242 research outputs found
Graphical representation of canonical proof: two case studies
An interesting problem in proof theory is to find representations of proof that do
not distinguish between proofs that are âmorallyâ the same. For many logics, the presentation
of proofs in a traditional formalism, such as Gentzenâs sequent calculus, introduces
artificial syntactic structure called âbureaucracyâ; e.g., an arbitrary ordering
of freely permutable inferences. A proof system that is free of bureaucracy is called
canonical for a logic. In this dissertation two canonical proof systems are presented,
for two logics: a notion of proof nets for additive linear logic with units, and âclassical
proof forestsâ, a graphical formalism for first-order classical logic.
Additive linear logic (or sumâproduct logic) is the fragment of linear logic consisting
of linear implication between formulae constructed only from atomic formulae and
the additive connectives and units. Up to an equational theory over proofs, the logic
describes categories in which finite products and coproducts occur freely. A notion of
proof nets for additive linear logic is presented, providing canonical graphical representations
of the categorical morphisms and constituting a tractable decision procedure
for this equational theory. From existing proof nets for additive linear logic without
units by Hughes and Van Glabbeek (modified to include the units naively), canonical
proof nets are obtained by a simple graph rewriting algorithm called saturation. Main
technical contributions are the substantial correctness proof of the saturation algorithm,
and a correctness criterion for saturated nets.
Classical proof forests are a canonical, graphical proof formalism for first-order
classical logic. Related to Herbrandâs Theorem and backtracking games in the style
of Coquand, the forests assign witnessing information to quantifiers in a structurally
minimal way, reducing a first-order sentence to a decidable propositional one. A similar
formalism âexpansion tree proofsâ was presented by Miller, but not given a method
of composition. The present treatment adds a notion of cut, and investigates the possibility
of composing forests via cut-elimination. Cut-reduction steps take the form
of a rewrite relation that arises from the structure of the forests in a natural way.
Yet reductions are intricate, and initially not well-behaved: from perfectly ordinary
cuts, reduction may reach unnaturally configured cuts that may not be reduced. Cutelimination
is shown using a modified version of the rewrite relation, inspired by the
game-theoretic interpretation of the forests, for which weak normalisation is shown,
and strong normalisation is conjectured. In addition, by a more intricate argument,
weak normalisation is also shown for the original reduction relation
At the intersection of temporal & modal interpretation
This work is chiefly concerned with the semantics of linguistic categories including tense, modality and negation and the relationships between them. In particular, how do they interact in order to âdisplaceâ discourse and to talk about situations remote from the time & place where they\u27re produced? What gets conventionally encoded in linguistic expressions (semantics)? And what\u27s the role of discourse context and extralinguistic factors (pragmatics) in performing these operations? The current thesis contains three connected (but independent) components; each explores different sets of data in view of understanding particular types of displacement phenomena â that is, how, in a given discourse context, reference is established to different possible worlds and different times. In other words, we are concerned with the interactions between temporal reference, modal reference and negation/polarity, and the linguistic phenomena that these give rise to. Methodologically, these projects also engage with diachronic considerations in view of explaining variation and change across spatially and temporally separate language varieties. This is motivated by the desiderata formulated by the amphichronic program --- that is, I assume that studying ostensible changes in language use over time has something to teach us about synchronic systems and vice versa, all in the service of developing an understanding of human language as a cognitive system. Each of these three component âessaysâ considers data from a number of languages spoken in Aboriginal Australia --- particularly YolĆu Matha and Australian Kriol --- on the basis of both published and original data, collected on-site in the Top End in consultation with native speakers. While there is a rich tradition of Australian language description, little Australian language data has been brought to bear on the development of formal theories of meaning. Data from these languages promise to challenge and enrich the methodological and theoretical toolbox of formal semantics. Equally, it is a general contention throughout this work that formal perspectives hold exceptional promise in terms of better understanding the range of linguistic diversity exhibited across Australian languages and developing typologies of the expression of grammatical categories. âThe emergence of apprehensionality in Australian Kriolâ considers the semantics of the adverb bambai in Australian Kriol, a creole language spoken by indigenous populations across northern Australia. Derived from English archaism by-and-by, cognates of bambai are found across contact varieties in the south Pacific. Kriol has retained the âtemporal frameâ use that is found in other South Pacific contact varieties (roughly `soon afterward\u27), although has also developed an identifiable âapprehensionalâ use. Apprehensionals, an understudied if cross-linguistically well-documented category, are taken to modalize their prejacent while implicating their speaker\u27s negative attitude vis-Ă -vis the possibility described in the prejacent. This essay proposes an unified analysis of the meaning contribution of bambai, analyzing the item as unambiguous and claiming that, synchronically, the apprehensional reading âemergesâ reliably in discourse contexts where the truth of its prejacent is not presumed settled as a result of standard assumptions about pragmatic reasoning. Diachronically, it is shown that a similar set of processes led to the generalisation and conventionalization of bambai\u27s meaning components. âThe semantics of the Negative Existential Cycleâ represents a semantic treatment of another little-theorized but cross-linguistically attested cyclic change as it is instantiated in a number of Australian (Pama-Nyungan) language (sub)families. The Cycle involves the recruitment of a âspecialâ nominal negative element which diachronically displaces an older sentential negator. In this essay, the privative---a nominal case marking described in many Australian languages---is analysed as a negative quantifier. The Cycle, then, is understood as the progressive generalisation in the quantificational domain of a negative quantifier: privatives scope over nominalized event descriptions and ultimately over full sentences, at which stage they have encroached into the domain of âstandardâ negation. âReality status & the YolĆu verbal paradigmâ contains a description of and formal proposal for strategies of expressing temporal and modal categories in Western Dhuwal(a), a YolĆu language of northern Arnhem Land. Crucially, this language exhibits a number of puzzling phenomena --- in particular, cyclic tense and the neutralization of reality status marking in negative sentences. As a consequence of these phenomena, the four inflectional categories that constitute WD\u27s verbal paradigm have been treated as unanalyzable from a compositional perspective. Further, neither of these phenomena has received attention in the formal semantic literature. Consequently, this essay represents the first formal proposal for the semantics WD inflectional paradigm (as instantiating a cyclic tense system and an irrealis mood which is licensed by negation) as well as the first formal analysis of these two typological phenomena
Ordered geometry in Hilbertâs Grundlagen der Geometrie
The Grundlagen der Geometrie brought Euclidâs ancient axioms up to the standards
of modern logic, anticipating a completely mechanical verification of their theorems.
There are five groups of axioms, each focused on a logical feature of Euclidean geometry.
The first two groups give us ordered geometry, a highly limited setting where
there is no talk of measure or angle. From these, we mechanically verify the Polygonal
Jordan Curve Theorem, a result of much generality given the setting, and subtle
enough to warrant a full verification.
Along the way, we describe and implement a general-purpose algebraic language
for proof search, which we use to automate arguments from the first axiom group. We
then follow Hilbert through the preliminary definitions and theorems that lead up to
his statement of the Polygonal Jordan Curve Theorem. These, once formalised and
verified, give us a final piece of automation. Suitably armed, we can then tackle the
main theorem
On past participle agreement in transitive clauses in French
This paper provides a Minimalist analysis of past participle agreement in French in transitive
clauses. Our account posits that the head v of vP in such structures carries an (accusativeassigning) structural case feature which may apply (with or without concomitant agreement)
to case-mark a clause-mate object, the subject of a defective complement clause, or an
intermediate copy of a preposed subject in spec-CP. In structures where a goal is extracted
from vP (e.g. via wh-movement) v also carries an edge feature, and may also carry a
specificity feature and a set of (number and gender) agreement features. We show how these
assumptions account for agreement of a participle with a preposed specific clausemate object
or defective-clause subject, and for the absence of agreement with an embedded object, with
the complement of an impersonal verb, and with the subject of an embedded (finite or nonfinite) CP complement. We also argue that the absence of agreement marking (in expected
contexts) on the participles faitmade and laissélet in infinitive structures is essentially viral in
nature. Finally, we claim that obligatory participle agreement with reflexive and reciprocal
objects arises because the derivation of reflexives involves A-movement and concomitant
agreement
One-to-many-relations in morphology, syntax, and semantics (Volume 7)
The standard view of the form-meaning interfaces, as embraced by the great majority of contemporary grammatical frameworks, consists in the assumption that meaning can be associated with grammatical form in a one-to-one correspondence. Under this view, composition is quite straightforward, involving concatenation of form, paired with functional application in meaning. In this book, we discuss linguistic phenomena across several grammatical sub-modules (morphology, syntax, semantics) that apparently pose a problem to the standard view, mapping out the potential for deviation from the ideal of one-to-one correspondences, and develop formal accounts of the range of phenomena. We argue that a constraint-based perspective is particularly apt to accommodate deviations from one-to-many correspondences, as it allows us to impose constraints on full structures (such as a complete word or the interpretation of a full sentence) instead of deriving such structures step by step
Liberal economics and a liberal education in Canada : leading theorists, apologists, and "imaginary expressions" of value
The structure of this dissertation is roughly three parts. Chapters one and two focus on the historical and current idea of higher education in Canada. Chapters three and four expand the scope, situating the Canadian university in the geo-political context of globalisation, globalisation's impact on labour generally, and intellectual labour specifically. Chapter four introduces theoretical debates within Marxism in order to formulate, in chapter five, a critical position better suited to resisting the political and ideological forces acting to reconstruct the Canadian university in the image of the globalised market
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