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This thesis studies homogeneous models by looking at their topos of finitely supported sets. Of various well-known toposes, such as cubical sets and simplicial sets, it is shown that they can be presented as a topos of this kind. It is argued that these examples can be understood as coming from a homogeneous model, in the sense of model theory, but formulated in topos-theoretical terms. It is shown that the connecting structure between the two is a factorizing prime site and the notion of a principal model. This is further substantiated by demonstrating that the Fraïssé-Hrushovski construction can be formulated in terms of such sites. The notion of finite support is abstracted and a general calculus of orbits for toposes of supported sets is developed. It is shown that these form a factorizing prime site and that they can be compared to the age of structure as in classical model theory, and can be given a topos-theoretic characterization in terms of the hyperconnected-localic factorization of a geometric morphism. The theory is then applied by revisiting and generalizing the Ryll-Nardzewski theorem to the new setting and by constructing a very general class of Fraenkel-Mostowki models of intuitionistic set theory. Lastly, a notion of a Kan fibration for such toposes is studied and shown to generalize the familiar notions in simplicial sets and cubical sets. Under particular assumptions, it is proved that these Kan fibrations are closed under dependent product and hence interpret a version of Martin-Löf intensional dependent type theory. For simplicial sets, a different approach using simplicial Moore paths and its connection to the present work is announced.

Robert Audi’s 2005 book The Good in the Right is a formidable representative of the recently resurgent theory ethical intuitionism (EI). Its renown is not unearned; Audi has developed a novel version of EI that preserves some of the most appealing features of Rossian EI while making a few key changes that help it to stand up better to some criticisms. This paper will explore Audi’s position and articulate a variety of objections to it that, taken together, prevent it from being a serious contender on the metaethical stage. The first section will discuss some of the difficulties with Audi’s attempt to establish a plausible and coherent account of the kind of self-evidence necessary to support EI. The next section will express two objections to Audi’s views concerning the role that reflection (as opposed to inference) plays in coming to justified conclusions. The last section will argue that Audi’s attempt to lend additional support to Rossian principles on the basis of a semi-utilitarian theory of value unambiguously fails.

This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity.I begin Part I by rehearsing an argument--which is due to Linnebo and which I partially endorse--that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first-order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting.I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. I argue that there is an important range of potential infinities, which I call infinities with branching possibilities, for which S4 is the right modal logic. I further argue that the usual necessity operator is not sufficiently expressive for reasoning about branching potential infinities. A new operator is needed, which I call the inevitability operator. (In tense logic, the same operator has been called the strong future tense). I show that first-order S4 with the inevitability operator is unaxiomatizable. This makes it improbable that there can be a faithful translation of non-modal mathematical discourse into modal language. I suggest that this means standard mathematical theories of infinite sequences are thus committed to actual infinities.In Part II, I apply the modal account of potential infinity to develop a theory of free choice sequences. Free choice sequences are a concept from intuitionistic mathematics. They can be thought of as potentially infinite sequences of natural numbers whose values are freely chosen one at a time by an (idealized) mathematician. They figure prominently in results of intuitionistic mathematics that contradict classical mathematics. I propose an axiomatic theory of free choice sequences in a modal extension of classical second-order arithmetic, called MC, with the aim of providing modal analogues of key ideas and results from the intuitionistic analysis. In this theory I define the temporal-potential continuum, which serves as an ersatz intuitionistic continuum in my modal theory.I show that the temporal-potential continuum exhibits many characteristically intuitionistic properties. The main results are these: it is not the case that every real number is determinately rational or irrational; the natural order on real numbers is not linear; a bounded monotone sequence of rationals need not be Cauchy; if two disjoint sets A and B decompose the temporal-potential continuum, then both A and B are topologically open; finally, I introduce a notion of sharp discontinuity and show there is no function on the temporal-potential continuum which is sharply discontinuous.

Two common beliefs in historical linguistics will be reviewed here: (1) that a once succesful change which is now fully integrated in the grammar must have followed a pattern of sustained growth (a “constant rate” S-curve) all along its history; and (2) that only increasing frequencies are associated with grammaticalization processes in any significant or symptomatic way. The first belief is based on the linguist’s intuitions about contemporary use, since a gramatically unsurprising phenomenon is not likely to be considered obsolescent or on the verge of abandonment. The second belief draws on an equivalence (also of intuitive nature, though observable in many particular changes) between grammaticalization as a process resulting in the emergence of processing routines and increased use of thus grammaticalized elements and constructions, a view that, albeit not necessarily wrong itself, has left in the shadows the role of failed changes in grammaticalization continua. Quantitative analysis of large amounts of data allows us to challenge both generalizations. By looking closely at the grammaticalization and diffusion of the definite article el before completives headed by que (‘that’) in Spanish, we will show that a clearly recessive phenomenon might go perfectly unnoticed to both speakers and linguists, thus revealing the potential frailty of intuition and introspective judgment as sources of access to historically conditioned grammatical facts. We will also claim that a failed evolution might in turn motivate the failure of a related (in fact, derived) construction at the far end of a grammaticalization continuum, which is tantamount to assuming that frequency drops can be both significant and symptomatic in the last stages of secondary grammaticalization, and possibly as well herald the reuse or exaptation of highly grammaticalized forms.