A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...) of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper (...) are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume's Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege's Theorem. (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Introduction to a special issue based on a summer school on set theory and high-order logic.

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's (...) Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)

In this paper, I develop an argument for the thesis that ‘maximality is extrinsic’, on which a whole physical object is not a whole of its kind in virtue of its intrinsic properties. Theodore Sider has a number of arguments that depend on his own simple argument that maximality is extrinsic. However, Peter van Inwagen has an argument in defence of his Duplication Principle that, I will argue, can be extended to show that Sider's simple argument fails. However, van Inwagen's (...) argument fails against a more complex, sophisticated argument that maximality is extrinsic. I use van Inwagen's own commitments to various forms of causation and metaphysical possibility to argue that maximality is indeed extrinsic, although not for the mundane reasons that Sider suggests. I then argue that moral properties are extrinsic properties. Two physically identical things can have different moral properties in a physical world. This argument is a counterexample to a classical ethical supervenience idea (often attributed to G.E. Moore) that if there is identity of physical properties in a physical world, then there is identity in moral properties as well. I argue moral value is ‘border sensitive’ and extrinsic for Kantians, utilitarians, and Aristotelians. (shrink)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...) relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (shrink)

A semantics for quantified modal logic is presented that is based on Kleene's notion of realizability. This semantics generalizes Flagg's 1985 construction of a model of a modal version of Church's Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church's Thesis and a variant of a modal set theory (...) due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra's generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott's graph model (for the untyped lambda calculus) which witnesses the failure of the stability of non-identity. (shrink)

David Lewis insists that restrictivist composition must be motivated by and occur due to some intuitive desiderata for a relation R among parts that compose wholes, and insists that a restrictivist’s relation R must be vague. Peter van Inwagen agrees. In this paper, I argue that restrictivists need not use such examples of relation R as a criterion for composition, and any restrictivist should reject a number of related mereological theses. This paper critiques Lewis and van Inwagen (and others) on (...) their respective mereological metaphysics, and offers a Golden Mean between their two opposite extremes. I argue for a novel account of mereology I call Modal Mereology that is an alternative to Classical Mereology. A modal mereologist can be a universalist about the possible composition of wholes from parts and a restrictivist about the actual composition of wholes from parts. I argue that puzzles facing Modal Mereology (e.g., puzzles concerning Cambridge changes and the Problem of the Many, and how to demarcate the actual from the possible) are also faced in similar forms by classical universalists. On my account, restricted composition is rather motivated by and occurs due to a possible whole’s instantiating an actual type. Universalists commonly believe in such types and defend their existence from objections and puzzles. The Modal Mereological restrictivist can similarly defend the existence of such types (adding that such types are the only wholes) from similar objections and puzzles. (shrink)

Aristotle’s best human life is attained through theoretical contemplation, and Confucius’ is attained through practical cultivation of the social self. However, I argue that in the best human life for both Confucius and Aristotle, a form of theoretical contemplation must occur and can only occur with an ethical commitment to community life. Confucius, like Aristotle, sees that the best contemplation comes after later-life, greater-learning and is central to ethical and community life. Aristotle, like Confucius, sees the best contemplation as presupposing (...) full ethical commitment to community life. So, I argue for the theses that: on Aristotle’s view, the best human contemplation requires one be fully morally good; on Confucius’ view, to be fully morally good requires the best human contemplation; being fully morally good for both requires commitment to the good of others and the community. (shrink)

Many philosophers believe that a moral theory, given all the relevant facts, should be able to determine what is morally right and wrong. It is commonly argued that Aristotle’s ethical theory suffers from a fatal flaw: it places responsibility for determining right and wrong with the virtuous agent who has phronesis rather than with the theory itself. It is also commonly argued that Immanuel Kant’s ethical theory does provide a concept of right that is capable of determining right and wrong (...) in specific cases. I argue, however, that Kant never gives a determinate moral theory of right. Rather, I argue that Kant’s moral theory is similar in many ways to that of Aristotle, in that it still holds that a moral agent with phronesis, rather than the theory, determines what is right. Kant’s practical philosophy was not so much meant to tell us right and wrong as to prevent bad moral theory from corrupting our moral common sense, and it is our moral common sense that determines right and wrong naturally. (shrink)

Normal 0 false false false EN-US X-NONE X-NONE Normal 0 false false false EN-US ZH-TW X-NONE It is commonly believed that impartial utilitarian moral theories have significant demands that we help the global poor, and that the partial virtue ethics of Mencius and Aristotle do not. This ethical partiality found in these virtue ethicists has been criticized, and some have suggested that the partialistic virtue ethics of Mencius and Aristotle are parochial (i.e., overly narrow in their scope of concern). I (...) believe, however, that the ethics of Mencius and Aristotle are both more cosmopolitan than many presume and also are very demanding. In this paper, I argue that the ethical requirements to help the poor and starving are very demanding for the quintessentially virtuous person in Mencius and Aristotle. The ethical demands to help even the global poor are demanding for Mencius’ jun-zi ( 君子 chün-tzu / junzi ) and Aristotle’s megalopsuchos . I argue that both the jun-zi and megalopsuchos have a wide scope of concern for the suffering of poor people. I argue that the relevant virtues of the jun-zi and megalopsuchos are also achievable for many people. The moral views of Mencius and Aristotle come with strong demands for many of us to work harder to alleviate global poverty. . (shrink)

Two puzzles with regard to the Kritik der reinen Vernunft (KrV) are incongruent counterparts and causality. In De mundi sensibilis atque intelligibilis forma et principiis (MSI), Kant indicates that the experience of things like left and right hands, so-called incongruent counterparts, involve certain pure intuitions, and hence constitute one line of evidence for the claim that the concept of space itself is a pure intuition. In KrV, Kant again argues that the concept of space itself is a pure intuition, but (...) does not cite the experience of incongruent counterparts as evidence for this claim. Since there is ostensibly nothing in KrV which tells against the existence of the experiences of incongruent counterparts, the natural question is: “Why, in KrV, does Kant not cite the experience of incongruent counterparts as evidence for the claim that the concept of space is a pure intuition?” The problem with causality is as follows. One of the most primary and basic claims of KrV is that empirical experience is structured by non-empirical concepts, such as substantiality and causality. In a portion of KrV entitled the Transcendental Deduction, Kant gives section 20 the heading “All sensible intuitions stand under the categories, as conditions under which alone their the manifold can come together in one consciousness”. Since categories are those concepts which structure empirical experience, section 20 has demonstrated that all sensible intuitions are subject to substantiality and causality. However, there is a later portion of KrV entitled the Second Analogy with the heading “Principle of temporal sequence according to the law of causality: All alterations occur in accordance with the law of the connection of cause and effect”. Thus, the natural question is: “What does the Second Analogy demonstrate about causality which the Transcendental Deduction did not already demonstrate about causality?”. (shrink)

In this book, Sean Noah Walsh applies Herbert Marcuse’s observations on counterrevolution to recent developments in education politics. Seemingly disparate issues such as the exercise of state power to reorganize curricula, the derision of intellectuals, the permeation of consumerism into the collegiate experience, and the expansion of online teaching belong to the same strategy in which the faculties of dissent are neutralized before they can develop and dissent is established as the paramount political obscenity.

Many philosophers believe that a moral theory, given all the relevant facts, should be able to determine what is morally right and wrong. It is commonly argued that Aristotle’s ethical theory suffers from a fatal flaw: it places responsibility for determining right and wrong with the virtuous agent who has phronesis rather than with the theory itself. It is also commonly argued that Immanuel Kant’s ethical theory does provide a concept of right that is capable of determining right and wrong (...) in specific cases. I argue, however, that Kant never gives a determinate moral theory of right. Rather, I argue that Kant’s moral theory is similar in many ways to that of Aristotle, in that it still holds that a moral agent with phronesis, rather than the theory, determines what is right. Kant’s practical philosophy was not so much meant to tell us right and wrong as to prevent bad moral theory from corrupting our moral common sense, and it is our moral common sense that determines right and wrong naturally. (shrink)

It is commonly believed that impartial utilitarian moral theories have significant demands that we help the global poor, and that the partial virtue ethics of Mencius and Aristotle do not. This ethical partiality found in these virtue ethicists has been criticized, and some have suggested that the partialistic virtue ethics of Mencius and Aristotle are parochial. I believe, however, that the ethics of Mencius and Aristotle are both more cosmopolitan than many presume and also are very demanding. In this paper, (...) I argue that the ethical requirements to help the poor and starving are very demanding for the quintessentially virtuous person in Mencius and Aristotle. The ethical demands to help even the global poor are demanding for Mencius’ jun-zi and Aristotle’s megalopsuchos. I argue that both the jun-zi and megalopsuchos have a wide scope of concern for the suffering of poor people. I argue that the relevant virtues of the jun-zi and megalopsuchos are also achievable for many people. The moral views of Mencius and Aristotle come with strong demands for many of us to work harder to alleviate global poverty. (shrink)

This book critically examines Leo Strauss s claim that the philosophers of antiquity, especially Plato, wrote esoterically, hiding the highest truths exclusively between the lines.