Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N → ∞ and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is O (N ). The model problems are integrated numerically in the stiff regime where the time-step t satisfies N t = O(1). The convergence of the algorithms is studied in this case in the limit N → ∞ and t → 0. For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically.

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For over thirty years a monster has lumbered through the philosophical landscape laying waste to naturalistic, broadly dispositional approaches to rule-following, meaning, and content. The monster is terrifying; the mere mention of his name is often enough to inspire retreat. If the forces of naturalism are ultimately to prevail, Kripkenstein's monster must be killed, once and for all. Here I attempt this dangerous and perhaps foolhardy task. But why now? Couldn't the monster have been killed back in the eighties? In principle: yes. But in reality: perhaps not. Many attacks were launched, and while some did significant damage, none fully succeeded. Additionally, several adventurers have rallied to the monster's cause in the intervening years, bolstering his already formidable attacks. And let us not dwell on the many raiding parties forever lost in the bog of Wittgenstein exegesis surrounding the monster's lair.Fantastical analogies aside, here I defend dispositionalism in the face of Kripke's influential anti-dispositionalist arguments in Wittgenstein on Rules and Private Language. 1 Leaving aside both Wittgenstein exegesis and the "sceptical" solution, I will focus only on showing how dispositionalism, broadly construed, withstands Kripke's attacks. Two of Kripke's arguments-those concerning finitude and error-purport to show that dispositionalism has problems extensionally accounting for the meanings we deploy and the rules that we follow, while the third-that concerning normativity-purports to show that even if the extensional challenge is met, dispositionalism cannot hope to account for the normativity involved in meaning and rule-following.

There seems to be logical disagreement: philosophers and logicians apparently disagree about whether inference rules like modus ponens are valid and about whether contradictions are ever true. Against this, some philosophers have claimed that despite appearances, advocates of different logics aren't really disagreeing with each other. According to this way of thinking, the logical constants mean different things in the mouths of the warring parties-call this the Change of Logic, Change of Meaning (CLCM) thesis.The CLCM thesis has traditionally been popular amongst philosophers who think that logic is analytic or true in virtue of meaning, but even W.V. Quine, canonical critic of both analyticity and meaning, accepted a version of the thesis. In fact, in Quine's writings we find the most developed statement of an important argument for the CLCM thesis that I call the translation argument. 1 But despite a distinguished history, the CLCM thesis and the translation argument have recently fallen on hard times: when J.C. Beall and Greg Restall presented a novel form of logical pluralism, they took pains to distinguish their brand of pluralism from the CLCM-type, and even despite their efforts, influential commentators like Hartry Field and Graham Priest expressed concerns that the new brand of pluralism collapsed into the older, supposedly discredited CLCM-type. 2 The CLCM thesis is often treated as, if not dead, as being on life support. 3 Against this consensus, I will argue that, pace recent trends in the philosophy of logic, the CLCM thesis is true and a reformulation of Quine's translation argument is sound. Like Quine, I will proceed from a general metasemantic background, without assuming any particular metasemantics for the logical constants. My central concern is philosophical, but at 1

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