Logblog: Richard Zach's Logic Blog
In February of last year, BIRS had an amazing workshop on "Mathematical Methods in Philosophy". We (i.e., Aldo Antonelli, Alasdair Urquhart, and I) collected some of the very exciting contributions from that workshop in a Special Issue of the new Review of Symbolic Logic, and that issue is now online! We even managed to get a nice picture of the participants into the Introduction.
As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.
Just as set theory can be divorced from Ernst Zermelo's original axiomatization of it, counterpart theory can be divorced from the eight postulates that were originally stipulated by David Lewis (1968, p. 114) to constitute it. These were postulates governing some of the properties and relations holding among possible worlds and their inhabitants. In particular, counterpart theory can be divorced from Lewis's postulate P2, the stipulation that individuals are ‘world bound’—that none exists in more than one possible world.
In standard modal logics, the worlds are 2-valued in the following sense: there are 2 values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case. The worlds could be many-valued. This paper presents one simple approach to a major family of many-valued modal logics, together with an illustration of why this family is philosophically interesting.
A new formal theory DT of truth extending PA is introduced, whose language is that of PA together with one new unary predicate symbol T (x), for truth applied to Gödel numbers of suitable sentences in the extended language. Falsity of x, F(x), is defined as truth of the negation of x; then, the formula D(x) expressing that x is the number of a determinate meaningful sentence is defined as the disjunction of T(x) and F(x). The axioms of DT are those of PA extended by (I) full induction, (II) strong compositionality axioms for D, and (III) the recursive defining axioms for T relative to D. By (II) is meant that a sentence satisfies D if and only if all its parts satisfy D; this holds in a slightly modified form for conditional sentences. The main result is that DT has a standard model. As an improvement over earlier systems developed by the author, DT meets a number of leading criteria for formal theories of truth that have been proposed in the recent literature and comes closer to realizing the informal view that the domain of the truth predicate consists exactly of the determinate meaningful sentences.
We introduce an epistemic theory of truth according to which the same rational degree of belief is assigned to Tr(???) and ?. It is shown that if epistemic probability measures are only demanded to be finitely additive (but not necessarily ?-additive), then such a theory is consistent even for object languages that contain their own truth predicate. As the proof of this result indicates, the theory can also be interpreted as deriving from a quantitative version of the Revision Theory of Truth.
Stephen Read (2002, 2006) has recently discussed Bradwardine's theory of truth and defended it as an appropriate way to treat paradoxes such as the liar. In this paper, I discuss Read's formalisation of Bradwardine's theory of truth and provide a class of models for this theory. The models facilitate comparison of Bradwardine's theory with contemporary theories of truth.
There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.
First, a brief historical trace of the developments in confirmation theory leading up to Goodman’s infamous “grue” paradox is presented. Then, Goodman’s argument is analyzed from both Hempelian and Bayesian perspectives. A guiding analogy is drawn between certain arguments against classical deductive logic, and Goodman’s “grue” argument against classical inductive logic. The upshot of this analogy is that the “New Riddle” is not as vexing as many commentators have claimed (especially, from a Bayesian inductive-logical point of view). Specifically, the analogy reveals an intimate connection between Goodman’s problem, and the “problem of old evidence”. Several other novel aspects of Goodman’s argument are also discussed (mainly, from a Bayesian perspective).
This is a paper about the constituents of arguments. It argues that several different kinds of truth-bearer may be taken to compose arguments, but that none of the obvious candidates—sentences, propositions, sentence/truth-value pairs etc.—make sense of logic as it is actually practiced. The paper goes on to argue that by answering the question in different ways, we can generate different logics, thus ensuring a kind of logical pluralism that is different from that of J. C. Beall and Greg Restall.
From the Introduction:
Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics. The theory and meta-theory of formal systems were pioneered by philosophers and philosophically minded mathematicians (Frege, Russell, Hilbert, Gödel, Tarski, among many others), and philosophers have continued to be significantly involved in the technical development of proof theory and to a certain degree also in the development of model theory and set theory. On the other hand, philosophers use formal models to test the implications of their theories in tractable cases. Philosophical inquiry can also uncover new mathematical structures and problems, as with recent work on paradoxes about truth. Areas outside mathematical logic have also been important in recent philosophical work, for example, probability and game theory in inductive logic, epistemology, and the philosophy of science. Formal epistemology is an emerging field of research in philosophy, encompassing formal approaches to ampliative inference (including inductive logic), game theory, decision theory, computational learning theory, and the foundations of probability theory.
It in fact seems that technical mathematical work is currently enjoying something of a renaissance in philosophy. And so the idea of a workshop on just such topics held at a conference center for the mathematical sciences was developed. From February 18–23, 2007, 40 researchers who apply mathematical methods to current issues in philosophy congregated at the Banff International Research Station (BIRS) in the Canadian Rockies for a workshop on ‘Mathematical Methods in Philosophy’.
These mathematical methods come mainly from the fields of mathematical logic and probability theory, and the areas of application include philosophical logic, metaphysics, epistemology, philosophy of mathematics, and philosophy of science. It is a fortuitous coincidence that the Association of Symbolic Logic now has a third journal, The Review of Symbolic Logic, the scope of which more or less covers the topics of the BIRS Workshop, and so it is only fitting that this special issue of the Review collects some of the papers presented at the workshop.