DOCSLIB.ORG

Inquisitive Semantics and Intermediate Logics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Inquisitive Semantics and Intermediate Logics Inquisitive Semantics and Intermediate Logics. MSc Thesis (Afstudeerscriptie) written by Ivano A. Ciardelli (born February 3rd, 1985 in Sesto san Giovanni, Italy) under the supervision of Prof Dr Jeroen Groenendijk, Prof Dr Dick de Jongh, and Dr Floris Roelofsen, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: September 3, 2009 Prof Dr Johan van Benthem Prof Dr Jeroen Groenendijk Dr Rosalie Iemhoff Prof Dr Dick de Jongh Dr Floris Roelofsen Dr Yde Venema [::: ] Sono allora un soldatino di piombo appeso a un filo sul mare aperto I miei pensieri passano come velieri [::: ] Leonardo Martellini, Velieri To my schoolteacher Mauro, who taught me how to doubt i Contents Acknowledgements............................. iv Overview of the thesis...........................v 1 An introduction to inquisitive semantics1 1.1 Information states and the classical update perspective......1 1.2 Limitations of the classical picture.................2 1.3 The inquisitive programme: propositions as proposals......3 1.4 Sources of inquisitiveness: questions, disjunctions and indefinites4 1.5 Historical notes............................5 2 Propositional inquisitive semantics7 2.1 Propositional inquisitive semantics and its properties.......7 2.1.1 Indices, States, and Support.................7 2.1.2 Inquisitive meanings.....................9 2.1.3 Inquisitiveness and Informativeness............. 12 2.2 Inquisitive semantics and intuitionistic Kripke semantics..... 15 2.3 Inquisitive semantics over an arbitrary common ground..... 16 2.4 Support as `knowing how'...................... 18 2.5 Expressive completeness and disjunctive normal form....... 19 3 Inquisitive logic 22 3.1 Inquisitive Logic........................... 22 3.1.1 Definitions and basic properties............... 22 3.1.2 Disjunction Property, Deduction Theorem, and Compact- ness.............................. 23 3.2 Axiomatizing InqL .......................... 25 3.2.1 Intermediate logics and negative variants......... 26 3.2.2 Completeness by canonical model.............. 30 3.2.3 Completeness via disjunctive-negative translation..... 39 3.3 InqL as the disjunctive-negative fragment of IPL.......... 41 3.4 Schematic Fragment of Inquisitive Logic.............. 44 3.4.1 Medvedev logic........................ 44 3.4.2 Sch(InqL)=ML ........................ 45 3.4.3 Characterization of the intermediate logics whose negative variant is InqL ........................ 46 ii 3.4.4 More on Medvedev logic................... 48 3.5 Independence of the connectives.................. 52 4 The inquisitive hierarchy 55 4.1 Generalizing the pair semantics................... 55 4.2 Axiomatizing the inquisitive hierarchy............... 58 4.3 A Plea for the Generalized Semantics................ 59 5 Intermediate logics with negative atoms 61 5.1 Negative closure of intermediate logics............... 62 5.2 Stability of intermediate logics................... 64 6 First-order inquisitive semantics 77 6.1 First-order inquisitive semantics and the maximality problem.. 77 6.2 Propositional possibility semantics................. 81 6.2.1 Propositions......................... 81 6.2.2 Resolutions.......................... 83 6.2.3 Strong entailment...................... 85 6.2.4 Disjunctive normal form and expressive completeness... 86 6.2.5 Inquisitiveness, informativeness, suggestiveness...... 87 6.2.6 Assertions, questions, and conjectures........... 90 6.2.7 Axiomatizing strong entailment............... 94 6.3 First-order possibility semantics................... 97 6.3.1 Propositions......................... 97 6.3.2 Entailment and strong entailment............. 99 6.3.3 Resolutions.......................... 101 6.3.4 Inquisitiveness, informativeness, suggestiveness...... 103 6.3.5 Assertions, questions, and conjectures........... 104 6.4 Might meets the logical constants.................. 106 6.5 An assessment of possibility semantics............... 109 6.6 Notes on first-order inquisitive logic................ 110 6.6.1 Definition and basic properties............... 110 6.6.2 IQL ⊆ InqQL ⊆ CQL ..................... 111 6.6.3 Correspondence theorem................... 114 7 Conclusions 117 References 118 iii Acknowledgements My gratitude goes in the first place to my supervisors, thanks to whom the preparation of this thesis has been a truly exciting, interesting and fun experi- ence. Thanks to Jeroen Groenendijk for introducing me to the ideas of inquisitive semantics, for stimulating and encouraging my work on it, and for actively considering and criticizing each of the preliminary versions of the ideas contained here: rarely have I met such an affable and open-minded person. Thanks to Dick de Jongh for his experienced and patient supervision, as well as for his wise advice and his support in my many moments of doubt and indecision. Thanks to Floris Roelofsen for a number of things: for several brilliant in- tuitions that were crucial to the development of the proof presented in section 3.2.2; for patiently answering a horde of silly questions that I have been gen- erating in virtue of my inexperience in scientific writing; but especially for the many, many pleasant hours we spent together discussing, attempting proofs, laughing, planning, eating, and climbing trees. Also, thanks to the supernatural room on the top floor of the philosophy department for incubating so many ideas: its contribution to the present thesis is invaluable. The ILLC has provided me with an extraordinarily rich, diverse and highly stimulating scientific environment. For this experience I am grateful to all the ILLC staff and to my fellow master's students, but special thanks I owe to: Yde Venema, for the clarity and pleasantness of his teaching; to In´esCrespo, for excellently sustaining the role of personal counselor I saddled her with; to Matthew Wampler-Doty, for crucially pointing out the existence of a similarity between inquisitive semantics and Medvedev frames; to Lex Hendriks, for con- tributing useful information and references; and, last but not least, to Benedikt L¨owe and Tanja Kassenaar, for the truly admirable organization of the program, which made everything easy even for a complete bureaucratic bungler like me. I also wish to thank three persons who have been especially influential in my own cultural and scientific growth, namely my schoolteacher Mauro Pellegrinelli and my university professors Bruno Franchi and Simone Martini. Finally, I would like to thank the Dutch Ministry of Education and the Beth Foundation for their generous financial support, which allowed me to con- centrate exclusively on my studies in these two years, and the Philosophy of Language Group directed by Paul Dekker and Martin Stokhof for funding my trip to the conference TARK at Stanford, thus giving me the opportunity to present some of the results contained in this thesis. iv Overview of the thesis The present thesis is concerned with inquisitive semantics and its logic. In- quisitive semantics is a system (or rather a class of systems) for modelling the effect of utterances in a cooperative dialogue; the crucial feature of inquisitive semantics is that it regards propositions as real proposals of one or more pos- sible updates for the common ground of a conversation, thus allowing for the representation of both inquisitive and informative content as two aspects of a unique notion of meaning. We start with a general overview of the structure and the contents of the thesis. Chapter 1: an introduction to inquisitive semantics. We introduce the reader to the ideas that motivate and guide the development of an inquisitive semantics. We first mention a traditional approach to the modelling of the exchange of information in a dialogue. We then illustrate a different perspec- tive on cooperative information exchange, namely the propositions-as-proposals view which informs the construction of inquisitive semantics. We conclude the chapter with some historical notes, clarifying how the present thesis relates to previous and ongoing work on the subject. Chapter 2: propositional inquisitive semantics. We set out to imple- ment the ideas expounded in the previous chapter for a propositional language. We evaluate formulas over sets of valuations, conceived of as information states. The semantics is based upon the relation of support between states and propo- sitional formulas: via the notion of support, we associate each formula with a set of possibilities, defined as maximal states supporting the formula. This set of possibilities represents the proposal put forward by a formula. We identify two effects of a proposal: an informative effect consisting in the suggestion to eliminate some possible worlds from the common ground, and an inquisitive effect consisting in the specification of alternative updates. We define assertions and questions as formulas that serve only one of these purposes and discuss the properties of these classes of formulas, in particular showing that each formula can always be decomposed into a question and an assertion, its assertive part coinciding with its classical meaning. We discuss several other properties of the semantics, such as expressive com- pleteness and normal form results, point out a tight connection linking inquisi- tive semantics to intuitionistic Kripke semantics, and suggest an intuitive inter- pretation of the notion of support as `knowing how'. Chapter 3: inquisitive logic.
Load more

Recommended publications
  • Radical Inquisitive Semantics∗
    Radical inquisitive semantics∗ Jeroen Groenendijk and Floris Roelofsen ILLC/Department of Philosophy University of Amsterdam http://www.illc.uva.nl/inquisitive-semantics October 6, 2010 Contents 1 Inquisitive semantics: propositions as proposals 2 1.1 Logic and conversation . .3 1.2 Compliance . .4 1.3 Counter-compliance . .5 2 Radical inquisitive semantics 6 2.1 Atoms, disjunction, and conjunction . .8 2.2 Negation . 12 2.3 Questions . 15 2.4 Implication . 18 3 Refusal, supposition and the Ramsey test 23 3.1 Refusing a proposal . 24 3.2 Suppositions . 26 ∗This an abridged version of a paper in the making. Earlier versions were presented in a seminar at the University of Massachusetts Amherst on April 12 and 21, 2010, at the Journ´ees S´emantiqueet Mod´elisation in Nancy (March 25, 2010), at the Formal Theories of Communication workshop at the Lorentz Center in Leiden (February 26, 2010), and at colloquia at the ILLC in Amsterdam (February 5, 2010) and the ICS in Osnabruck¨ (January 13, 2010). 1 3.3 Conditional questions . 27 3.4 The Ramsey test . 28 3.5 Conditional questions and questioned conditionals . 30 4 Minimal change semantics for conditionals 32 4.1 Unconditionals . 33 4.2 Dependency . 34 4.3 Minimal change with disjunctive antecedents . 36 4.4 Alternatives without disjunction . 39 4.5 Strengthening versus simplification . 40 5 Types of proposals 41 5.1 Suppositionality, informativeness, and inquisitiveness . 41 5.2 Questions, assertions, and hybrids . 42 6 Assessing a proposal 42 6.1 Licity, acceptability, and objectionability . 42 6.2 Support and unobjectionability . 44 6.3 Resolvability and counter-resolvability .
    [Show full text]
  • Vagueness: Insights from Martin, Jeroen, and Frank
    Vagueness: insights from Martin, Jeroen, and Frank Robert van Rooij 1 Introduction My views on the semantic and pragmatic interpretation of natural language were in- fluenced a lot by the works of Martin, Jeroen and Frank. What attracted me most is that although their work is always linguistically relevant, it still has a philosophical point to make. As a student (not in Amsterdam), I was already inspired a lot by their work on dynamics semantics, and by Veltman’s work on data semantics. Afterwards, the earlier work on the interpretation of questions and aswers by Martin and Jeroen played a signifcant role in many of my papers. Moreover, during the years in Amster- dam I developped—like almost anyone else who ever teached logic and/or philosophy of language to philosophy students at the UvA—a love-hate relation with the work of Wittgenstein, for which I take Martin to be responsible, at least to a large extent. In recent years, I worked a lot on vagueness, and originally I was again influenced by the work of one of the three; this time the (unpublished) notes on vagueness by Frank. But in recent joint work with Pablo Cobreros, Paul Egr´eand Dave Ripley, we developed an analysis of vagueness and transparant truth far away from anything ever published by Martin, Jeroen, or Frank. But only recently I realized that I have gone too far: I should have taken more seriously some insights due to Martin, Jeroen and/or Frank also for the analysis of vagueness and transparant truth. I should have listened more carefully to the wise words of Martin always to take • Wittgenstein seriously.
    [Show full text]
  • Inquisitive Semantics OUP CORRECTED PROOF – FINAL, //, Spi
    OUP CORRECTED PROOF – FINAL, //, SPi Inquisitive Semantics OUP CORRECTED PROOF – FINAL, //, SPi OXFORD SURVEYS IN SEMANTICS AND PRAGMATICS general editors: Chris Barker, NewYorkUniversity, and Christopher Kennedy, University of Chicago advisory editors: Kent Bach, San Francisco State University; Jack Hoeksema, University of Groningen;LaurenceR.Horn,Yale University; William Ladusaw, University of California Santa Cruz; Richard Larson, Stony Brook University; Beth Levin, Stanford University;MarkSteedman,University of Edinburgh; Anna Szabolcsi, New York University; Gregory Ward, Northwestern University published Modality Paul Portner Reference Barbara Abbott Intonation and Meaning Daniel Büring Questions Veneeta Dayal Mood Paul Portner Inquisitive Semantics Ivano Ciardelli, Jeroen Groenendijk, and Floris Roelofsen in preparation Aspect Hana Filip Lexical Pragmatics Laurence R. Horn Conversational Implicature Yan Huang OUP CORRECTED PROOF – FINAL, //, SPi Inquisitive Semantics IVANO CIARDELLI, JEROEN GROENENDIJK, AND FLORIS ROELOFSEN 1 OUP CORRECTED PROOF – FINAL, //, SPi 3 Great Clarendon Street, Oxford, ox dp, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Ivano Ciardelli, Jeroen Groenendijk, and Floris Roelofsen The moral rights of the authors have been asserted First Edition published in Impression: Some rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, for commercial purposes, without the prior permission in writing of Oxford University Press, or as expressly permitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rights organization. This is an open access publication, available online and distributed under the terms ofa Creative Commons Attribution – Non Commercial – No Derivatives .
    [Show full text]
  • Reasoning on Issues: One Logic and a Half
    Reasoning on issues: one logic and a half Ivano Ciardelli partly based on joint work with Jeroen Groenendijk and Floris Roelofsen KNAW Colloquium on Dependence Logic — 3 March 2014 Overview 1. Dichotomous inquisitive logic: reasoning with issues 2. Inquisitive epistemic logic: reasoning about entertaining issues 3. Inquisitive dynamic epistemic logic: reasoning about raising issues Overview 1. Dichotomous inquisitive logic: reasoning with issues 2. Inquisitive epistemic logic: reasoning about entertaining issues 3. Inquisitive dynamic epistemic logic: reasoning about raising issues Preliminaries Information states I Let W be a set of possible worlds. I Definition: an information state is a set of possible worlds. I We identify a body of information with the worlds compatible with it. I t is at least as informed as s in case t ⊆ s. I The state ; compatible with no worlds is called the absurd state. w1 w2 w1 w2 w1 w2 w1 w2 w3 w4 w3 w4 w3 w4 w3 w4 (a) (b) (c) (d) Preliminaries Issues I Definition: an issue is a non-empty, downward closed set of states. I An issue is identified with the information needed to resolve it. S I An issue I is an issue over a state s in case s = I. I The alternatives for an issue I are the maximal elements of I. w1 w2 w1 w2 w1 w2 w1 w2 w3 w4 w3 w4 w3 w4 w3 w4 (e) (f) (g) (h) Four issues over fw1; w2; w3; w4g: only alternatives are displayed. Part I Dichotomous inquisitive logic: reasoning with issues Dichotomous inquisitive semantics Definition (Syntax of InqDπ) LInqDπ consists of a set L! of declaratives and a set L? of interrogatives: 1.
    [Show full text]
  • Negation, Alternatives, and Negative Polar Questions in American English
    Negation, alternatives, and negative polar questions in American English Scott AnderBois Scott [email protected] February 7, 2016 Abstract A longstanding puzzle in the semantics/pragmatics of questions has been the sub- tle differences between positive (e.g. Is it . ?), low negative (Is it not . ?), and high negative polar questions (Isn't it . ?). While they are intuitively ways of ask- ing \the same question", each has distinct felicity conditions and gives rise to different inferences about the speaker's attitude towards this issue and expectations about the state of the discourse. In contrast to their non-interchangeability, the vacuity of double negation means that most theories predict all three to be semantically identical. In this chapter, we build on the non-vacuity of double negation found in inquisitive seman- tics (e.g. Groenendijk & Roelofsen (2009), AnderBois (2012), Ciardelli et al. (2013)) to break this symmetry. Specifically, we propose a finer-grained version of inquisitive semantics { what we dub `two-tiered' inquisitive semantics { which distinguishes the `main' yes/no issue from secondary `projected' issues. While the main issue is the same across positive and negative counterparts, we propose an account deriving their distinc- tive properties from these projected issues together with pragmatic reasoning about the speaker's choice of projected issue. Keywords: Bias, Negation, Polar Questions, Potential QUDs, Verum Focus 1 Introduction When a speaker wants to ask a polar question in English, they face a choice between a bevy of different possible forms. While some of these differ dramatically in form (e.g. rising declaratives, tag questions), even focusing more narrowly on those which only have interrogative syntax, we find a variety of different forms, as in (1).
    [Show full text]
  • A Dynamic Semantics of Single-Wh and Multiple-Wh Questions*
    Proceedings of SALT 30: 376–395, 2020 A dynamic semantics of single-wh and multiple-wh questions* Jakub Dotlacilˇ Floris Roelofsen Utrecht University University of Amsterdam Abstract We develop a uniform analysis of single-wh and multiple-wh questions couched in dynamic inquisitive semantics. The analysis captures the effects of number marking on which-phrases, and derives both mention-some and mention-all readings as well as an often neglected partial mention- some reading in multiple-wh questions. Keywords: question semantics, multiple-wh questions, mention-some, dynamic inquisitive semantics. 1 Introduction The goal of this paper is to develop an analysis of single-wh and multiple-wh questions in English satisfying the following desiderata: a) Uniformity. The interpretation of single-wh and multiple-wh questions is derived uniformly. The same mechanisms are operative in both types of questions. b) Mention-some vs mention-all. Mention-some and mention-all readings are derived in a principled way, including ‘partial mention-some readings’ in multiple-wh questions (mention-all w.r.t. one of the wh-phrases but mention-some w.r.t. the other wh-phrase). c) Number marking. The effects of number marking on wh-phrases are captured. In particular, singular which-phrases induce a uniqueness requirement in single-wh questions but do not block pair-list readings in multiple-wh questions. To our knowledge, no existing account fully satisfies these desiderata. While we cannot do justice to the rich literature on the topic, let us briefly mention a number of prominent proposals. Groenendijk & Stokhof(1984) provide a uniform account of single-wh and multiple-wh ques- tions, but their proposal does not successfully capture mention-some readings and the effects of number marking.
    [Show full text]
  • Epistemic Implicatures and Inquisitive Bias: a Multidimensional Semantics for Polar Questions
    EPISTEMIC IMPLICATURES AND INQUISITIVE BIAS: A MULTIDIMENSIONAL SEMANTICS FOR POLAR QUESTIONS by Morgan Mameni B.A., Simon Fraser University, 2008 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN THE DEPARTMENT OF LINGUISTICS c Morgan Mameni 2010 SIMON FRASER UNIVERSITY Summer 2010 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Morgan Mameni Degree: Master of Arts Title of Thesis: Epistemic Implicatures and Inquisitive Bias: A Multidimensional Semantics for Polar Questions Examining Committee: Dr. John Alderete (Chair) Dr. Nancy Hedberg (Senior Supervisor) Associate Professor of Linguistics, SFU Dr. Chung-Hye Han (Supervisor) Associate Professor of Linguistics, SFU Dr. Francis Jeffry Pelletier (Supervisor) Professor of Linguistics and Philosophy, SFU Dr. Hotze Rullmann (Supervisor) Associate Professor of Linguistics, UBC Dr. Philip P. Hanson (External Examiner) Associate Professor of Philosophy, SFU Date Approved: July 2, 2010 ii Abstract This thesis motivates and develops a semantic distinction between two types of polar in- terrogatives available to natural languages, based on data from Persian and English. The first type, which I call an ‘impartial interrogative,’ has as its pragmatic source an ignorant information state, relative to an issue at a particular stage of the discourse. The second type, which I call a ‘partial interrogative’ arises from a destabilized information state, whereby the proposition supported by the information state conflicts with contextually available data.
    [Show full text]
  • Inquisitive Semantics: a New Notion of Meaning
    Inquisitive semantics: a new notion of meaning Ivano Ciardelli Jeroen Groenendijk Floris Roelofsen March 26, 2013 Abstract This paper presents a notion of meaning that captures both informative and inquisitive content, which forms the cornerstone of inquisitive seman- tics. The new notion of meaning is explained and motivated in detail, and compared to previous inquisitive notions of meaning. 1 Introduction Recent work on inquisitive semantics has given rise to a new notion of mean- ing, which captures both informative and inquisitive content in an integrated way. This enriched notion of meaning generalizes the classical, truth-conditional notion of meaning, and provides new foundations for the analysis of linguistic discourse that is aimed at exchanging information. The way in which inquisitive semantics enriches the notion of meaning changes our perspective on logic as well. Besides the classical notion of en- tailment, the semantics also gives rise to a new notion of inquisitive entailment, and to new logical notions of relatedness, which determine, for instance, whether a sentence compliantly addresses or resolves a given issue. The enriched notion of semantic meaning also changes our perspective on pragmatics. The general objective of pragmatics is to explain aspects of inter- pretation that are not directly dictated by semantic content, in terms of general features of rational human behavior. Gricean pragmatics has fruitfully pursued this general objective, but is limited in scope. Namely, it is only concerned with what it means for speakers to behave rationally in providing information. In- quisitive pragmatics is broader in scope: it is both speaker- and hearer-oriented, and is concerned more generally with what it means to behave rationally in co- operatively exchanging information rather than just in providing information.
    [Show full text]
  • A Dynamic Semantics of Single and Multiple Wh-Questions
    A dynamic semantics of single and multiple wh-questions Jakub Dotlacilˇ Floris Roelofsen Utrecht University University of Amsterdam SALT, August 20, 2020 1 Desiderata The goal of this paper is to develop an analysis of single and multiple wh-questions • in English satisfying the following desiderata: (i) Uniformity. The interpretation of single and multiple wh-questions is derived uniformly. The same mechanism is operative in both types of questions. (ii) Mention-some vs mention-all. Mention-some and mention-all readings are derived in a principled way, and contraints on the availability of mention-some readings in single and multiple wh-questions are captured. (iii) Number marking. The effects of number marking on wh-phrases is captured. In particular, singular which-phrases induce a uniqueness requirement in single wh-questions but do not block pair-list readings in multiple wh-questions. To our knowledge, no existing account fully satisfies these three desiderata. • For instance: (we cannot do justice here to the rich literature on the topic) • – Groenendijk and Stokhof(1984) provide a uniform account of single and mul- tiple wh-questions, but their proposal does not successfully capture mention- some readings and the effects of number marking. – Dayal(1996) captures e ffects of number marking in single and multiple wh- questions, but her account does not succesfully capture mention-some readings and is not uniform, in the sense that it involves different question operators for single and multiple wh-questions, respectively. Moreover, as pointed out by Xiang(2016, 2020b), the so-called domain exhaus- tivity presupposition that Dayal predicts for multiple wh-questions is not borne out (examples will be given later).
    [Show full text]
  • A Note on Representing Uncertainty in Inquisitive Semantics Dylan Bumford July 7, 2014
    A note on representing uncertainty in Inquisitive Semantics Dylan Bumford July 7, 2014 A classical probability space hΩ, χ, pi consists of a set Ω (referred to as the sample space), a σ-algebra χ ⊆ ´(Ω) (i.e. a family of sets closed under union and complementation), and a measure p: χ → [0, 1] mapping subsets of the sample space Ω to real numbers in such a way that for any E ∈ χ and any set of disjoint Ei ∈ χ: • p(E) ≥ 0 • p(Ω) = 1 S P • p( Ei) = p(Ei) This classical conception fits cleanly around a possible worlds semantics for classical logic. To see this, let · : L → ´(W ) be the standard Tarskian interpretation function mapping classical formulas J K to the sets of worlds in which they’re true. Then from any probability function pW : ´(W ) → [0, 1] defined on W , the set of possible worlds, we can read off a probability function p: L → [0, 1] defined on classical formulas by composing pW with · . That is, we set p(φ) = pW ( φ ). Under conditions of certainty, this transformationJ willK respect classical entailmentJ andK validity: if Γ ` φ, then every probability function that assigns 1 to each γ ∈ Γ will also assign 1 to φ. The converse is also true: if every probability function that assigns 1 to each γ ∈ Γ also assigns 1 to φ, then Γ ` φ. As a result, this notion of probabilistic validity is sound and complete with respect to classical logic (Leblanc 1983). More generally, the compositional procedure that routes probabilistic interpretation through classical possible worlds semantics will verify the following equations, for any formula φ and any set of mutually inconsistent formulas φi: • p(φ) ≥ 0 • p(T) = 1 W P • p( φi) = p(φi) For the class of real-valued interpretation functions satisfying these constraints, Adams(1975) established that probability preservation is sound and complete with respect to classical logic.
    [Show full text]
  • Puzzling Response Particles: an Experimental Study on the German Answering System
    Semantics & Pragmatics Volume 10, Article 19, 2017 https://doi.org/10.3765/sp.10.19 This is an early access version of Claus, Berry, A. Marlijn Meijer, Sophie Repp & Manfred Krifka. 2017. Puzzling response particles: An experimental study on the German answering system. Semantics and Pragmatics 10(19). https://doi.org/10.3765/sp.10. 19. This version will be replaced with the final typeset version in due course. Note that page numbers will change, so cite with caution. ©2017 Berry Claus, A. Marlijn Meijer, Sophie Repp, and Manfred Krifka This is an open-access article distributed under the terms of a Creative Commons Attribution License (https://creativecommons.org/licenses/by/3.0/). early access Puzzling response particles: An experimental study on the German answering system* Berry Claus A. Marlijn Meijer Humboldt-Universität zu Berlin Universität zu Köln Sophie Repp Manfred Krifka Universität zu Köln Humboldt-Universität zu Berlin Abstract This paper addresses the use and interpretation of the German re- sponse particles ja, nein, and doch. In four experiments, we collected acceptability- judgement data for the full paradigm of standard German particles in responses to positive and negative assertions. The experiments were designed to test the empirical validity of two recent accounts of response particles, Roelofsen & Farkas(2015) and Krifka(2013), which view response particles as propositional anaphors. The results for responses to negative antecedents were unpredicted and inconsistent with either account. A further unexpected finding was that there was large interindividual variation in the acceptability patterns for affirming responses to negative antecedents to the extent that most speakers found ja more acceptable whereas some found nein more acceptable.
    [Show full text]
  • Proof-Theoretic Semantics and Inquisitive Logic
    Proof-Theoretic Semantics and Inquisitive Logic Will Stafford December 19, 2020 Abstract Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This article resolves one of the questions left open by this recent result by showing the exten- sional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of the logic of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restrict- ing proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic. Contents 1 Introduction 2 2 Logics Without Closure Under Substitution3 2.1 Weak Logics.........................................3 2.2 Equality Between Weak Logics with the Disjunction Property.............4 3 Inquisitive Logic8 4 Proof-Theoretic Validity 12 4.1 Atomic Systems....................................... 12 4.2 Prawitz's Definition of Proof-Theoretic Validity..................... 15 4.3 PCS Semantics....................................... 19 5 Results about PCS semantics 21 5.1 Relations between proof-theoretic systems........................ 21 5.2 S1 Satisfies the Kreisel-Putnam axiom......................... 23 6 Proof and Consequences of Main Theorem 25 7 Conclusion 26 1 1 Introduction Proof-Theoretic Validity was proposed by Prawitz (1971) as an explication of Gentzen's famous claim that the introduction rules can be viewed as definitions of the connectives.1 Prawitz's formal definition of proof-theoretic validity (henceforth PTV) is complex but, roughly, a proof is valid if it stands in the correct relationship to the introduction rules (see Definition 4.8).
    [Show full text]