Universality of the Future Chronological Boundary

The purpose of this note is to establish, in a categorical manner, the universality of the Geroch-Kronheimer-Penrose causal boundary when considering the types of causal structures that may profitably be put on any sort of boundary for a spacetime. Actually, this can only be done for the future causal boundary (or the past causal boundary) separately; furthermore, only the chronology relation, not the causality relation, is considered, and the GKP topology is eschewed. The final result is that there is a unique map, with the proper causal properties, from the future causal boundary of a spacetime onto any ``reasonable" boundary which supports some sort of chronological structure and which purports to consist of a future completion of the spacetime. Furthermore, the future causal boundary construction is categorically unique in this regard.Comment: 25 pages, AMS-TeX; 2 figures, PostScript (separate); captions (separate); submitted to Class. Quantum Grav, slight revision: bottom lines legible, figures added, expanded discussion and example

Debunking logical grounding: distinguishing metaphysics from semantics

Many philosophers take purportedly logical cases of ground (such as a true disjunction being grounded in its true disjunct(s)) to be obvious cases, and indeed such cases have been used to motivate the existence of and importance of ground. I argue against this. I do so by motivating two kinds of semantic determination relations. Intuitions of logical ground track these semantic relations. Moreover, our knowledge of semantics for (e.g.) first order logic can explain why we have such intuitions. And, I argue, neither semantic relation can be a species of ground, even on a quite broad conception of what ground is. Hence, without a positive argument for taking so-called ‘logical ground’ to be something distinct from a semantic determination relation, we should cease treating logical cases as cases of ground.Accepted manuscrip

Debunking Logical Ground: Distinguishing Metaphysics from Semantics

Many philosophers take purportedly logical cases of ground ) to be obvious cases, and indeed such cases have been used to motivate the existence of and importance of ground. I argue against this. I do so by motivating two kinds of semantic determination relations. Intuitions of logical ground track these semantic relations. Moreover, our knowledge of semantics for first order logic can explain why we have such intuitions. And, I argue, neither semantic relation can be a species of ground even on a quite broad conception of what ground is. Hence, without a positive argument for taking so-called ‘logical ground’ to be something distinct from a semantic determination relation, we should cease treating logical cases as cases of ground

Deontic logic as a study of conditions of rationality in norm-related activities

The program put forward in von Wright's last works defines deontic logic as ``a study of conditions which must be satisfied in rational norm-giving activity'' and thus introduces the perspective of logical pragmatics. In this paper a formal explication for von Wright's program is proposed within the framework of set-theoretic approach and extended to a two-sets model which allows for the separate treatment of obligation-norms and permission norms. The three translation functions connecting the language of deontic logic with the language of the extended set-theoretical approach are introduced, and used in proving the correspondence between the deontic theorems, on one side, and the perfection properties of the norm-set and the ``counter-set'', on the other side. In this way the possibility of reinterpretation of standard deontic logic as the theory of perfection properties that ought to be achieved in norm-giving activity has been formally proved. The extended set-theoretic approach is applied to the problem of rationality of principles of completion of normative systems. The paper concludes with a plaidoyer for logical pragmatics turn envisaged in the late phase of Von Wright's work in deontic logic

Proving soundness of combinatorial Vickrey auctions and generating verified executable code

Using mechanised reasoning we prove that combinatorial Vickrey auctions are soundly specified in that they associate a unique outcome (allocation and transfers) to any valid input (bids). Having done so, we auto-generate verified executable code from the formally defined auction. This removes a source of error in implementing the auction design. We intend to use formal methods to verify new auction designs. Here, our contribution is to introduce and demonstrate the use of formal methods for auction verification in the familiar setting of a well-known auction

Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects