This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".

According to actualism about possible worlds everything that exists is actual. Possible worlds and individuals are actually existing abstract parts of the actual world. Aristotelian actualism is a view that there are only actual individuals but no possible ones, nor their individual abstract representatives. Because of that, our actualist account of modality should differ depending on whether it concerns actual individuals or possible ones. The main goal of the dissertation is to develop a metaphysical framework for Aristotelian actualism. Chapter 1 (...) explains basic issues associated with the possible world approach to modality. I overview modal realist and actualist views on possible worlds and explain why I support the actualist approach. Subsequently, I introduce a distinction between Platonic and Aristotelian actualism, and discuss some semantic issues associated with actualism as such. In Chapter 2 I argue that Aristotelian actualism, modeled on linguistic ersatzism, is preferable over its Platonic counterpart. Subsequently, I propose a metaphysical framework for Aristotelian ersatzism which is based on a claim that our modal concepts work differently for actual and possible individuals. In order to explain that claim I introduce three specific differences concerning modal features of actual and possible individuals: (a) Representational Difference, according to which actual and possible individuals are represented differently by possible worlds; (b) Metaphysical Difference, according to which actual and possible individuals are represented by possible worlds as having different metaphysical nature; (c) Modal Difference, which says while there are singular and contingent possibilities involving actual individuals, all possibilities about possible individuals are general and necessary. I propose to interpret those differences in terms of the doctrines of haecceitism, antihaecceitism and existentialism. There is however no consensus on how those views should be characterized. Chapters 3, 4, and 5 focus on providing a precise characterization of those doctrines. Chapter 3 focuses on the doctrines of modal haecceitism and antihaecceitism, which I view as opposite accounts of how possible worlds represent possibilities. According to modal haecceitism what possible worlds say about particular individuals does not supervene on what they say qualitatively. Modal antihaecceitism is a denial of such a claim. Chapter 4 concerns metaphysical haecceitism and antihaecceitism, which I take to be alternative accounts of the fundamental structure of reality. For the metaphysical haecceitist reality contains irreducible singular facts, while for the metaphysical antihaecceitist reality is purely qualitative and general. Chapter 5 focuses on an argument between existentialists and antiexistentialists. Existentialists claim that there are contingent singular propositions, while antiexistentialists deny that. I defend existentialism against antiexistentialist counterarguments, as well as criticize some of the antiexistentialist accounts of singular propositions modeled on the notion of individual essence. In Chapter 6, by appealing to the results of investigations conducted in Chapters 3, 4 and 5, I reconsider Representational, Metaphysical and Modal Differences. According to a view that I propose: (a) Representational Difference entails (extreme) modal haecceitism for actual individuals but (extreme) modal antihaecceitism for possible individuals; (b) Metaphysical Difference entails metaphysical haecceitism (individualism) for actual individuals, but metaphysical antihaecceitism (generalism) for possible individuals; finally (c) Modal Difference entails existentialism: while there are singular and contingent possibilities involving actual individuals, all possibilities about possible individuals are general and necessary. In Chapter 6, I also explain the implications of those views for the various issues, including transworld identity, essentialism, or the modal status of modal space. Lastly, Chapter 7 overviews some semantic and metaphysical applications of Aristotelian ersatzism. I explain how it manages to accommodate Kripkean semantics and how it is able to account for the possibilities of indiscernibles, alien individuals and iterated modalities. I also address some possible objections to my proposal, including an issue of implicit representation and the Humphrey objection. (shrink)