Presupposition and negation

This paper is an attempt to show that given the available observations on the behaviour of negation and presuppositions there is no simpler explanation than to assume that natural language has two distinct negation operators, the minimal negation which preserves presuppositions and the radical negation which does not. The three-valued logic emerging from this distinction, and especially its model-theory, are discussed in detail. It is, however, stressed that the logic itself is only epiphenomenal on the structures and processes involved in the interpretation of sentences. Horn (1985) brings new observations to bear, related with metalinguistic uses of negation, and proposes a “pragmatic” ambiguity in negation to the effect that in descriptive (or “straight”) use negation is the classical bivalent operator, whereas in metalinguistic use it is non-truthfunctional but only pragmatic. Van der Sandt (to appear) accepts Horn's observations but proposes a different solution: he proposes an ambiguity in the argument clause of the negation operator (which, for him, too, is classical and bivalent), according to whether the negation takes only the strictly asserted proposition or covers also the presuppositions, the (scalar) implicatures and other implications (in particular of style and register) of the sentence expressing that proposition. These theories are discussed at some length. The three-valued analysis is defended on the basis of partly new observations, which do not seem to fit either Horn's or Van der Sandt's solution. It is then placed in the context of incremental discourse semantics, where both negations are seen to do the job of keeping increments out of the discourse domain, though each does so in its own specific way. The metalinguistic character of the radical negation is accounted for in terms of the incremental apparatus. The metalinguistic use of negation in denials of implicatures or implications of style and register is regarded as a particular form of minimal negation, where the negation denies not the proposition itself but the appropriateness of the use of an expression in it. This appropriateness negation is truth-functional and not pragmatic, but it applies to a particular, independently motivated, analysis of the argument clause. The ambiguity of negation in natural language is different from the ordinary type of ambiguity found in the lexicon. Normally, lexical ambiguities are idiosyncratic, highly contingent, and unpredictable from language to language. In the case of negation, however, the two meanings are closely related, both truth-conditionally and incrementally. Moreover, the mechanism of discourse incrementation automatically selects the right meaning. These properties are taken to provide a sufficient basis for discarding the, otherwise valid, objection that negation is unlikely to be ambiguous because no known language makes a lexical distinction between the two readings

On the semantics and logic of declaratives and interrogatives

In many natural languages, there are clear syntactic and/or intonational differences between declarative sentences, which are primarily used to provide information, and interrogative sentences, which are primarily used to request information. Most logical frameworks restrict their attention to the former. Those that are concerned with both usually assume a logical language that makes a clear syntactic distinction between declaratives and interrogatives, and usually assign different types of semantic values to these two types of sentences. A different approach has been taken in recent work on inquisitive semantics. This approach does not take the basic syntactic distinction between declaratives and interrogatives as its starting point, but rather a new notion of meaning that captures both informative and inquisitive content in an integrated way. The standard way to treat the logical connectives in this approach is to associate them with the basic algebraic operations on these new types of meanings. For instance, conjunction and disjunction are treated as meet and join operators, just as in classical logic. This gives rise to a hybrid system, where sentences can be both informative and inquisitive at the same time, and there is no clearcut division between declaratives and interrogatives. It may seem that these two general approaches in the existing literature are quite incompatible. The main aim of this paper is to show that this is not the case. We develop an inquisitive semantics for a logical language that has a clearcut division between declaratives and interrogatives. We show that this language coincides in expressive power with the hybrid language that is standardly assumed in inquisitive semantics, we establish a sound and complete axiomatization for the associated logic, and we consider a natural enrichment of the system with presuppositional interrogatives

Gender and interpretation in Greek: Comments on Merchant (2014)

Merchant (2014, “Gender mismatches under nominal ellipsis”, Lingua, 151: 9–32) makes the following two claims about nominal ellipsis in (Modern) Greek. (i) There are three classes of MASCULINE-FEMININE noun pairs that differ in whether nominal ellipsis with gender mismatch is possible. (ii) Nominal ellipsis with gender mismatch is possible in predicative positions but not in argument positions. We take issue with both of these claims. Our qualms about (i) are relatively minor. It appears that his primary data are hard to replicate, but we present novel sets of data involving focus constructions that also demonstrate that Greek has three classes of MASCULINE-FEMININE noun pairs. As for (ii), we argue that it is empirically inaccurate and nominal ellipsis with gender mismatch is in fact possible in argument positions as well. This is problematic for the analysis Merchant develops, as it is tailored to derive (ii). Furthermore, we argue that his analysis does not give a straightforward account of our observations about focus constructions. We put forward an alternative account of the interpretation of gendered nouns according to which there are three types of nouns with gender inferences: (a) those that have gender inferences in both assertive and presuppositional dimensions of meaning, (b) those that only have gender inferences in the presuppositional dimension of meaning, and (c) those that do not have gender inferences in their semantics but through competition with the opposite gender (gender competition)

Presupposed ignorance and exhaustification: how scalar implicatures and presuppositions interact

We investigate the interactions between scalar implicatures and presuppositions in sentences containing both a scalar item and presupposition trigger. We first critically discuss Gajewski and Sharvit’s previous approach. We then closely examine two ways of integrating an exhaustivity-based theory of scalar implicatures with a trivalent approach to presuppositions. The empirical side of our discussion focuses on two novel observations: (i) the interactions between prosody and monotonicity, and (ii) what we call presupposed ignorance. In order to account for these observations, our final proposal relies on two mechanisms of scalar strengthening, the Presupposed Ignorance Principle and an exhaustivity operator which lets the presuppositions of negated alternatives project

Proof-checking mathematical texts in controlled natural language

The research conducted for this thesis has been guided by the vision of a computer program that could check the correctness of mathematical proofs written in the language found in mathematical textbooks. Given that reliable processing of unrestricted natural language input is out of the reach of current technology, we focused on the attainable goal of using a controlled natural language (a subset of a natural language defined through a formal grammar) as input language to such a program. We have developed a prototype of such a computer program, the Naproche system. This thesis is centered around the novel logical and linguistic theory needed for defining and motivating the controlled natural language and the proof checking algorithm of the Naproche system. This theory provides means for bridging the wide gap between natural and formal mathematical proofs. We explain how our system makes use of and extends existing linguistic formalisms in order to analyse the peculiarities of the language of mathematics. In this regard, we describe a phenomenon of this language previously not described by other logicians or linguists, the implicit dynamic function introduction, exemplified by constructs of the form "for every x there is an f(x) such that ...". We show how this function introduction can lead to a paradox analogous to Russell's paradox. To tackle this problem, we developed a novel foundational theory of functions called Ackermann-like Function Theory, which is equiconsistent to ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) and can be used for imposing limitations to implicit dynamic function introduction in order to avoid this paradox. We give a formal account of implicit dynamic function introduction by extending Dynamic Predicate Logic, a formalism developed by linguists to account for the dynamic nature of natural language quantification, to a novel formalism called Higher-Order Dynamic Predicate Logic, whose semantics is based on Ackermann-like Function Theory. Higher-Order Dynamic Predicate Logic also includes a formal account of the linguistic theory of presuppositions, which we use for clarifying and formally modelling the usage of potentially undefined terms (e.g. 1/x, which is undefined for x=0) and of definite descriptions (e.g. "the even prime number") in the language of mathematics. The semantics of the controlled natural language is defined through a translation from the controlled natural language into an extension of Higher-Order Dynamic Predicate Logic called Proof Text Logic. Proof Text Logic extends Higher-Order Dynamic Predicate Logic in two respects, which make it suitable for representing the content of mathematical texts: It contains features for representing complete texts rather than single assertions, and instead of being based on Ackermann-like Function Theory, it is based on a richer foundational theory called Class-Map-Tuple-Number Theory, which does not only have maps/functions, but also classes/sets, tuples, numbers and Booleans as primitives. The proof checking algorithm checks the deductive correctness of proof texts written in the controlled natural language of the Naproche system. Since the semantics of the controlled natural language is defined through a translation into the Proof Text Logic formalism, the proof checking algorithm is defined on Proof Text Logic input. The algorithm makes use of automated theorem provers for checking the correctness of single proof steps. In this way, the proof steps in the input text do not need to be as fine-grained as in formal proof calculi, but may contain several reasoning steps at once, just as is usual in natural mathematical texts. The proof checking algorithm has to recognize implicit dynamic function introductions in the input text and has to take care of presuppositions of mathematical statements according to the principles of the formal account of presuppositions mentioned above. We prove two soundness and two completeness theorems for the proof checking algorithm: In each case one theorem compares the algorithm to the semantics of Proof Text Logic and one theorem compares it to the semantics of standard first-order predicate logic. As a case study for the theory developed in the thesis, we illustrate the working of the Naproche system on a controlled natural language adaptation of the beginning of Edmund Landau's Grundlagen der Analysis.Beweisprüfung mathematischer Texte in kontrollierter natürlicher Sprache Die Forschung, die für diese Dissertation durchgeführt wurde, basiert auf der Vision eines Computerprogramms, das die Korrektheit von mathematischen Beweisen, die in der gewöhnlichen mathematischen Fachsprache verfasst sind, überprüfen kann. Da die zuverlässige automatische Bearbeitung von uneingeschränktem natürlich-sprachlichen Input außer Reichweite der gegenwärtigen Technologie ist, haben wir uns auf das erreichbare Ziel fokussiert, eine kontrollierte natürliche Sprache (eine Teilmenge der natürlichen Sprache, die durch eine formale Grammatik definiert ist) als Eingabesprache für ein solches Programm zu verwenden. Wir haben einen Prototypen eines solchen Computerprogramms, das Naproche-System, entwickelt. Die vorliegende Dissertation beschreibt die neuartigen logischen und linguistischen Theorien, die benötigt werden, um die kontrollierte natürliche Sprache und den Beweisprüfungs-Algorithmus des Naproche-Systems zu definieren und zu motivieren. Diese Theorien stellen Methoden zu Verfügung, die dazu verwendet werden können, die weite Kluft zwischen natürlichen und formalen mathematischen Beweisen zu überbrücken. Wir erklären, wie unser System existierende linguistische Formalismen verwendet und erweitert, um die Besonderheiten der mathematischen Fachsprache zu analysieren. In diesem Zusammenhang beschreiben wir ein Phänomen dieser Fachsprache, das bisher von Logikern und Linguisten nicht beschrieben wurde – die implizite dynamische Funktionseinführung, die durch Konstruktionen der vorm "für jedes x gibt es ein f(x), so dass ..." veranschaulicht werden kann. Wir zeigen, wie diese Funktionseinführung zu einer der Russellschen analogen Antinomie führt. Um dieses Problem zu lösen, haben wir eine neuartige Grundlagentheorie für Funktionen entwickelt, die Ackermann-artige Funktionstheorie, die äquikonsistent zu ZFC (Zermelo-Fraenkel-Mengenlehre mit Auswahlaxiom) ist und verwendet werden kann, um der impliziten dynamischen Funktionseinführung Grenzen zu setzen, die zur Vermeidung dieser Antinomie führen. Wir beschreiben die implizite dynamische Funktionseinführung formal, indem wir die Dynamische Prädikatenlogik – ein Formalismus, der von Linguisten entwickelt wurde, um die dynamischen Eigenschaften der natürlich-sprachlichen Quantifizierung zu erfassen – zur Dynamischen Prädikatenlogik Höherer Stufe erweitern, deren Semantik auf der Ackermann-artigen Funktionstheorie basiert. Die Dynamische Prädikatenlogik Höherer Stufe formalisiert auch die linguistische Theorie der Präsuppositionen, die wir verwenden, um den Gebrauch potentiell undefinierter Terme (z.B. der Term 1/x, der für x=0 undefiniert ist) und bestimmter Kennzeichnungen (z.B. "die gerade Primzahl") in der mathematischen Fachsprache zu modellieren. Die Semantik der kontrollierten natürlichen Sprache wird definiert durch eine Übersetzung dieser in eine Erweiterung der Dynamischen Prädikatenlogik Höherer Stufe mit der Bezeichnung Beweistext-Logik. Die Beweistext-Logik erweitert die Dynamische Prädikatenlogik Höherer Stufe in zwei Hinsichten: Sie stellt Funktionalitäten für die Repräsentation von vollständigen Texten, und nicht nur von Einzelaussagen, zur Verfügung, und anstatt auf der Ackermann-artigen Funktionstheorie zu basieren, basiert sie auf einer reichhaltigeren Grundlagentheorie – der Klassen-Abbildungs-Tupel-Zahlen-Theorie, die neben Abbildungen/Funktionen auch noch Klassen/Mengen, Tupel, Zahlen und boolesche Werte als Grundobjekte zur Verfügung stellt. Der Beweisprüfungs-Algorithmus prüft die deduktive Korrektheit von Beweistexten, die in der kontrollierten natürlichen Sprache des Naproche-Systems verfasst sind. Da die Semantik dieser kontrollierten natürlichen Sprache durch eine Übersetzung in die Beweistext-Logik definiert ist, ist der Beweisprüfungs-Algorithmus für Beweistext-Logik-Input definiert. Der Algorithmus verwendet automatische Beweiser für die Überprüfung einzelner Beweisschritte. Dadurch müssen die Beweisschritte in dem Eingabetext nicht so kleinschrittig sein wie in formalen Beweiskalkülen, sondern können mehrere Deduktionsschritte zu einem Schritt vereinen, so wie dies auch in natürlichen mathematischen Texten üblich ist. Der Beweisprüfungs-Algorithmus muss die impliziten Funktionseinführungen im Eingabetext erkennen und Präsuppositionen von mathematischen Aussagen auf Grundlage der oben erwähnten Präsuppositionstheorie behandeln. Wir beweisen zwei Korrektheits- und zwei Vollständigkeitssätze für den Beweisprüfungs-Algorithmus: Jeweils einer dieser Sätze vergleicht den Algorithmus mit der Semantik der Beweistext-Logik und jeweils einer mit der Semantik der üblichen Prädikatenlogik erster Stufe. Als Fallstudie für die in dieser Dissertation entwickelte Theorie veranschaulichen wir die Funktionsweise des Naproche-Systems an einem an die kontrollierte natürliche Sprache angepassten Anfangsabschnitt von Edmund Landaus Grundlagen der Analysis