Epistemic Modality, Mind, and Mathematics
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On Proof Interpretations and Linear Logic
On Proof Interpretations and Linear Logic On Proof Interpretations and Linear Logic Paulo Oliva Queen Mary, University of London, UK ([email protected]) Heyting Day Utrecht, 27 February 2015 To Anne Troelstra and Dick de Jongh 6 ?? - !A ::: - !A - ! ILLp ILLp ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations IL = Intuitionistic predicate logic 6 ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) ?? - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) - !A ::: - !A On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Overview - ! IL ::: - IL interpretations 6 ?? - ! ILLp ILLp - !A ::: - !A IL = Intuitionistic predicate logic ILLp = Intuit. predicate linear logic (!-free) On Proof Interpretations and Linear Logic Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic On Proof Interpretations and Linear Logic Proof Interpretations Outline 1 Proof Interpretations 2 Linear Logic 3 Unified Interpretation of Linear Logic E.g. the set of prime numbers is infinite The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. the set of even numbers is finite Need to fix the formal system On Proof Interpretations and Linear Logic Proof Interpretations Formulas as Sets of Proofs Formulas as set of proofs A 7! fπ : π ` Ag The book \Proofs from THE BOOK" contains six beautiful proofs of this Non-theorems map to the empty set E.g. -
On Root Modality and Thematic Relations in Tagalog and English*
Proceedings of SALT 26: 775–794, 2016 On root modality and thematic relations in Tagalog and English* Maayan Abenina-Adar Nikos Angelopoulos UCLA UCLA Abstract The literature on modality discusses how context and grammar interact to produce different flavors of necessity primarily in connection with functional modals e.g., English auxiliaries. In contrast, the grammatical properties of lexical modals (i.e., thematic verbs) are less understood. In this paper, we use the Tagalog necessity modal kailangan and English need as a case study in the syntax-semantics of lexical modals. Kailangan and need enter two structures, which we call ‘thematic’ and ‘impersonal’. We show that when they establish a thematic dependency with a subject, they express necessity in light of this subject’s priorities, and in the absence of an overt thematic subject, they express necessity in light of priorities endorsed by the speaker. To account for this, we propose a single lexical entry for kailangan / need that uniformly selects for a ‘needer’ argument. In thematic constructions, the needer is the overt subject, and in impersonal constructions, it is an implicit speaker-bound pronoun. Keywords: modality, thematic relations, Tagalog, syntax-semantics interface 1 Introduction In this paper, we observe that English need and its Tagalog counterpart, kailangan, express two different types of necessity depending on the syntactic structure they enter. We show that thematic constructions like (1) express necessities in light of priorities of the thematic subject, i.e., John, whereas impersonal constructions like (2) express necessities in light of priorities of the speaker. (1) John needs there to be food left over. -
The Logic of Brouwer and Heyting
THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds. -
Finite Type Arithmetic
Finite Type Arithmetic Computable Existence Analysed by Modified Realisability and Functional Interpretation H Klaus Frovin Jørgensen Master’s Thesis, March 19, 2001 Supervisors: Ulrich Kohlenbach and Stig Andur Pedersen Departments of Mathematics and Philosohy University of Roskilde Abstract in English and Danish Motivated by David Hilbert’s program and philosophy of mathematics we give in the context of natural deduction an introduction to the Dialectica interpretation and compare the interpre- tation with modified realisability. We show how the interpretations represent two structurally different methods for unwinding computable information from proofs which may use certain prima facie non-constructive (ideal) elements of mathematics. Consequently, the two inter- pretations also represent different views on what is to be regarded as constructive relative to arithmetic. The differences show up in the interpretations of extensionality, Markov’s prin- ciple and restricted forms of independence-of-premise. We show that it is computationally a subtle issue to combine these ideal elements and prove that Markov’s principle is computa- tionally incompatible with independence-of-premise for negated purely universal formulas. In the context of extracting computational content from proofs in typed classical arith- metic we also compare in an extensional context (i) the method provided by negative trans- lation + Dialectica interpretation with (ii) the method provided by negative translation + ω A-translation + modified realisability. None of these methods can be applied fully to E-PA , since E-HAω is not closed under Markov’s rule, whereas the method based on the Dialectica interpretation can be used if only weak extensionality is required. Finally, we present a new variant of the Dialectica interpretation in order to obtain (the well-known) existence property, disjunction property and other closure results for typed in- tuitionistic arithmetic and extensions hereof. -
The Consistency of Arithmetic
The Consistency of Arithmetic Timothy Y. Chow July 11, 2018 In 2010, Vladimir Voevodsky, a Fields Medalist and a professor at the Insti- tute for Advanced Study, gave a lecture entitled, “What If Current Foundations of Mathematics Are Inconsistent?” Voevodsky invited the audience to consider seriously the possibility that first-order Peano arithmetic (or PA for short) was inconsistent. He briefly discussed two of the standard proofs of the consistency of PA (about which we will say more later), and explained why he did not find either of them convincing. He then said that he was seriously suspicious that an inconsistency in PA might someday be found. About one year later, Voevodsky might have felt vindicated when Edward Nelson, a professor of mathematics at Princeton University, announced that he had a proof not only that PA was inconsistent, but that a small fragment of primitive recursive arithmetic (PRA)—a system that is widely regarded as implementing a very modest and “safe” subset of mathematical reasoning—was inconsistent [11]. However, a fatal error in his proof was soon detected by Daniel Tausk and (independently) Terence Tao. Nelson withdrew his claim, remarking that the consistency of PA remained “an open problem.” For mathematicians without much training in formal logic, these claims by Voevodsky and Nelson may seem bewildering. While the consistency of some axioms of infinite set theory might be debatable, is the consistency of PA really “an open problem,” as Nelson claimed? Are the existing proofs of the con- sistency of PA suspect, as Voevodsky claimed? If so, does this mean that we cannot be sure that even basic mathematical reasoning is consistent? This article is an expanded version of an answer that I posted on the Math- Overflow website in response to the question, “Is PA consistent? do we know arXiv:1807.05641v1 [math.LO] 16 Jul 2018 it?” Since the question of the consistency of PA seems to come up repeat- edly, and continues to generate confusion, a more extended discussion seems worthwhile. -
Subverting the Classical Picture: (Co)Algebra, Constructivity and Modalities∗
Informatik 8, FAU Erlangen-Nürnberg [email protected] Subverting the Classical Picture: (Co)algebra, Constructivity and Modalities∗ Tadeusz Litak Contents 1. Subverting the Classical Picture 2 2. Beyond Frege and Russell: Modalities Instead of Predicates 6 3. Beyond Peirce, Tarski and Kripke: Coalgebras and CPL 11 3.1. Introduction to the Coalgebraic Framework ............. 11 3.2. Coalgebraic Predicate Logic ...................... 14 4. Beyond Cantor: Effectivity, Computability and Determinacy 16 4.1. Finite vs. Classical Model Theory: The Case of CPL ........ 17 4.2. Effectivity, Choice and Determinacy in Formal Economics ..... 19 4.3. Quasiconstructivity and Modal Logic ................. 21 5. Beyond Scott, Montague and Moss: Algebras and Possibilities 23 5.1. Dual Algebras, Neighbourhood Frames and CA-baes ........ 23 5.2. A Hierachy of (In)completeness Notions for Normal Modal Logics . 24 5.3. Possibility Semantics and Complete Additivity ........... 26 5.4. GQM: One Modal Logic to Rule Them All? ............. 28 6. Beyond Boole and Lewis: Intuitionistic Modal Logic 30 6.1. Curry-Howard Correspondence, Constructive Logic and 2 ..... 31 6.2. Guarded Recursion and Its Neighbours ................ 33 6.3. KM, mHC, LC and the Topos of Trees ................ 34 6.4. Between Boole and Heyting: Negative Translations ......... 35 7. Unifying Lewis and Brouwer 37 7.1. Constructive Strict Implication .................... 37 7.2. Relevance, Resources and (G)BI .................... 40 ∗Introductory overview chapter for my cumulative habilitation thesis as required by FAU Erlangen-Nürnberg regulations (submitted in 2018, finally approved in May 2019). Available online at https://www8.cs.fau. de/ext/litak/gitpipe/tadeusz_habilitation.pdf. 1 8. Beyond Brouwer, Heyting and Kolmogorov: Nondistributivity 43 8.1. -
Contextual Epistemic Logic Manuel Rebuschi, Franck Lihoreau
Contextual Epistemic Logic Manuel Rebuschi, Franck Lihoreau To cite this version: Manuel Rebuschi, Franck Lihoreau. Contextual Epistemic Logic. C. Degrémont, L. Keiff & H. Rückert. Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman, King’s College Publication, pp.305-335, 2008, Tributes. hal-00133359 HAL Id: hal-00133359 https://hal.archives-ouvertes.fr/hal-00133359 Submitted on 11 Jan 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Contextual Epistemic Logic Manuel Rebuschi Franck Lihoreau L.H.S.P. – Archives H. Poincar´e Instituto de Filosofia da Linguagem Universit´eNancy 2 Universidade Nova de Lisboa [email protected] [email protected] Abstract One of the highlights of recent informal epistemology is its growing theoretical emphasis upon various notions of context. The present paper addresses the connections between knowledge and context within a formal approach. To this end, a “contextual epistemic logic”, CEL, is proposed, which consists of an extension of standard S5 epistemic modal logic with appropriate reduction axioms to deal with an extra contextual operator. We describe the axiomatics and supply both a Kripkean and a dialogical semantics for CEL. -
Epistemic Modality and Evidentiality in Gitksan at the Semantics-Pragmatics Interface By
Epistemic Modality and Evidentiality in Gitksan at the Semantics-Pragmatics Interface by Tyler Roy Gösta Peterson B.Mus., University of British Columbia, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Linguistics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April, 2010 c Tyler Roy Gösta Peterson 2010 Abstract The aim of this dissertation is to provide an empirically driven, theoretically informed investigation of how speakers of Gitksan, a Tsimshianic language spoken in the northwest coast of Canada, express knowledge about the world around them. There are three main goals that motivate this investigation, summarized below: (1) (i.) To provide the first detailed description of the evidential and modal system in Gitksan. (ii.) To provide a formal semantic and pragmatic account of this system that adequately explains the meanings of the modals and evidentials, as well as how they are used in discourse. (iii.) To identify and examine the specific properties the Gitksan evidential/modal system brings to bear on current theories of semantics and pragmatics, as well as the consequences this analysis has on the study of modality and evidentiality cross-linguistically. In addition to documenting the evidential and modal meanings in Gitksan, I test and work through a variety of theoretical tools from the literature designed to investigate evidentiality and modality in a language. This begins by determining what level of mean- ing the individual evidentials in Gitksan operate on. The current state of research into the connection between evidentiality and epistemic modality has identified two different types of evidentials defined by the level of meaning they operate on: propositional and illocutionary evidentials. -
On the Unusual Effectiveness of Proof Theory in Epistemology
WORKSHOP ON RECENT TRENDS IN PROOF THEORY, Bern, July 9-11, 2008 On the unusual effectiveness of proof theory in epistemology Sergei Artemov The City University of New York Bern, July 11, 2008 This is a report about the extending the ideas and methods of Proof Theory to a new, promising area. It seems that this is a kind of development in which Proof Theory might be interested. This is a report about the extending the ideas and methods of Proof Theory to a new, promising area. It seems that this is a kind of development in which Proof Theory might be interested. Similar stories about what the Logic of Proofs brings to foundations, constructive semantics, combinatory logic and lambda-calculi, the- ory of verification, cryptography, etc., lie mostly outside the scope of this talk. Mainstream Epistemology: tripartite approach to knowledge (usually attributed to Plato) Knowledge Justified True Belief. ∼ A core topic in Epistemology, especially in the wake of papers by Russell, Gettier, and others: questioned, criticized, revised; now is generally regarded as a necessary condition for knowledge. Logic of Knowledge: the model-theoretic approach (Kripke, Hin- tikka, . .) has dominated modal logic and formal epistemology since the 1960s. F F holds at all possible worlds (situations). ! ∼ Logic of Knowledge: the model-theoretic approach (Kripke, Hin- tikka, . .) has dominated modal logic and formal epistemology since the 1960s. F F holds at all possible worlds (situations). ! ∼ Easy, visual, useful in many cases, but misses the mark considerably: What if F holds at all possible worlds, e.g., a mathematical truth, say P = NP , but the agent is simply not aware of the fact due to " lack of evidence, proof, justification, etc.? Logic of Knowledge: the model-theoretic approach (Kripke, Hin- tikka, . -
Interactions of Metaphysical and Epistemic Concepts
Interactions of metaphysical and epistemic concepts Th`esepr´esent´ee`ala Facult´edes lettres et sciences humaines Universit´ede Neuchˆatel- CH Pour l’obtention du grade de docteur `eslettres Par Alexandre Fernandes Batista Costa Leite Accept´eesur proposition du jury: Prof. Jean-Yves B´eziau, Universit´ede Neuchˆatel, directeur de th`ese Prof. Pascal Engel, Universit´ede Gen`eve, rapporteur Prof. Paul Gochet, Universit´ede Li`ege,rapporteur Prof. Arnold Koslow, The City University of New York, rapporteur Soutenue le 03 Juillet 2007 Universit´ede Neuchˆatel 2007 lnlTT Faculté des lettres et sciences humaines Le doyen r EspaceLouis-Agassiz 1 I CH-2000Neuchâtel IMPRIMATUR La Facultédes lettreset scienceshumaines de I'Universitéde Neuchâtel,sur les rapportsde Mon- sieurJean-Yves Béziau, directeur de thèse,professeur assistant de psychologieà I'Université de Neu- châtel; M. PascalEngel, professeur à I'Universitéde Genève; M. Paul Gochet,professeur à I'Universitéde Liège;M. ArnoldKoslow, professeur à CityUniversity of NewYork autorise I'impres- sionde la thèseprésentée par Monsieur Alexandre Costa Leite, en laissantà I'auteurla responsabilité desopinions énoncées. .\A \.\ i \,^_ Neuchâtel,le 3 juillet2007 Le doyen J.-J.Aubert r Téléphone'.+41 32718 17 0O r Fax: +4132718 17 01 . E-mail: [email protected] www.unine.ch/lettres Abstract This work sets out the results of research on topics at the intersection of logic and philosophy. It shows how methods for combining logics can be applied to the study of epistemology and metaphysics. In a broader per- spective, it investigates interactions of modal concepts in order to create a bridge between metaphysics and epistemology. -
30. Tense Aspect Mood 615
30. Tense Aspect Mood 615 Richards, Ivor Armstrong 1936 The Philosophy of Rhetoric. Oxford: Oxford University Press. Rockwell, Patricia 2007 Vocal features of conversational sarcasm: A comparison of methods. Journal of Psycho- linguistic Research 36: 361−369. Rosenblum, Doron 5. March 2004 Smart he is not. http://www.haaretz.com/print-edition/opinion/smart-he-is-not- 1.115908. Searle, John 1979 Expression and Meaning. Cambridge: Cambridge University Press. Seddiq, Mirriam N. A. Why I don’t want to talk to you. http://notguiltynoway.com/2004/09/why-i-dont-want- to-talk-to-you.html. Singh, Onkar 17. December 2002 Parliament attack convicts fight in court. http://www.rediff.com/news/ 2002/dec/17parl2.htm [Accessed 24 July 2013]. Sperber, Dan and Deirdre Wilson 1986/1995 Relevance: Communication and Cognition. Oxford: Blackwell. Voegele, Jason N. A. http://www.jvoegele.com/literarysf/cyberpunk.html Voyer, Daniel and Cheryl Techentin 2010 Subjective acoustic features of sarcasm: Lower, slower, and more. Metaphor and Symbol 25: 1−16. Ward, Gregory 1983 A pragmatic analysis of epitomization. Papers in Linguistics 17: 145−161. Ward, Gregory and Betty J. Birner 2006 Information structure. In: B. Aarts and A. McMahon (eds.), Handbook of English Lin- guistics, 291−317. Oxford: Basil Blackwell. Rachel Giora, Tel Aviv, (Israel) 30. Tense Aspect Mood 1. Introduction 2. Metaphor: EVENTS ARE (PHYSICAL) OBJECTS 3. Polysemy, construal, profiling, and coercion 4. Interactions of tense, aspect, and mood 5. Conclusion 6. References 1. Introduction In the framework of cognitive linguistics we approach the grammatical categories of tense, aspect, and mood from the perspective of general cognitive strategies. -
University of Groningen Expressions of Epistemic Modality in Mainland
University of Groningen Expressions of epistemic modality in Mainland Scandinavian Beijering, Karin IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2012 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Beijering, K. (2012). Expressions of epistemic modality in Mainland Scandinavian Groningen: s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 11-02-2018 Expressions of epistemic modality in Mainland Scandinavian A study into the lexicalization-grammaticalization-pragmaticalization interface KARIN BEIJERING The research reported on in this thesis has been carried out under the auspices of the Netherlands National Graduate School of Linguistics (LOT – Landelijke Onderzoekschool Taalwetenschap) and the Center for Language and Cognition Groningen (CLCG) of the Faculty of Arts of the University of Groningen. Publication of this dissertation was financially supported by the University of Groningen.