Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.

Hossack's 'The Metaphysics of Knowledge' develops a theory of facts, entities in which universals are combined with universals or particulars, as the foundation of his metaphysics. While Hossack argues at length that there must be negative facts, facts in which the universal 'negation' is combined with universals or particulars, his conclusion that there are also general facts, facts in which the universal 'generality' is combined with universals, is reached rather more swiftly. In this paper I present Hossack with three arguments (...) for his conclusion. They all draw, as does Hossack's theory of facts, on views Russell expressed in various writings. Two arguments are based on Russell's explanation of universals as aspects of resemblance; the third on Russell's observation that general propositions do not follow logically from exclusively particular premises. Comparison with other metaphysics of generality show them to be wanting and Russell's and Hossack's accounts superior. (shrink)

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a (...) new hierarchical account that solves the problem. (shrink)

In this paper, I will show to what extent we can use our modern understanding of the Square of Opposition in order to make sense of Kant 's double standard solution to the cosmological antinomies. Notoriously, for Kant, both theses and antitheses of the mathematical antinomies are false, while both theses and antitheses of the dynamical antinomies are true. Kantian philosophers and interpreters have criticized Kant 's solution as artificial and prejudicial. In the paper, I do not dispute such claims, (...) but I show that our modern understanding of the Square of Opposition enables us to more naturally deliver the result Kant was aiming at. Accordingly, the paper does not pretend to be exegetically accurate. It is an attempt to revise the antinomies with the help of standard classical logic. And although such a revision entails some re-interpretation, in the end, it will actually help to unveil some of Kant 's thoughts. (shrink)

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. (shrink)

Although resolution-based inference is perhaps the industry standard in automated theorem proving, there have always been systems that employed a different format. For example, the Logic Theorist of 1957 produced proofs by using an axiomatic system, and the proofs it generated would be considered legitimate axiomatic proofs; Wang’s systems of the late 1950’s employed a Gentzen-sequent proof strategy; Beth’s systems written about the same time employed his semantic tableaux method; and Prawitz’s systems of again about the same time are often (...) said to employ a natural deduction format. [See Newell, et al (1957), Beth (1958), Wang (1960), and Prawitz et al (1960)]. Like sequent proof systems and tableaux proof systems, natural deduction systems retain.. (shrink)

It has been suggested that quantum particles are genuinelyvague objects (Lowe 1994a). The present work explores thissuggestion in terms of the various metaphysical packages that areavailable for describing such particles. The formal frameworksunderpinning such packages are outlined and issues of identityand reference are considered from this overall perspective. Indoing so we hope to illuminate the diverse ways in whichvagueness can arise in the quantum context.

We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...) full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory. (shrink)

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going (...) to discover them by analysing the content of the book. (shrink)

If the words in Spinoza’s Ethics are considered as symbols, then certain words in the definitions of the Ethics can be replaced with symbols from set theory and we can reexamine Spinoza’s first definitions within a logical–mathematical frame. The authors believe that, some aspects of Spinoza’s work can be explained and illustrated through mathematics. A semantic relation between the definitions of the philosopher and set theory is presented. It is explained each chosen symbol. The ontological argument is developed through modal (...) logic. And finally, we present some conclusions drawn from this work. (shrink)

Wittgenstein's Tractatus carefully distinguished the concept all from\nthe notion of a truth-function, and thereby from the quantifiers.\nI argue that Wittgenstein's rationale for this distinction is lost\nunless propositional functions are understood within the context\nof his picture theory of the proposition. Using a model Tractatus\nlanguage, I show how there are two distinct forms of generality implicit\nin quantified Tractatus propositions. Although the explanation given\nin the Tractatus for this distinction is ultimately flawed, the distinction\nitself is a genuine one, and the forms of generality that (...) Wittgenstein\nindicated can be seen in the quantified sentences of contemporary\nlogic. (shrink)

This is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the (...) Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here. (shrink)

In , Frank Ramsey separates paradoxes into two groups, now taken to be the logical and the semantical. But he also revises the logical system developed in Whitehead and Russellthe intensional paradoxess interest in these problems seriously, then the intensional paradoxes deserve more widespread attention than they have historically received.

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...) when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’. (shrink)