Rigid and flexible quantification in plural predicate logic

Noun phrases with overt determiners, such as \u3ci\u3esome apples\u3c/i\u3e or \u3ci\u3ea quantity of milk\u3c/i\u3e, differ from bare noun phrases like \u3ci\u3eapples\u3c/i\u3e or \u3ci\u3emilk\u3c/i\u3e in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable fixed throughout a quantificational context while the other allows them to vary. Inspired by Dynamic Plural Logic and Dependence Logic, we propose Plural Predicate Logic as an extension of Predicate Logic to formalize this difference. We suggest that temporal \u3ci\u3efor\u3c/i\u3e-adverbials are sensitive to aspect because of the way they manipulate quantificational contexts, and that analogous manipulations occur with spatial \u3ci\u3efor\u3c/i\u3e-adverbials, habituals, and the quantifier \u3ci\u3eall\u3c/i\u3e

Introduction

Introduction to genericity in the nominal, verbal and sentential domain

Rigid and Flexible Quantification in Plural Predicate Logic

Noun phrases with overt determiners, such as some apples or a quantity of milk, differ from bare noun phrases like apples or milk in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable fixed throughout a quantificational context while the other allows them to vary. Inspired by Dynamic Plural Logic and Dependence Logic, we propose Plural Predicate Logic as an extension of Predicate Logic to formalize this difference. We suggest that temporal for-adverbials are sensitive to aspect because of the way they manipulate quantificational contexts, and that analogous manipulations occur with spatial for-adverbials, habituals, and the quantifier all

Axiomatic Architecture of Scientific Theories

The received concepts of axiomatic theory and axiomatic method, which stem from David Hilbert, need a systematic revision in view of more recent mathematical and scientific axiomatic practices, which do not fully follow in Hilbert’s steps and re-establish some older historical patterns of axiomatic thinking in unexpected new forms. In this work I motivate, formulate and justify such a revised concept of axiomatic theory, which for a variety of reasons I call constructive, and then argue that it can better serve as a formal representational tool in mathematics and science than the received concept

A hierarchy of languages, logics, and mathematical theories

We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence

Tárgyas szerkezetek elemzése tenzorfelbontással - áttekintő cikk

Áttekintjük a tenzorfelbontás számítógépes nyelvészeti alkalmazásait, különösen az igei argumentumstruktúrára vonatkozókat, és olyan asszociációs mértékekre hívjuk fel a figyelmet, amelyeket eddig nem használtak erre a feladatra

Learning Functional Prepositions

In first language acquisition, what does it mean for a grammatical category to have been acquired, and what are the mechanisms by which children learn functional categories in general? In the context of prepositions (Ps), if the lexical/functional divide cuts through the P category, as has been suggested in the theoretical literature, then constructivist accounts of language acquisition would predict that children develop adult-like competence with the more abstract units, functional Ps, at a slower rate compared to their acquisition of lexical Ps. Nativists instead assume that the features of functional P are made available by Universal Grammar (UG), and are mapped as quickly, if not faster, than the semantic features of their lexical counterparts. Conversely, if Ps are either all lexical or all functional, on both accounts of acquisition we should observe few differences in learning. Three empirical studies of the development of P were conducted via computer analysis of the English and Spanish sub-corpora of the CHILDES database. Study 1 analyzed errors in child usage of Ps, finding almost no errors in commission in either language, but that the English learners lag in their production of functional Ps relative to lexical Ps. That no such delay was found in the Spanish data suggests that the English pattern is not universal. Studies 2 and 3 applied novel measures of phrasal (P head + nominal complement) productivity to the data. Study 2 examined prepositional phrases (PPs) whose head-complement pairs appeared in both child and adult speech, while Study 3 considered PPs produced by children that never occurred in adult speech. In both studies the productivity of Ps for English children developed faster than that of lexical Ps. In Spanish there were few differences, suggesting that children had already mastered both orders of Ps early in acquisition. These empirical results suggest that at least in English P is indeed a split category, and that children acquire the syntax of the functional subset very quickly, committing almost no errors. The UG position is thus supported. Next, the dissertation investigates a \u27soft nativist\u27 acquisition strategy that composes the distributional analysis of input, minimal a priori knowledge of the possible co-occurrence of morphosyntactic features associated with functional elements, and linguistic knowledge that is presumably acquired via the experience of pragmatic, communicative situations. The output of the analysis consists in a mapping of morphemes to the feature bundles of nominative pronouns for English and Spanish, plus specific claims about the sort of knowledge required from experience. The acquisition model is then extended to adpositions, to examine what, if anything, distributional analysis can tell us about the functional sequences of PPs. The results confirm the theoretical position according to which spatiotemporal Ps are lexical in character, rooting their own extended projections, and that functional Ps express an aspectual sequence in the functional superstructure of the PP

Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D

We study locality preserving automorphisms of operator algebras on $D$-dimensional uniform lattices of prime $p$-dimensional qudits (QCA), specializing in those that are translation invariant (TI) and map every prime $p$-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries, and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Further, we prove that for any $D$ the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for $D=3$ and any odd prime $p$, and show that the Witt group of the finite field $\mathbb F_p$ is a subgroup of the group $\mathfrak C(D = 3, p)$ of all TI Clifford QCA modulo trivial ones. That is, $\mathfrak C(D = 3, p \equiv 1 \mod 4) \supseteq \mathbb Z_2 \times \mathbb Z_2$ and $\mathfrak C(D = 3, p \equiv 3 \mod 4) \supseteq \mathbb Z_4$. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of the paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.Comment: 38 pages and a calculation note in Mathematica, (v2, v3) a new section on boundary antihermitian form