We present a valid deductive cosmological argument for the necessary existence of a powerful and intelligent creator of the actual universe. Whereas traditional cosmological arguments had to employ a strong version of the principle of sufficient reason that held that every fact actually has an explanation, our argument can make do with the weak version of Duns Scotus according to which every fact possibly has an explanation. As a result, our argument is less vulnerable to the charge of begging the question than are these traditional cosmological arguments.

A number of philosophers have attempted to solve the problem of null-probability possible events in Bayesian epistemology by proposing that there are infinitesimal probabilities. Hájek (2003) (more tentatively) and Easwaran (2014) (more definitively) have argued that because there is no way to specify a particular hyperreal extension of the real numbers, solutions to the regularity problem involving infinitesimals, or at least hyperreal infinitesimals, involve an unsatisfactory ineffability or arbitrariness. The arguments depend on the alleged impossibility of picking out a particular hyperreal extension of the real numbers and/or of a particular value within such an extension due to the use of the Axiom of Choice. However, it is false that the Axiom of Choice precludes a specification of a hyperreal extension-such an extension can indeed be specified. Moreover, for all we know, it is possible to explicitly specify particular infinitesimals within such an extension. Nonetheless, I prove that because any regular probability measure that has infinitesimal values can be replaced by one that has all the same intuitive features but other infinitesimal values, the heart of the arbitrariness objection remains.

A necessary being is a concrete entity that cannot fail to exist. An example of such a being might be the God of classical theism or the universe of necessitarians. Necessary Existence offers and carefully defends a number of novel arguments for the thesis that there exists at least one necessary being, while inviting the reader to a future investigation of what the neccessary being(s) is (are) like. The arguments include a defense of a classic contingency argument, a series of new modal arguments from possible causes, an argument from abstract objects, and a Gödelian argument from perfections. Furthermore, arguments against the possibility of a necessary being are critically examined. Among these arguments are old and new arguments from conceivability, a subtraction argument, problems with causation, and an argument from parsimony. Necessary Existence also includes a defense of the axioms of S5 modal logic, which is a framework for understanding several arguments for necessary existents.

Infinity is paradoxical in many ways. A particular large family of paradoxes is examined that on its face iswidely varied. Some involve deterministic supertasks, such as Thomson’s Lamp where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. Yet others involve paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin-flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin-flips to predict future ones with only finitely many mistakes. It turns out that these, and a number of other paradoxes have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes can all be solved with a single move: embrace causal finitism, the view that it is impossible for a single output to have an infinite causal history. The book exposits such paradoxes, defends causal finitism at length, and ends up considering connections with the philosophy of physics, where causal finitism favors, but does not require, discretist theories of space and time, and the philosophy of religion, where we get a cosmological argument reminiscent of the Kalām argument for the existence of God.

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