The Evolution of Principia Mathematica; Bertrand Russell's Manuscripts and Notes for the Second Edition

This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ‘inferential definitions’ of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logical constants rather than as a semantic foundation for logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations.     

Typos of Principia Mathematica

Principia Mathematic goes to great lengths to hide its order/type indices and to make it appear as if its incomplete symbols behave as if they are singular terms. But well-hidden as they are, we cannot understand the proofs in Principia unless we bring them into focus. When we do, some rather surprising results emerge – which is the subject of this paper.     

The Versatility of Universality in Principia Mathematica

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed.     

Quantification Theory in *9 of Principia Mathematica

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition.     

The Definability of the set of natural numbers in the 1925 Principia Mathematica

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Gödel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered.     

Quantification Theory in *8 of Principia Mathematica and the Empty Domain

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell successfully antedates Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain.     

罗素悖论的预言者——施罗德与策梅罗

罗素悖论是逻辑学中极为重要的一个悖论,它对现代逻辑的推进和完善起到了关键的作用.但学界对罗素悖论提出的背景和逻辑学界的研究状况了解不多.在罗素悖论提出之前,逻辑学家施罗德、策梅罗等就已经展开了对罗素悖论及类型理论雏形的讨论.不少逻辑学家和哲学家,如弗雷格、罗素、胡塞尔和维纳等都对这两人的贡献进行了评价与分析.     

On the Continuity and Origin of Identity in Distributed Ledgers: Learning from Russell's Paradox

This article studies the origin and continuity of the identity of the entities inscribed in a distributed ledger. Specifically, it focuses on the differences between the identities of the entities that exist in a distributed ledger and those of the entities that exist outside the ledger but must be represented in the ledger in order to interact with it. It suggests that a distributed ledger that contains representations of entities that exist outside the ledger can yield a continuum of interconnected existing and past identities that is constantly redefined to represent new conceptual entities. This continuum can be understood as a metasortal—or a sortal of sortals—that resembles the mathematical structure of a set of sets. Further, the article presents the dilemma that arises when representing the identities of entities in a distributed ledger, and it draws an analogy between this dilemma and Russell's Paradox.     

罗素《哲学问题》序言

罗素的《哲学问题》是几代哲学学生的必读之物。在该书中,他并没有探讨所有的哲学问题,而仅限于他认为自己可以肯定而且能有所建树的那些问题,他主要涉及的是知识论,即考察我们能说知道或有理由相信的那部分哲学分支。在此基础上,他得出了某些令人瞩目的有关所有事物的终极类别的结论。他并没有探讨伦理学以及有关心灵和行为的各种经典问题,如自我的本性或意志自由等问题。     

The No-No Paradox Is a Paradox

The No-No Paradox consists of a pair of statements, each of which ‘says’ the other is false. Roy Sorensen claims that the No-No Paradox provides an example of a true statement that has no truthmaker: Given the relevant instances of the T-schema, one of the two statements comprising the ‘paradox’ must be true (and the other false), but symmetry constraints prevent us from determining which, and thus prevent there being a truthmaker grounding the relevant assignment of truth values. Sorensen's view is mistaken: situated within an appropriate background theory of truth, the statements comprising the No-No Paradox are genuinely paradoxical in the same sense as is the Liar (and thus, on Sorensen's view, must fail to have truth values). This result has consequences beyond Sorensen's semantic framework. In particular, the No-No Paradox, properly understood, is not only a new paradox, but also provides us with a new type of paradox, one which depends upon a general background theory of the truth predicate in a way that the Liar Paradox and similar constructions do not.     

The Self in Pain: The Paradox of Memory. The Paradox of Testimony

Using the 7-year psychotherapy of a Holocaust survivor, this paper explores the sometimes contradictory aspects of approaches to trauma. Conceptualizing a “self in pain” as an alternative to contemporary conceptualizations of the traumatized person as having a damaged, dissociated or collapsed self leads to a corresponding alternative clinical approach. The paradoxes of traumatic memory and testimony necessitate an adaptational emphasis and the emergence of a “doubled” in contrast to a dissociated self. The decision to respect this “doubled” self involves a privileging of “reality” over “psychic reality” which then, paradoxically enables this patient to develop a phantasy life.     

The ontological foundation of Russell's theory of modality

Prominent thinkers such as Kripke and Rescher hold that Russell has no modal logic, even that Russell was indisposed toward modal logic. In Part I, I show that Russell had a modal logic which he repeatedly described and that Russell repeatedly endorsed Leibniz's multiplicity of possible worlds. In Part II, I describe Russell's theory as having three ontological levels. In Part III, I describe six Parmenidean theories of being Russell held, including: literal in 1903; universal in 1912; timeless in 1914; transcendental in 1918–1948. The transcendental theory underlies the primary level of Russell's modal logic. In Part IV, I examine Rescher's view that Russell and modal logic did not mix.The United States Naval Academy Research Council kindly provided a summer 1988 research grant for work on this essay. This essay was presented at the Bertrand Russell Society Meeting during the December 1988 Eastern Division Meeting of the American Philosophical Association in Washington, D.C.