In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in prover9 (a first-order automated reasoning system which is the successor to otter). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in prover9's first-order syntax, and (2) how prover9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.

Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topic-neutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

While the subtitle “A History of Truth ” is far too ambitious for this brief slice through the history of selected topics in deductive thought, this project does offer, in the words of the pioneers, a few passages that explain how modern mathematics has arrived at an understanding of elementary propositional logic. Different cultures have held differing views about deduction, i.e. whether a

Supposing that truths require truthmakers, that true propositions are those which correspond to facts, is there a distinctive domain of facts that make true the relational truths? Or is it rather that, if we had collected the facts required to make true the other truths, the non-relational ones, that we would then have enough facts to make all truths true? If the former is the case, let us say that anti-reductionism about relational facts is true; if the latter, that reductionism about relational facts is true. Let us say that a fact is relational if it makes true some relational proposition (a proposition that asserts that a relation holds between some objects 1), that it is irreducibly relational if, in addition, it does not make true any nonrelational propositions, and that it is monadic if it is not irreducibly relational (if it makes true some proposition that does not assert that a relation holds between some objects). Anti-reductionism (as we will say for short) holds that there are irreducibly relational facts — reductionism (as we will say for short) that while there may be relational facts, there are no irreducibly relational ones. This is a very fine definition, but is it an interesting issue? Yes, for three reasons: 1. Reasons internal to truthmaker theory. If you are one of those metaphysicians who believes the antecedent of the first sentence of this paper, that truths require truthmakers, you will naturally be interested in what manner of things you are thereby committed to. Different truthmaker theorists offer different ontologies of facts. These ontologies deal with relational facts in different ways, so independent arguments for one or other view of relational facts give us some grip on which truthmaker ontology is likely to be the right one. In particular, irreducibly relational facts could seem nominalistically unrespectable.

The propositional and relational syllogistic ∗