This article connects value-sensitive design to Gibson’s affordance theory: the view that we perceive in terms of the ease or difficulty with which we can negotiate space. Gibson’s ideas offer a nonsubjectivist way of grasping culturally relative values, out of which we develop a concept of political affordances, here understood as openings or closures for social action, often implicit. Political affordances are equally about environments and capacities to act in them. Capacities and hence the severity of affordances vary with age, (...) health, social status and more. This suggests settings are selectively permeable, or what postphenomenologists call multistable. Multistable settings are such that a single physical location shows up differently – as welcoming or hostile – depending on how individuals can act on it. In egregious cases, authoritarian governments redesign politically imbued spaces to psychologically cordon both them and the ideologies they represent. Selective permeability is also orchestrated according to business interests, which is symptomatic of commercial imperatives increasingly dictating what counts as moral and political goods. (shrink)

For a long time, academic institutes stigmatized activism and dissociated it from academic practice. It was looked down upon and considered to be disruptive and western institutes continued silencing critical thinking and practice, and encouraged what they named 'critical distance'. These practices of exclusion must push us, city inhabitants, to ask: what is the point of excluding activism from academic practice? How can we bridge between theory and activism? How can we decenter city planning? If cities belong to the people, (...) why are public authorities trying to erase the public's print in the urban realm? Who gets to speak and why? In our group as the ‘Radicals’, we discussed the gap between urban theory and practice, while encouraging the audience to reflect on how urban concepts adopt multiple definitions within different geographies and communities. (shrink)

The positions of Frege, Russell and Wittgenstein on the priority of complexes over (propositional) functions are sketched, challenging those who take the "judgment centered" aspects of the Tractatus to be inherited from Frege not Russell. Frege's views on the priority of judgments are problematic, and unlike Wittgenstein's. Russell's views on these matters, and their development, are discussed in detail, and shown to be more sophisticated than usually supposed. Certain misreadings of Russell, including those regarding the relationship between propositional functions and (...) universals, are exposed. Wittgenstein's and Russell's views on logical grammar are shown to be very similar. Russell's type theory does not countenance types of genuine entities nor metaphysical truths that cannot be put into words, contrary to conventional wisdom. I relate this to the debate over "inexpressible truths" in the Tractatus. I lastly comment on the changes to Russell's views brought about by Wittgenstein's influence. (shrink)

Interesuje nas, prednost koju Delez daje sintagmi?revolucionarna institucija? u odnosu na manje originalnu sintagmu?institucionalna revolucija?. Delez nekoliko puta upotrebljava u svojim tekstovima?revolucionarna institucija?, sigurno sasvim svestan da ova sintagma ima jednu prilicno haoticnu i nejasnu istoriju tokom post-revolucionarnog perioda, ali i da je ona savrseno u duhu Sanzistovih namera. Treba da proverimo da li ova sintagma najbolje opisuje jednu Delezovu imaginarnu teoriju institucije, ali i njegov angazman u teoriji uopste. Preeliminarna teskoca, koja automatski dovodi u pitanje i devalvira nasu intervenciju, (...) jeste Delezovo odbijanje da on s?m zaista tematizuje instituciju i svoj napor, te da zaista odgovori na pitanja koja s?m postavlja pedesetih godina proslog veka:?sta je to sto objasnjava instituciju?? i?koje bi to trebalo da budu savrsene institucije, to jest one koje se suprotstavljaju svakom ugovoru i pretpostavljaju tek najmanji broj zakona??. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. (...) Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...) what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

In this paper I present some difficulties for Lewis’s and similar theories of counterfactuals, and suggest that the problem lies in the notion of absolute similarity. In order to explain the problem, I discuss the relation between Lewis’s and Goodman’s theory, and show that the two theories are not related in the way Lewis thought they were.

U ovom tekstu se obavlja jedno?naknadno citanje? onih opstih mesta koja ukazuju na partikularni, licni interes koji se krije iza maske objektivnosti, nuzde i istine. Kao primeri za takvu strukturu ovde se uzimaju drustveni prevrati, koji su svi redom stupali na snagu uz obavezno pozivanje na istinu, a koji su, po svom politickom karakteru, bili fasisticki, liberalno-demokratski ili komunisticki. Preispitivanjem njihovog odnosa prema istini i prema?velu iza kog se istina krije?, dalje se pokusava nesto reci o njima samima. Uporedo sa (...) razmatranjem fasistickih, liberalno-demokratskih i komunistickih rezima, ovde se razmatra i specificnost subjekta kakav se u svakom od ovih rezima konstituise. (shrink)

This paper is about the distinction between arguments and conditionals, and the corresponding distinction between premises and antecedents. I will also propose a further distinction between two different kinds of argument, and, correspondingly, two kinds of premise that I will call "assumption" and "hypothesis." The distinction between assumptions, hypotheses, and antecedents is easily made in artificial languages, and we are already familiar with it from our first logic courses (although not necessarily under those names, since there is no standard terminology (...) for the distinction). After explaining their differences in artificial languages, I will argue that there are ordinary-language counterparts of these three notions, meaning that some formal properties of the artificial notions nicely capture some features of the ordinary-language counterparts and their behavior in contexts of reasoning. My next crucial claim is that these three notions often get confused in ordinary language, which leads to problems for translation into symbols. I will suggest a solution to the translation problem by pointing to some distinctive characteristics of the three notions that link them to their artificial-language counterparts. Next, I will argue that this confusion is behind some well-known philosophical problems and puzzles. I will apply the distinctions in order to explain away some famous paradoxes: the direct argument (also known as or-to-if inference), a standard argument for fatalism, and McGee's counterexample to modus ponens. As Stalnaker also solved the first two of these paradoxes by using his theory of reasonable inference, I will elucidate the similarities between our solutions, and also explain why my distinctions apply more broadly, to some cases involving indicative and counterfactuals conditionals, where reasonable inference does not apply. (shrink)

An incorrect interpretation of Goodman’s theory of counterfactuals is persistently being offered in the literature. I find that strange. Even more so since the incorrectness is rather obvious. In this paper I try to figure out why is that happening. First I try to explain what Goodman did say, which of his claims are ignored, and what he did not say but is sometimes ascribed to him. I emphasize one of the bad features of the interpretation: it gives counterfactuals some (...) formal properties that neither Goodman nor (usually) the interpreter would accept. The usual interpretation (UI), which I claim should not be ascribed to Goodman, considers a counterfactual A>C true iff A, together with natural laws and all contingent truths cotenable with it, entails C. (UI) makes valid the law of conditional excluded middle, which Goodman clearly rejected. Among possible reasons for which the interpreters might find (UI) adequate is that (UI), as I argue, smuggles in the idea of minimal change, which is otherwise attractive, natural to many, but not to be found anywhere in Goodman’s paper. At the end I stress the significance of Goodman’s theory by arguing that we still need some of his notions to test the adequacy of our modern theories. (shrink)

The presented study uses the Lukan parable of the Good Samaritan (10, 27-35) in order to present the shifts in the meaning depending on the reading contexts. After the basic structure of the original meaning is established, the pragmatic nuances of the parable are illustrated. The research subsequently throws light on the paradigmatic interpretations in both the medieval and the contemporary contexts. It concludes by exemplifying that the polyvalence of meaning is not only dependent upon the genuine literary structure of (...) the parable, but also on the innate ability of the Christian organism (Church) to actualize certain features of the sacred text in the concrete life-settings. (shrink)

This paper examines the roles of dialogue in the process of communication with stakeholders. The conceptual frameworks of corporate social responsibility (CSR) and stakeholder relationships frequently present the initiation of a dialogue with stakeholders as a way for an organization to respond to criticisms of its social and environmental policies and actions. The paper discusses dialogue in the stakeholder and CSR literature. This is followed by the analysis of in-depth semi-structured interviews in the empirical section. Theoretical discussion and empirical examples (...) demonstrate that the role of stakeholder dialogue in implementing CSR strategy is crucial in many ways. The case examples taken from the interviews with the heads of two large companies and two non-governmental organizations illustrate the practical difficulties that may be encountered in applying the principle and practice of dialogue to the implementation of CSR programs in the real world. (shrink)

Some enlightenment regarding the project to mechanise reason. The assembly line of machine learning: data, algorithm, model. The training dataset: the social origins of machine intelligence. The history of AI as the automation of perception. The learning algorithm: compressing the world into a statistical model. All models are wrong, but some are useful. World to vector: the society of classification and prediction bots. Faults of a statistical instrument: the undetection of the new. Adversarial intelligence vs. statistical intelligence: labour in the (...) age of AI. (shrink)

This book aims to develop certain aspects of Gottlob Frege’s theory of meaning, especially those relevant to intensional logic. It offers a new interpretation of the nature of senses, and attempts to devise a logical calculus for the theory of sense and reference that captures as closely as possible the views of the historical Frege. (The approach is contrasted with the less historically-minded Logic of Sense and Denotation of Alonzo Church.) Comparisons of Frege’s theory with those of Russell and others (...) are given. It is in the end shown that developing Frege’s theory in these ways reveals serious problems hitherto largely unnoticed, including those possibly rendering a Fregean intensional logic inconsistent even if his naïve class theory is excluded. (shrink)

A summary of Russell’s logical atomism, understood to include both a metaphysical view and a certain methodology for doing philosophy. The metaphysical view amounts to the claim that the world consists of a plurality of independently existing things exhibiting qualities and standing in relations. The methodological view recommends a process of analysis, whereby one attempts to define or reconstruct more complex notions or vocabularies in terms of simpler ones. The origins of this theory, and its influence and reception are also (...) discussed. (shrink)

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic.

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient (...) sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms. (shrink)

Globally, consumers are increasingly turning to sustainable consumption practices. This article emphasizes the importance of social and cultural context in the study of sustainable consumption, drawing on social representations. It attempts to explain and empirically demonstrate how sustainable consumption is socially represented. The aim of the study was to investigate the construction of representations of sustainable consumption as knowledge and its appropriation in relation to the purchase and consumption of food. Online focus groups were employed in a cross-sectional study conducted (...) in Slovenia and Austria. The results of the study not only show how the “global” concept of sustainable consumption is appropriated and reflected in practices in a specific national and cultural context, but also highlight the importance of social representations in terms of how their meanings can influence the emergence of new practices. Furthermore, they show how sustainable consumption can occasionally be seen in actions that precede reflection or exist in a more abstract form unrelated to actions. The results offer several implications for practitioners seeking to promote sustainable consumption. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell (...) did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

Abstract:Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more clearly the primary meta-ontological (...) and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy (...) due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

In 1903, in _The Principles of Mathematics_ (_PoM_), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was short-lived; rejected before _PoM_ even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about (...) why he abandoned this view. In this paper, I speculate as to what those reasons were, and evaluate them in light of recent work and interest in plural logic. (shrink)

This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature between objects and functions. (...) Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve these paradoxes, (...) including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. (shrink)

This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...) and denotation, etc., and summarizes the correspondence between Frege and Russell and the light it sheds on the philosophical logic of both. (shrink)

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used to manufacture (...) paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

I trace changes to Frege's understanding of numbers, arguing in particular that the view of arithmetic based in geometry developed at the end of his life (1924–1925) was not as radical a deviation from his views during the logicist period as some have suggested. Indeed, by looking at his earlier views regarding the connection between numbers and second-level concepts, his understanding of extensions of concepts, and the changes to his views, firstly, in between Grundlagen and Grundgesetze, and, later, after learning (...) of Russell's paradox, this position is a natural position for him to have retreated to, when properly understood. (shrink)

Attempts to evaluate a belief or argument on the basis of its cause or origin are usually condemned as committing the genetic fallacy. However, I sketch a number of cases in which causal or historical factors are logically relevant to evaluating a belief, including an interesting abductive form that reasons from the best explanation for the existence of a belief to its likely truth. Such arguments are also susceptible to refutation by genetic reasoning that may come very close to the (...) standard examples given of supposedly fallacious genetic reasoning. (shrink)

Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are also (...) discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic. (shrink)

Russell discovered the classes version of Russell's Paradox in spring 1901, and the predicates version near the same time. There is a problem, however, in dating the discovery of the propositional functions version. In 1906, Russell claimed he discovered it after May 1903, but this conflicts with the widespread belief that the functions version appears in _The Principles of Mathematics_, finished in late 1902. I argue that Russell's dating was accurate, and that the functions version does not appear in the (...) _Principles_. I distinguish the functions and predicates versions, give a novel reading of the _Principles_, section 85, as a paradox dealing with what Russell calls _assertions_, and show that Russell's logical notation in 1902 had no way of even formulating the functions version. The propositional functions version had its origins in the summer of 1903, soon after Russell's notation had changed in such a way as to make a formulation possible. (shrink)

Gli autori intendono proporre le premesse necessarie al chiarimento del rapporto, incerto e da sempre dibattuto, che lega architettura e filosofia attraverso il riferimento al concetto di “Filosofia architettonica” proposto da Peter Eisenman. Una prima premessa riguarda le varie tipologie di approccio alla filosofia riscontrabili nel campo dell’architettura, fino a delineare una forma di consapevolezza filosofica nell’architettura contemporanea. La seconda premessa riguarda la possibilità di spiegare il rapporto tra filosofia e architettura attraverso i casi di collaborazione concreta tra architetti e (...) filosofi. (shrink)

Interesuje nas, prednost koju Delez daje sintagmi?revolucionarna institucija? u odnosu na manje originalnu sintagmu?institucionalna revolucija?. Delez nekoliko puta upotrebljava u svojim tekstovima?revolucionarna institucija?, sigurno sasvim svestan da ova sintagma ima jednu prilicno haoticnu i nejasnu istoriju tokom post-revolucionarnog perioda, ali i da je ona savrseno u duhu Sanzistovih namera. Treba da proverimo da li ova sintagma najbolje opisuje jednu Delezovu imaginarnu teoriju institucije, ali i njegov angazman u teoriji uopste. Preeliminarna teskoca, koja automatski dovodi u pitanje i devalvira nasu intervenciju, (...) jeste Delezovo odbijanje da on s?m zaista tematizuje instituciju i svoj napor, te da zaista odgovori na pitanja koja s?m postavlja pedesetih godina proslog veka:?sta je to sto objasnjava instituciju?? i?koje bi to trebalo da budu savrsene institucije, to jest one koje se suprotstavljaju svakom ugovoru i pretpostavljaju tek najmanji broj zakona??. (shrink)

In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...) treating apparent terms for numbers using these definitions. (shrink)

In he Problems of Philosophy and other works of the same period, Russell claims that every proposition must contain at least one universal. Even fully general propositions of logic are claimed to contain “abstract logical universals”, and our knowledge of logical truths claimed to be a species of a priori knowledge of universals. However, these views are in considerable tension with Russell’s own philosophy of logic and mathematics as presented in Principia Mathematica. Universals generally are qualities and relations, but if, (...) for example, PM’s disjunction (∨) is a relation, what is it a relation between? There is no obvious answer to this given Russell’s other philosophical commitments at this time, although hints are left in some of the pre-PM manuscripts. In this paper, I explore this tension in Russell's philosophy and relate it to developments both before and after Problems. (shrink)

Analytic philosophy has been perhaps the most successful philosophical movement of the twentieth century. While there is no one doctrine that defines it, one of the most salient features of analytic philosophy is its reliance on contemporary logic, the logic that had its origin in the works of George Boole and Gottlob Frege and others in the mid‐to‐late nineteenth century. Boolean algebra, the heart of Boole's contributions to logic, has also come to represent a cornerstone of modern computing. Frege had (...) broad philosophical interests, and his writings on the nature of logical form, meaning and truth remain the subject of intense theoretical discussion, especially in the analytic tradition. Frege's works, and the powerful new logical calculi developed at the end of the nineteenth century, influenced many of its most seminal figures, such as Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap. (shrink)

This paper discusses certain problems arising within the treatment of the senses of functions in Alonzo Church's Logic of Sense and Denotation. Church understands such senses themselves to be "sense-functions," functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or "deviant" sense-functions. Certain absurdities result, e.g., an argument can be found for equating any two (...) senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege's original theory. (shrink)

Russell claims in his autobiography and elsewhere that he discovered his 1905 theory of descriptions while attempting to solve the logical and semantic paradoxes plaguing his work on the foundations of mathematics. In this paper, I hope to make the connection between his work on the paradoxes and the theory of descriptions and his theory of incomplete symbols generally clearer. In particular, I argue that the theory of descriptions arose from the realization that not only can a class not be (...) thought of as a single thing, neither can the meaning/intension of any expression capable of singling out one collection (class) of things as opposed to another. If this is right, it shows that Russell’s method of solving the logical paradoxes is wholly incompatible with anything like a Fregean dualism between sense and reference or meaning and denotation. I also discuss how this realization lead to modifications in his understanding of propositions and propositional functions, and suggest that Russell’s confrontation with these issues may be instructive for ongoing research. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of arithmetical (...) concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)