An optimal construction of Hanf sentences

We give the first elementary construction of equivalent formulas in Hanf normal form. The triply exponential upper bound is complemented by a matching lower bound

Counting Incompossibles

We often speak as if there are merely possible people—for example, when we make such claims as that most possible people are never going to be born. Yet most metaphysicians deny that anything is both possibly a person and never born. Since our unreflective talk of merely possible people serves to draw non-trivial distinctions, these metaphysicians owe us some paraphrase by which we can draw those distinctions without committing ourselves to there being merely possible people. We show that such paraphrases are unavailable if we limit ourselves to the expressive resources of even highly infinitary first-order modal languages. We then argue that such paraphrases are available in higher-order modal languages only given certain strong assumptions concerning the metaphysics of properties. We then consider alternative paraphrase strategies, and argue that none of them are tenable. If talk of merely possible people cannot be paraphrased, then it must be taken at face value, in which case it is necessary what individuals there are. Therefore, if it is contingent what individuals there are, then the demands of paraphrase place tight constraints on the metaphysics of properties: either (i) it is necessary what properties there are, or (ii) necessarily equivalent properties are identical, and having properties does not entail even possibly being anything at all

Gluing together proof environments: Canonical extensions of LF type theories featuring locks

© F. Honsell, L. Liquori, P. Maksimovic, I. Scagnetto This work is licensed under the Creative Commons Attribution License.We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF/p,is the canonical version of the system LLF p, presented earlier by the authors. The second system, CLLF p?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms

On elimination of quantifiers in some non-classical mathematical theories

Elimination of quantifiers is shown to fail dramatically for a group of well-known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back

Developments from enquiries into the learnability of the pattern languages from positive data

AbstractThe pattern languages are languages that are generated from patterns, and were first proposed by Angluin as a non-trivial class that is inferable from positive data [D. Angluin, Finding patterns common to a set of strings, Journal of Computer and System Sciences 21 (1980) 46–62; D. Angluin, Inductive inference of formal languages from positive data, Information and Control 45 (1980) 117–135]. In this paper we chronologize some results that developed from the investigations on the inferability of the pattern languages from positive data

Generalised Indiscernibles, Dividing Lines, and Products of Structures

Generalised indiscernibles highlight a strong link between model theory and structural Ramsey theory. In this paper, we use generalised indiscernibles as tools to prove results in both these areas. More precisely, we first show that a reduct of an ultrahomogenous $\aleph_0$-categorical structure which has higher arity than the original structure cannot be Ramsey. In particular, the only nontrivial Ramsey reduct of the generically ordered random $k$-hypergraph is the linear order. We then turn our attention to model-theoretic dividing lines that are characterised by collapsing generalised indiscernibles, and prove, for these dividing lines, several transfer principles in (full and lexicographic) products of structures. As an application, we construct new algorithmically tame classes of graphs