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History and Philosophy of Logic and Mathematics
 
 
 

The Arché Research Group on History and Philosophy of Logic and Mathematics organises two weekly seminars and regular workshops. It is led by Professor Stephen Read, with the help of Professor Gabriel Uzquiano and Dr Mark Thakkar, and includes doctoral students working in the history and philosophy of logic and mathematics.

Seminars:

HPLM Seminar, every Tuesday (during Arché semesters) 3 – 5pm in Edgecliffe room G03. This is a forum for series of seminars discussing a recent book (or series of articles) of topical interest, and for talks by members of the group or by visitors. For details of the current programme, see the Arché Calendar.
MLRG (Medieval Logic Reading Group), every Friday (during University semesters) 10.15 - 11.45 in the Arché Seminar Room. This is a forum for reading (usually in English translation) through a recently published text on medieval logic, or simiar material. For details of the current programme, see MLRG webpage.

Workshops:

November 2015: Inferentialism

June 2016: Cardinality, Worlds and Paradox Workshop

May 2017: Probationes Propositionum

Current Research:

Stephen Read (Professor Emeritus): My future research will focus on three main areas: 1) Inferentialism and proof-theoretic semantics, in particular, exploring the possibilities of extending an inferentialist account of meaning to non-logical and categorematic terms, and investigating the historical antecedents of non-denotational accounts of meaning in the middle ages; 2) the logical paradoxes, in particular, extending the “multiple-meanings” solution (due to Bradwardine) to other paradoxes, of signification, of knowledge, of properties and of propositions; 3) editing and translating texts on the logical paradoxes by Paul of Venice, Walter Segrave and John Dumbleton.

Kevin Scharp (Reader): The areas of philosophy on which I focus are logic, philosophy of language, metaphysics, philosophy of science, and the history of analytic philosophy. Much of my work concerns the concept of truth and the paradoxes to which it gives rise, like the liar paradox. I argue that truth is best understood as an inconsistent concept, and I propose a detailed theory of inconsistent concepts that can be used to show exactly how the liar and other paradoxes derive from the concept of truth. Truth also happens to be a useful concept, but its inconsistency inhibits its utility; as such, it should be replaced with consistent concepts that can do truth’s job without giving rise to paradoxes. I offer a pair of replacements, which I dub ascending truth and descending truth, along with an axiomatic theory of them and a new kind of possible-worlds semantics for this theory. As for the nature of truth, I develop Davidson’s idea that it is best understood as the core of a measurement system for rational phenomena (e.g., belief, desire, and meaning). Finally, I propose a semantic theory that treats truth predicates as assessment-sensitive (i.e., their extension is relative to a context of assessment), and demonstration of how this theory solves the problems posed by the liar and other paradoxes.

I also write about realism and fundamentality in metaphysics, measurement in philosophy of science, and figures like Sellars, Quine, and Davidson in the history of analytic philosophy. I'm currently working on several projects, including: (i) a short monograph on the semantics for reasons locutions (e.g., safety is a reason for Sarah to drive slowly), (ii) an account of philosophical methodology that emphasizes engineering new concepts to address perennial philosophical problems, and (iii) a novel version of atheism that utilizes the resources of formal epistemology. 

Gabriel Uzquiano (Professorial Fellow): I plan to do more research on the intensional paradoxes (Prior, Kripke, Kaplan) as well as ramifications of Cantorian arguments for the prospects of absolute generality. Some of this work will involve, on the one hand, a clarification of the role impredicativity is supposed to play in the arguments, and, on the other, an exploration of the availability of a higher-order logic as a framework for regimenting our talk of properties and propositions. In general, I’m interested in the interplay between logic and metaphysics as exemplified, for example, by the recent debate between necessitism and contingentism in modal metaphysics.

Mark Thakkar (Leverhulme Early Career Fellow): I primarily work on 14th-century philosophy, and my current project is a critical edition and translation of two logical works written by John Wyclif in the 1360s. These works obviously contain material that should be of interest to historians of logic, but what is surprising is that the larger and more advanced work also contains some passages that may be of interest to historians of mathematics.

Matthew Cameron (doctoral student): My research focuses primarily on understanding the scope, aims and criteria of a theory of natural language semantics from the perspective of generative grammar, and its relation to theories of communication, speech acts and other aspects of mind and cognition. I'm particularly interested in methodological issues surrounding formal treatments of semantics in relation to context sensitivity. Currently, I'm investigating the syntax-semantics interface, especially the notion of logical form and the status of covert syntactic structure.

Ryo Ito (doctoral student): I have been working on Russell's conception of logic. In spite of his status as a founder of mathematical logic, his remarks on logic may well appear naive to the contemporary reader, as they often involve metaphysical/ontological notions which we find unnecessary for and irrelevant to logic. My thesis is aimed at explaining why he invokes such notions to offer an account of logic, thereby indicating how his conception of logic differs from ways in which we, contemporary philosophers, understand logic. Although my main research is thus an exegetical one, I am also interested in the contemporary philosophy of mathematics, hoping to obtain some useful insights from Russell's work on logicism and in particular his debate with Poincare.

Hasen Khudairi (doctoral student): My research is primarily in the philosophies of mind and mathematics, with an emphasis on the nature of epistemic modality. Topics on which I have written include the nature and significance of mathematical modality; the role of modal algebras and coalgebras in countenancing the nature of mental representation; the modal profile of $\Omega$-logic; the nature of indefinite extensibility; epistemicist approaches to the resolution of the alethic paradoxes; knowledge of logic; social choice theory; as well as a range of issues in the epistemology and ontology of mathematics.

Alessandro Rossi (doctoral student): My current research is about the relation between modal logic and modal metaphysics. My interests in modal metaphysics lie in the actualism vs possibilism  and necessitism vs contingentism disputes. On the other hand, the area of modal logic I am primarily concerned with is its model theory. In particular, I am working on a model-theoretic account of modal logic in which two notions of validity are simultaneously at work. The first one, called ‘Sigma’ or ‘Weak’ validity, refers to all possible worlds of any model excluding the actual world, whereas the second one – labelled ‘Strong Validity’ -  refers to all worlds simpliciter – hence, including the actual world. The main philosophical  idea governing this account of modal logic, currently at an early stage,  may be said to consist in providing actualism with a semantics assigning to the actual world a somewhat model-theoretically privileged role.   Moreover, I have strong interests in the philosophical significance of plural logic as well as in the question of its semantic determinacy.  

Roy Dyckhoff (Honorary Senior Lecturer in Computer Science): For the next few months I'll be looking at a combination of reasoning in coherent logic with the use of coherent logic for handling accessibility conditions for modal and intermediate logics, with special reference to termination conditions, and their automation. In general I am interested in constructive (and thus computational) approaches to logic and mathematics (and related topics such as formal linguistics), especially via Gentzen-style proof theory.