Logicism, Possibilism, and the Logic of Kantian Actualism

In this extended critical discussion of 'Kant's Modal Metaphysics' by Nicholas Stang (OUP 2016), I focus on one central issue from the first chapter of the book: Stang’s account of Kant’s doctrine that existence is not a real predicate. In §2 I outline some background. In §§3-4 I present and then elaborate on Stang’s interpretation of Kant’s view that existence is not a real predicate. For Stang, the question of whether existence is a real predicate amounts to the question: ‘could there be non-actual possibilia?’ (p.35). Kant’s view, according to Stang, is that there could not, and that the very notion of non-actual or ‘mere’ possibilia is incoherent. In §5 I take a close look at Stang’s master argument that Kant’s Leibnizian predecessors are committed to the claim that existence is a real predicate, and thus to mere possibilia. I argue that it involves substantial logical commitments that the Leibnizian could reject. I also suggest that it is danger of proving too much. In §6 I explore two closely related logical commitments that Stang’s reading implicitly imposes on Kant, namely a negative universal free logic and a quantified modal logic that invalidates the Converse Barcan Formula. I suggest that each can seem to involve Kant himself in commitment to mere possibilia

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S

Logical and Spiritual Reflections

Logical and Spiritual Reflections is a collection of six shorter philosophical works, including: Hume’s Problems with Induction; A Short Critique of Kant’s Unreason; In Defense of Aristotle’s Laws of Thought; More Meditations; Zen Judaism; No to Sodom. Of these works, the first set of three constitutes the Logical Reflections, and the second set constitutes the Spiritual Reflections. Hume’s Problems with Induction, which is intended to describe and refute some of the main doubts and objections David Hume raised with regard to inductive reasoning. It replaces the so-called problem of induction with a principle of induction. David Hume’s notorious skepticism was based on errors of observation and reasoning, with regard to induction, causation, necessity, the self and freewill. These are here pointed out and critically analyzed in detail – and more accurate and logical theories are proposed. The present work also includes refutations of Hempel’s and Goodman’s alleged paradoxes of induction. A Short Critique of Kant’s Unreason, which is a brief critical analysis of some of the salient epistemological and ontological ideas and theses in Immanuel Kant’s famous Critique of Pure Reason. It shows that Kant was in no position to criticize reason, because he neither sufficiently understood its workings nor had the logical tools needed for the task. Kant’s transcendental reality, his analytic-synthetic dichotomy, his views on experience and concept formation, and on the forms of sensibility (space and time) and understanding (his twelve categories), are here all subjected to rigorous logical evaluation and found deeply flawed – and more coherent theories are proposed in their stead. In Defense of Aristotle’s Laws of Thought, which addresses, from a phenomenological standpoint, numerous modern and Buddhist objections and misconceptions regarding the basic principles of Aristotelian logic. Many people seem to be attacking Aristotle’s Laws of Thought nowadays, some coming from the West and some from the East. It is important to review and refute such ideas as they arise. More Meditations, which is a sequel to the author’s earlier work, Meditations. It proposes additional practical methods and theoretical insights relating to meditation and Buddhism. It also discusses certain often glossed over issues relating to Buddhism – notably, historicity, idolatry, messianism, importation to the West. Zen Judaism, which is a frank reflection on the tensions between reason and faith in today’s context of knowledge, and on the need to inject Zen-like meditation into Judaism. This work also treats some issues in ethics and theodicy. No to Sodom, which is an essay against homosexuality, using biological, psychological, spiritual, ethical and political arguments

What Makes Logical Truths True?

The concern of deductive logic is generally viewed as the systematic recognition of logical principles, i.e., of logical truths. This paper presents and analyzes different instantiations of the three main interpretations of logical principles, viz. as ontological principles, as empirical hypotheses, and as true propositions in virtue of meanings. I argue in this paper that logical principles are true propositions in virtue of the meanings of the logical terms within a certain linguistic framework. Since these principles also regulate and control the process of deduction in inquiry, i.e., they are prescriptive for the use of language and thought in inquiry, I argue that logic may, and should, be seen as an instrument or as a way of proceeding (modus procedendi) in inquiry

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Two Squares of Opposition: for Analytic and Synthetic Propositions

In the paper I prove that there are two squares of opposition. The unconventional one is built up for synthetic propositions. There a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation

Between the old metaphysics and the new empiricism: Collingwood's defence of the autonomy of philosophy

Collingwood has failed to make a significant impact in the history of twentieth century philosophy either because he has been dismissed as a dusty old idealist committed to the very metaphysics the analytical school was trying to leave behind, or because his later work has been interpreted as advocating the dissolution of philosophy into history. I argue that Collingwood’s key philosophical works are a sustained attempt to defend the view that philosophy is an autonomous discipline with a distinctive domain of inquiry and that Collingwood’s attempt to defend the autonomy of philosophy is intimately connected to his defence of intensional notions against the kind of meaning scepticism which came to prevail from the 1920s. I defend the philosophical claim that there is a third way between the idealist metaphysics with which Collingwood is often associated and the neo-empiricist agenda which characterised analytic philosophy in mid-century by defending the hermeneutic thesis that Collingwood’s work is a sustained attempt to articulate a conception of philosophy as an epistemologically first science. Since there is a via media between the old metaphysics and the new empiricism there is no need to choose between a certain kind of armchair metaphysics and a scientifically informed ontology