frame to another, a fact which somehow was ignored by Galileo Galilee and Isaac Newton, and which remained ill-treated by the all post-Newtonian theories, including Einstein's relativity and quantum theories. (...) Except for the aforementioned correction of classical physics, IR has no axiomatic presumptions, nor arbitrary free parameters. Astonishingly, accounting for the aforementioned delays in information results in a set of simple and beautiful transformations, which explain and predict a great deal of physical phenomena. Most importantly, IR's transformations reveal the mysteries of dark matter, dark energy, and gravity. They also provide a unifying platform for the physics of the too-big (astrophysics and cosmology), and the too-small (small particles dynamics and quantum mechanics). The phenomena explained and predicted successfully by IR include The Michelson-Morley's null-result, "time-dilation" of decaying muons, the neutrino velocities measured by OPERA and other collaborations, particle diffraction in the double-slit experiment, Sagnac Effects, the quantization of orbits in Bohr's hydrogen atom, entanglement, quantum criticality, confinement, asymptotic freedom, solar light bending, gravitational redshift, the Pioneer anomaly, dark matter in galaxies, and the Schwarzschild's black hole. (shrink)

A Schnorr test relative to some oracle A may informally be called “universal” if it covers all Schnorr tests. Since no true universal Schnorr test exists, such an A cannot be computable. We prove that the sets with this property are exactly those with high Turing degree. Our method is closely related to the proof of Terwijn and Zambella’s characterization of the oracles which are low for Schnorr tests. We also consider the oracles which compute relativized Schnorr tests with the (...) weaker property of covering all computable reals. The degrees of these oracles strictly include the hyperimmune degrees and are strictly included in the degrees not computably traceable. (shrink)

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. -/- For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that (...) we call divisible amalgamation. -/- The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. -/- Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed. (shrink)

We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized to an oracle in C. We show that this is not always (...) true. However, after studying the problem from a general point of view, we derive sufficient conditions on C such that FO(Q) captures L C . These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures L NP . This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform. vol. 18, 1993]. (shrink)

The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP.As an application it is shown that whenever KP+AC proves a ΠP2 statement then it holds true in the segment Vτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal.

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res, as well as their tree-like versions, Res⁎. The contradictions we use are natural combinatorial principles: the Least number principle, LNPn and an ordered variant thereof, the Induction principle, IPn.LNPn is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNPn iterated d times provides a separation between Res and Res. We (...) prove the same result for the iterated relativization of IPn, where the tree-like variant Res⁎ is considered instead of Res.We go on to provide separations between the parameterized versions of Res and Res. Here we are able again to use the relativization of the LNPn, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res⁎ and Res⁎. Here, the relativization of IPn will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming. (shrink)

This paper describes two ways in which Plato addresses the compresence of opposites in appearances and shows how his treatment of the phenomenon is a response to Protagoras: (1) by relativizing the opposites so as to avoid making the contradictory claims that the same thing is characterized by F and not-F, and (2) by seeking the cause of the conflicting phenomena. Relativization, a rational strategy for avoiding contradiction which Plato inherits from Protagoras, enables the user to avoid refutation in (...) a dialectical encounter, though Plato thinks (like Heraclitus and Democritus) that knowledge requires explaining, rather than only resolving, the contradictions that arise from compresent opposition. But the metaphysics that would explain unlimited relativization is a metaphysics of relative-coming-to-be that, even though it guarantees infallibility of and contact with reality for each perceiver, ends in the impossibility of knowledge. For Plato, knowledge requires relativization limited by, and in the service of, explanation. (shrink)

Ontology is taken by Moulines as supervenient on science: what kinds of things there are is determined by our well-confirmed theories. But the fact is that today, science provides us with a multiplicity of well-confirmed theories, each having its own ontological commitments. The modest, ontological form of reduction advocated by Moulines (this volume) restores hope of putting some ontological order in the “huge chaotic supermarket of science”. In this paper I show that any claim on the amount of order obtained (...) by reduction does not only always remain “temporally qualified” but, worse for the reductionist with a taste for ontological order, that the very notion of orderliness must be relativized to the capacities and interests of knowers. (shrink)

I object to eliminativism about innateness and André Ariew’s identification of innateness with canalization, and I propose a new treatment of innateness. I first argue that the concept of innateness is serving a valuable function in a diverse set of research contexts, and in these contexts, claims about innateness are best understood as claims about the insensitivity of the appearance of a trait to certain variations in the environment. I then argue that innateness claims, like claims about canalization, should be (...) explicitly relativized to the specific range of environmental variations of interest to the scientist. My account characterizes an important way in which scientists are employing the concept and offers a way for scientists to carry on using the concept in their research while minimizing confusion and miscommunication. There is a fruitful research program, I claim, in which scientists employ the concept of innateness to help distinguish environmental factors of interest that have a causal influence on the appearance of a trait from those that do not. (shrink)

We discuss some logico-mathematical systems which deviate from classical logic and mathematics with respect to the concept of identity. In the first part of the paper we present very general formulations of the principle of identity and show how they can be ‘relativized’ to objects and to properties. Then, as an application, we study the particular cases of physics and logic . In the last part of the paper, we discuss the alphabar logics, that is, those logical systems which violate (...) a formulation of one of the most fundamental versions of the principle of identity; in these logics, there are formulas which are not deducible from themselves. (shrink)

It is commonly supposed that metaphysical modal claims are to be evaluated with respect to a single domain of possible worlds: a claim is metaphysically necessary just in case it is true in every possible world, and metaphysically possible just in case it is true in some possible world. We argue that the standard understanding is incorrect; rather, whether a given claim is metaphysically necessary or possible is relative to which world is indicatively actual. We motivate our view by attention (...) to discussions in Salmon 1989 and Fine 2005, in which various data are taken to support rejecting the transitivity of accessibility and modal monism ; we argue that relativized metaphysical modality can accommodate these data compatible with both standard modal logic and modal monism. Noting an analogy with two-dimensional semantics, we argue that metaphysical modality has a complex structure, reflecting what is counterfactually possible, relative to each indicatively actual world. In arguing for the need for relativization, we are broadly on the same side as Crossley and Humberstone and Davies and Humberstone ; our contribution here is, first, to offer distinctively metaphysical reasons for relativization, and second, to show that relativization can be incorporated in ways minimally departing from standard modal logic. (shrink)

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

On what I call absolutist essentialism about modality (AE), the metaphysical necessities are the propositions that are true in virtue of the essence (i.e. Aristotelian, absolute essence) of some entities. Other kinds of necessity can then be defined by restriction – e.g. the conceptual necessities are the propositions that are true in virtue of the essence of conceptual entities specifically. As an account of metaphysical modality and some other kinds (e.g. logical, conceptual), AE may have important virtues. However, when it (...) comes to accounting for further important kinds, like natural or normative necessity, it faces a challenge. Three main options have been defended: treat those kinds as further restricted forms of metaphysical necessity; define them as conditional forms of metaphysical necessity; treat them as primitive kinds. -/- In this paper, I propose a new option, which combines the main idea of AE (reducing necessities to essences) with an idea which has been developed largely independently: that of relative essence. On the proposed view, those kinds (e.g. natural necessity) that cannot be grounded in the essences (i.e. absolute essences) of the relevant entities (e.g. natural entities) may be grounded in their relative essences instead. Thus, I propose a generalized, or extended, version of AE, which I call relativized essentialism about modality (RE). In particular, RE offers prospects for a general framework for kinds of modality which is flexible enough to cover a large range of kinds (both absolute and relative ones) while remaining parsimonious and unified. (shrink)

Can we solve the problem of the essential indexical, and account for de se belief, by appealing to 'relativized propositions' (functions from rich indices to truth-values)? According to John Perry, we cannot. This paper offers a detailed examination and a critique of Perry's argument.

Building on a previously published contextualization of Marco Buzzoni’s Neo- Kantian account of scientific thought-experiments, this paper examines the explanatory power of this account. It is argued that Buzzoni’s account suffers from a number of shortcomings. Einstein’s clock-in-the-box thought experiment facilitates the demonstration of these deficits. In the light of both the identified inadequacies of Buzzoni’s account and the long-standing history of Kantian approaches to thought experiments, this paper finally sketches an alternative Neo-Kantian account. This alternative utilizes Michael Friedman’s reading (...) of Kant’s a priori within a Kuhnian account of thought experiments along the lines of conceptual constructivism as anticipated by Georg Lichtenberg and further developed recently by Tamar Gendler. (shrink)

This chapter contains sections titled: Abstract Introduction Relativism and Apparent Faultless Disagreement The Many Relativisms: Moderate vs. Radical Moderate Relativisms: Indexical vs. Non ‐ indexical Contextualism Radical Relativism: Content vs. Truth Relativism The Many “Relativism”s References.

Some have concluded that the only appropriate response to the problem of temporary intrinsics is the view that familiar, concrete objects persist through time by perduring, that is, by having temporal parts. Many, including myself, believe this view of persistence is false, and so reject this conclusion. However, the most common attempts to resolve the problem and yet defend the view that familiar, concrete objects endure are self-defeating. This has heretofore gone unnoticed. I consider the most familiar such attempts, based (...) on a strategy called tensing the copula , and present a general argument to demonstrate why this strategy – and any strategy based on relativization – fails. I then show how the considerations raised in this general argument undermine other attempts to resolve the problem while denying perdurance. All these attempts are undermined by an assumption essential to the problem of temporary intrinsics, to wit, that there are many moments of time and all have the same ontological status. As long as this assumption is maintained, the only solution to the problem is that familiar, concrete objects perdure. Thus, in order to defend the view that objects persist through time by enduring, one must adopt a different metaphysics of time (viz., presentism). I conclude that it is neither unreasonable nor impracticable to do so. (shrink)

A cause is proportional to its effect when, roughly speaking, it is at the right level of detail. There is a lively debate about whether proportionality is a necessary condition for causation. One of the main arguments against a proportionality constraint on causation is that many ordinary and seemingly perfectly acceptable causal claims cite causes that are not proportional to their effects. In this paper, I suggest that proponents of a proportionality constraint can respond to this objection by developing an (...) idea that is present in Yablo’s early work on proportionality, but which has strangely been ignored by both Yablo and others in the subsequent debate. My suggestion is that proportionality—and, indeed, causation itself—is relative to a domain of events. At the metaphysical level, this means that the causal relation has an extra relatum—namely, a domain of events. At the level of language, it introduces a new way in which causal claims are context-sensitive: what is expressed by a causal claim depends on the contextually relevant domain of events. As I argue, this suggestion allows us to accommodate the truth of ordinary causal claims while extending the explanatory benefits of a proportionality constraint. (shrink)

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $$\Sigma $$. We introduce the notion of a Lascar tuple for $$\Sigma $$ and by considering the space of types over a Lascar tuple for $$\Sigma $$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup (...) of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $$\Sigma $$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types. (shrink)

Carl G. Hempel's doctrine of essential epistemic relativization of inductive-statistical explanation seems to entail the unintelligibility of the notion of objective homogeneity of reference classes. This discussion note explores the question of whether, as a consequence, essential epistemic relativization also entails the unintelligibility of the doctrine of indeterminism.

We propose an account of the subject’s cognition and its relation to the world that allows for an articulation of the phenomenon of ideology. We argue that ideology is a form of what we call ‘a priori activity’: it transcendentally conditions the intelligibility of thought and practice. But we draw from strands of post-Kantian philosophy of science and social philosophy in repudiating Kant’s view that the a priori is necessary and fixed. Instead, we relativize the a priori: we argue that (...) it is contingent, and therefore revisable. More precisely, it is conditioned materially, in that it is enmeshed with and shaped by material social practice. We conclude with some remarks about the possibility of agency over the relativized, materially conditioned a priori—that is, about the possibility of critique. (shrink)

Content nonconceptualism and State conceptualism are motivated by constraints of content-attribution that pull in opposite directions, namely, the so-called cognitive significance requirement and what I would like to call here the same representing constraint. The solution to this apparent contradiction is the rejection of the real content view and the adherence to what I call here Content-pragmatism. In CPR, “proposition” is not as real as a mental state, but rather is a term of art that semanticists use, as a matter (...) of theoretical convenience, to classify mental states. What follows from CPR is Content-pluralism : there is no such a thing as the content of experience. Because both criticisms and the defense of SNC overlook CPR and CP, they are ineffective. In this paper, I will argue that the opposition between SNC and CNC is a real one, but only a relative one, that is, relative to the opposite constraints to be met, namely CSR and SRR. If we want to meet CSR, the best we can do is to let the content of experience be modeled as neo-Fregean content, namely a compound of non-conceptual modes of presentation of objects and properties. In this case, CNC prevails. In contrast, if we want to meet SRR, the best we can do is let the content of experience be modeled either as a Russellian compound of objects and properties or as a function from possible worlds to truth-values. In this case, SNC prevails. (shrink)

In this paper, I defend Peter Geach’s theory of Relative Identity against the charge that it cannot make sense of basic semantic notions.

Content nonconceptualism and State conceptualism are motivated by different readings of what I want to call here Bermúdez’s conditions on content-attribution (2007). In one read- ing, what is required is a neo-Fregean content to solve problems of cognitive significance at the nonconceptual level (Toribio, 2008; Duhau, 2011). In the other reading, what is required is a neo-Russellian or possible-world content to account for how conspecifics join attention and cooperate, contemplating the same things from different perspectives in the same perceptual field. (...) The solution to this apparent contradiction is the rejection of the real content view and the adherence to what I call here Content-pragmatism: there is no such thing as the content of experience. According to content pragmatism, “proposition” is not as real as a mental state, but rather, it is a term of art that semanticists use, as a matter of theoretical convenience, to classify mental states. What follows from Content-pragmatism is Content- pluralism: there are so many contents that are required to meet Bermúdez’s condition on content attribution. Because both criticism and the defense of State conceptualism overlook the real scope of Bermúdez’s condition on content attribution, they are ineffective. In this paper, I will argue that the opposition between State and Content nonconceptualism is a real one, but only a relative one, that is, relative to the opposite constraints to be met. If we want to solve the problems of cognitive significance, the best we can do is to let the content of experience be modeled as neo-Fregean content, namely, a compound of nonconceptual modes of presentation of objects and properties. In this case, Content-nonconceptualism prevails. In contrast, if we want to account for how conspecifics join attention to the same entities in the same perceptual field from different perspectives, the best we can do is let the content of experience be modeled either as a structured Russellian proposition or as a function from possible worlds to truth-values. In this case, State-nonconceptualism prevails. (shrink)

For each class in the piecewise-local subregular hierarchy, a relativized (tier-based) variant is defined. Algebraic as well as automata-, language-, and model-theoretic characterizations are provided for each of these relativized classes, except in cases where this is provably impossible. These various characterizations are necessarily intertwined due to the well-studied logic-automaton connection and the relationship between finite-state automata and (syntactic) semigroups. Closure properties of each class are demonstrated by using automata-theoretic methods to provide constructive proofs for the closures that do hold (...) and giving language-theoretic counterexamples for those that do not. The net result of all of this is that, rather than merely existing as an operationally-defined parallel set of classes, these relativized variants integrate cleanly with the other members of the piecewise-local subregular hierarchy from every perspective. Relativization may even prove useful in the characterization of star-free, as every star-free stringset is the preprojection of another (also star-free) stringset whose syntactic semigroup is not a monoid. (shrink)

While the rise of a ‘dictatorship of relativism’ was a longstanding concern for Joseph Ratzinger/Pope Benedict XVI, some commentators have suggested that—for better or for worse—the challenge posed by relativism appears to be less of a priority for Pope Francis. Indeed, Francis's remark, ‘Who am I to judge?’ appears to have become as much the defining soundbite for his papacy as the ‘dictatorship of relativism’ was for Benedict's. Contrary to these perceptions, this article argues that a critique of relativism is (...) in fact a central aspect of Pope Francis's magisterium. Francis exhibits continuity with Benedict by amplifying his concerns about the danger of attempting to reconstruct reality according to human desire, but Francis also develops his predecessor's teaching by introducing the category of ‘practical relativism’. With this development, relativism cannot be dismissed as a theoretical problem that only plagues those who refuse to assent to the truth intellectually. The willingness to set aside the truth to which one assents when it proves inconvenient represents a more pernicious—precisely because more subtle—form of relativism that Francis argues must be rooted out, not by argument, but through self‐examination and the practice of solidarity. (shrink)

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅. We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C ≥ |x|-c. The ‘only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results (...) on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees. (shrink)