Empiricism,probability, and knowledge of arithmetic: A preliminary defense

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 judgments 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 (Section 2.1), while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic (Section 2.2). 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 ω-rule (Section 3.1) or to the non-computability and non-continuity of probability assignments (Section 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments (Appendix A–Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4 Appendix B). In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar.     

Natural language semantics in biproduct dagger categories

Biproduct dagger categories serve as models for natural language. In particular, the biproduct dagger category of finite dimensional vector spaces over the field of real numbers accommodates both the extensional models of predicate calculus and the intensional models of quantum logic. The morphisms representing the extensional meanings of a grammatical string are translated to morphisms representing the intensional meanings such that truth is preserved. Pregroup grammars serve as the tool that transforms a grammatical string into a morphism. The chosen linguistic examples concern negation, relative noun phrases, comprehension and quantifiers.     

Can Hardcore Actualism Validate S5?

Hardcore actualism (HA) grounds all modal truths in the concrete constituents of the actual world (see, e.g., Borghini and Williams (2008), Jacobs (2010), Vetter (2015)). I bolster HA, and elucidate the very nature of possibility (and necessity) according to HA, by considering if it can validate S5 modal logic. Interestingly, different considerations pull in different directions on this issue. To resolve the tension, we are forced to think hard about the nature of the hardcore actualist’s modal reality and how radically this departs from possible worlds orthodoxy. Once we achieve this departure, the prospects of a hardcore actualist validation of S5 look considerably brighter. This paper thus strengthens hardcore actualism by arguing that it can indeed validate S5–arguably the most popular logic of metaphysical modality–and, in the process, it elucidates the very nature of modality according to this revisionary, but very attractive, modal metaphysics.     

科学发展与文化逻辑--从杨振宁论中国无科学的观点说起

每一个文明都有自己存在的理由和发展的逻辑线路,不同文明可以有不同的发展程度,但不能有凌驾于其他文明之上的话语解释权.近代科学作为近代西方文明的产物,也同时是古代多种文明融合的产物.科学发展离不开自身所属的文化逻辑,但反过来具备同等文化逻辑的文明未必就一定产生科学.科学作为文化的一种形态,它没有必要成为所有文明的必经之路.我们在尊重、发展作为现代化工具的科学技术的同时,要反对用科学话语来解释工具之上的东西.     

Philosophical foundations of partial belief models

This paper is an attempt to put forward a new kind of partial model for representing belief states. I first introduce some philosophical motivations for working with partial models. Then, I present the standard (total) model proposed by Hintikka, and the partial models studied by Humberstone and Holliday. I then show how to reduce Hintikka’s semantics in order to obtain a partial model which, however, differs from Humberstone’s and Holliday’s. The nature of such differences is assessed, and I provide motivations for using the newly proposed semantics rather than the existing ones. Finally, I review some promising philosophical applications of the ideas developed throughout the discussion.     

The early Russell on the metaphysics of substance in Leibniz and Bradley

While considerable ink has been spilt over the rejection of idealism by Bertrand Russell and G.E. Moore at the end of the 19th Century, relatively little attention has been directed at Russell’s A Critical Exposition of the Philosophy of Leibniz, a work written in the early stages of Russell’s philosophical struggles with the metaphysics of Bradley, Bosanquet, and others. Though a sustained investigation of that work would be one of considerable scope, here I reconstruct and develop a two-pronged argument from the Philosophy of Leibniz that Russell fancied—as late as 1907—to be the downfall of the traditional category of substance. Here, I suggest, one can begin to see Russell’s own reasons—arguments largely independent of Moore—for the abandonment of idealism. Leibniz, no less than Bradley, adhered to an antiquated variety of logic: what Russell refers to as the subject-predicate doctrine of logic. Uniting this doctrine with a metaphysical principle of independence—that a substance is prior to and distinct from its properties—Russell is able to demonstrate that neither a substance pluralism nor a substance monism can be consistently maintained. As a result, Russell alleges that the metaphysics of both Leibniz and Bradley has been undermined as ultimately incoherent. Russell’s remedy for this incoherence is the postulation of a bundle theory of substance, such that the category of “substance” reduces to the most basic entities—properties.     

Every Finitely Reducible Logic has the Finite Model Property with Respect to the Class of ♦-Formulae

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators.