Kripke’s paradox and the Church–Turing thesis

Kripke (1982, Wittgenstein on rules and private language. Cambridge, MA: MIT Press) presents a rule-following paradox in terms of what we meant by our past use of “plus”, but the same paradox can be applied to any other term in natural language. Many responses to the paradox concentrate on fixing determinate meaning for “plus”, or for a small class of other natural language terms. This raises a problem: how can these particular responses be generalised to the whole of natural language? In this paper, I propose a solution. I argue that if natural language is computable in a sense defined below, and the Church–Turing thesis is accepted, then this auxiliary problem can be solved.     

Sequent Calculi for Some Strict Implication Logics

We introduce various sequent systems for propositional logicshaving strict implication, and prove the completeness theoremsand the finite model properties of these systems.The cut-eliminationtheorems or the (modified) subformula properties are provedsemantically.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

Basic Logic,K4, and Persistence

We characterize the first-order formulas with one free variable that are preserved under bisimulation and persistence or strong persistence over the class of Kripke models with transitive frames and unary persistent predicates. This revised version was published online in June 2006 with corrections to the Cover Date.     

Looking for Whitehead

Saul Kripke in his revolutionary and influential series of lectures from the early 1970s (later published as the book Naming and Necessity) famously resurrected John Stuart Mill's theory of proper names. Kripke at the same time rejected Mill's theory of general terms. According to Kripke, many natural kind terms do not fit Mill's account of general terms and are closer to proper names. Unfortunately, Kripke and his followers ignored key passages in Mill's A System of Logic in which Mill enunciates a sophisticated and detailed theory of natural kind terms that anticipates and is in some ways superior to Kripke's.     

Paradigm Terms: The Necessity of Kind Term Identifications Generalized

Standard Kripke-Putnam semantics is widely taken to entail that theoretical identifications like ‘Brontosauruses are Apatosauruses’ or ‘Gold is ⁷⁹Au’ are necessary, if true. I offer a new diagnosis as to why this modal consequence ensues. Central to my diagnosis is the concept of a paradigm term. I argue that modal and epistemic peculiarities that are commonly considered as distinctive of natural kind expressions are in fact traits that are shared by paradigm terms in general. Philosophical semantics should broaden its focus from natural kind expressions to paradigm terms.     

Thomas Kuhn's Misunderstood Relation to Kripke-Putnam Essentialism

Kuhn's ‘taxonomic conception’ of natural kinds enables him to defend and re-specify the notion of incommensurability against the idea that it is reference, not meaning/use, that is overwhelmingly important. Kuhn's ghost still lacks any reason to believe that referentialist essentialism undercuts his central arguments in SSR – and indeed, any reason to believe that such essentialism is even coherent, considered as a doctrine about anything remotely resembling our actual science. The actual relation of Kuhn to Kripke-Putnam essentialism, is as follows: Kuhn decisively undermines it – drawing upon the inadequacies of such essentialism when faced with the failure of attempts to instantiate in history or contemporaneously its ‘thought-experiment’ – and leaves the field open instead for his own more ‘realistic’, deflationary way of thinking about the operation of ‘natural kinds’ in science. This revised version was published online in August 2006 with corrections to the Cover Date.