DOCSLIB.ORG

From Syllogism to Common Sense Normal Modal Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From Syllogism to Common Sense Normal Modal Logic From Syllogism to Common Sense Normal Modal Logic K r i p k e S e m a n t i c s Completeness Mehul Bhatt and Correspondence Theory Oliver Kutz Thomas Schneider Lecture 9 Department of Computer Science & Research Center on Spatial Cognition (SFB/TR 8) University of Bremen Examples of Modal Logics Examples of Modal Logics Classic Distinctions between Modalities Modern interpretations of modalities ‣ Mathematical Logic: ‣ Alethic modality: necessity, possibility, contingency, impossibility ‣ The logic of proofs GL: [] A means: In PA it is provable that ‘A’. ‣ distinguish further: logical - physical - metaphysical, etc. ‣ Computer Science: ‣ Temporal modality: always, some time, never ‣ Linear Temporal Logic LTL: Formal Verification ‣ Deontic modality: obligatory, permissible ‣ X A : in the next moment ‘A’ ‣ Epistemic modality: it is known that ‣ A U B: A is true until B becomes true ‣ Doxastic modality: it is believed that ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: Technically, all these modalities are treated ‣ G F A or also G (B -> F A) in the same way, by using unary modal operators ‣ Linguistics / KR / etc. Modal Logic: Some History A Hilbert system for Modal Logic K ‣ The following is the standard Hilbert system for the modal logic K. ‣ Modern modal logic typically begins with the systems devised by Axioms !1 → (!2 → !1) C. I. LEWIS, intended to model strict implication and avoid the (!1 → !2) → (!1 → (!2 → !3)) → (!1 → !3) paradoxes of material implication, such as the ‘ex falso quodlibet’. !1 → !1 ∨ !2 ‣ “ If it never rains in Copenhagen, then Elvis never died.” !2 → !1 ∨ !2 (!1 → !3) → (!2 → !3) → (!1 ∨ !2 → !3) ‣ (No variables are shared in example => relevant implication) (!1 → !2) → (!1 →¬!2) →¬!1 two classical tautologies ‣ For strict implication, we define A ~~> B by [] (A --> B) ¬¬!1 → !1 ! ∧ ! → ! instead of ! " p in INT ‣ These systems are however mutually incompatible, and no base 1 2 1 logic was given of which the other logics are extensions of. !1 ∧ !2 → !2 !1 → !2 → !1 ∧ !2 ‣ The modal logic K is such a base logic, named after SAUL KRIPKE, □(! → ") → (□! → □") new axiom of and which serves as a minimal logic for the class of all its (normal) Box Distribution extensions - defined next via a Hilbert system. ! ! → ! ! Rules 1 1 2 !2 □! new rule of Necessitation Table 1. AFregesystemsforthemodallogic# modal logic axioms #4 # + □! → □□! KB # + ! → □♢! GL # + □(□! → !) → □! $4 #4+ □! → ! Some More Modal frege systems $4GrzKripke$4+ Semantics□(□(! → □!) → !) → □! ‣ Hilbert systems for other modal logics are obtained by adding axioms. ‣ A Kripke frameTable consists 2. Frege of systems a set W for, the important set of `possible modal logics worlds’, and a modal logic axioms binary relation R between worlds. A valuation ! assigns propositional variables to worlds. A pointed model M is a frame, K4 K + ✷p ✷✷p x → together with a valuation and a distinguished world x. KB K + p ✷✸p As in classical logic, if we augment frames with assignments,wearriveatthe GL K + ✷(→✷p p) ✷p Mx = p q Mx = p and Mx = q → → notion of a model. | ∧ ⇐⇒ | | S4 K4+ ✷p p Mx = p q Mx = p or Mx = q → Definition 12. A model| ∨ for⇐⇒ the modal language| is| a pair (!, # ) where S4Grz S4+✷(✷(p ✷p) p) ✷p M = p q if M = p then M = q → → → x | → ⇐⇒ x | x | – ! =($, %) is a frame and ‣ More generally, in a fixed language, the class of all normal modal logics is defined Mx = p Mx = p – # : Var ($ ) |is a¬ mapping⇐⇒ assigning| to each propositional variable & a as any set of formulae that !→ # Mx = ✷p for all xRy : My = p set # (&) of worlds| ( ($ )⇐⇒denotes the power set of| $ ). ‣ (1) contains K (2) is closed under substitution and (3) Modus Ponens Mx = #✸p exists xRy : My = p ‣ In particular, any normal extension of K contains the Axiom of Box-Distribution: With the notion of| models⇐⇒ we can now define the| semantics of modal for- ✷(p q) (✷p ✷q) mulas: → → → Definition 13. Let ', ( be modal formulas, let ) =($, %, # ) be a model and * $ be a world. Inductively we define the notion of a formula to be satisfied ∈ in ) at world *: – ),* = & if * # (&) where & Var , – ),* ∣= ' if not∈ ),* = ', ∈ – ),* ∣= '¬ ( if ),* =∣ ' and ),* = ( – ),* ∣= ' ∧ ( if ),* ∣= ' or ),* =∣ ( – ),* ∣= □'∨ if for all +∣ $ with (*,∣ +) % we have ),+ = '. ∣ ∈ ∈ ∣ Amodalformula' is satisfiable if there exists a model ) =($, %, # )and aworld* $ such that ),* = '.Dually,' is a modal tautology if for every ∈ ∣ model ) =($, %, # )andevery* $ we have ),* = '. ∈ ∣ It can be shown that the Frege system from the previous sectionisindeeda proof system for the modal logic ,,i.e.,itissoundandcompleteforallmodal tautologies. 14 L10.2 Modal Tableaux (P) all propositional tautologies Modal Sat / Taut / Validity A Tableaux(K) (φ ψ) system( φ ψ) for Modal Logic K → → → φφ ψ L10.2 Modal Tableaux (MP) → ‣ Hilbert systemsψ are generally considered difficult to use in a ‣ Modal Sat: A modal formula is satisfiable if there practical way. φ exists a pointed model that satisfies it. (G) (P) all propositional tautologies ‣ There areφ many proof systems for Modal Logics. One of the most popular ones are Semantic Tableaux: (K) (φ ψ) ( φ ψ) ‣ Modal Taut: A formula→ is→ a modal → tautology if it is ‣ refutation based proof system satisfied in all pointed models. Figure 1: Modal logic K φφ ψ ‣ highly developed optimisation techniques (MP) → ψ T ‣ isallows system toK extractplus models (T) φdirectlyφ from proofs ‣ Modal Validity: A formula is valid in a class of → φ S4 ‣ ispopular system Tinplus particular (4) forφ Descriptionφ Logic based formalisms frames (G)if it a modal tautology relative to that class of → frames. φ ‣ often used for establishing upper bounds for the complexity of a SAT problem for a logic. Figure 2: Some other modal logics Check validity of Figure✷(p 1: Modalq) ( logic✷p K✷q) Box Distribution → → → is a finite sequence of natural numbers. In addition, every formula on the T is system K plus (T) φ φ tableau has a sign Z F, T that indicates the truth-value we currently → ∈ { } S4 is system T plus (4) φ φ expect for the formula in our reasoning. That is, a formula in the modal → tableaux is of the form σZA Figure 2: Some other modal logics where the prefix σ is a finite sequence of natural numbers, the sign Z is in F, T , and F is a formula of modal logic. At this point, we understand a { } isA a finiteTableaux sequence of natural system numbers. Infor addition, Modal every formula Logic on the K prefix σ as a symbolicA Tableaux name for a world insystem a Kripke structure. for Modal Logic K tableau has a sign Z F, T that indicates the truth-value we currently ∈ { } expect for the formula in our reasoning. That is, a formula in the modal Definition 1 (K prefix accessibility) For modal logic K, prefix σ is accessible ‣ In prefixed tableaux, every formula starts with a prefix and a sign tableaux is of the form from prefix σ if σ‣ isA of basic the form semanticσn for tableaux some natural for K number is givenn. as follows: Z " σZA ‣ We introduce prefixes (denotingModal Tableaux possible worlds) that keep track of accessibility. L10.3 ! ‣ For every formula of a class α with a top level operator and sign (T or ‣ Prefixes (denoting possible worlds) keep track of accessibility. ‣ Formulae in the tableaux are labelled with T or F. where the prefix σ is a finite sequence of natural numbers, the sign Z is in F for true and false) as indicated, we define two successor formulas α1 and ‣ A prefix ! is a finite sequence of natural numbers F, T , and F is a formula of modal logic. At this point, we understand a α2: ‣ We differentiate the following four kinds of formulas: { } prefix‣ Formulaeσ as a symbolic in a tableaux name forare alabelled world inwith a Kripke T or F. structure. α α1 α2 β β1 β2 ν ν0 π π0 TA B TA TB TA B TA TB T A TA T A TA ∧ ∨ ♦ Definition 1 (K prefix accessibility) For modal logic K, prefix σ is accessible FA B FA FB FA B FA FB ∨ ∧ F ♦A FA F A FA from prefix σ if σ is of the form σn for some natural number n. FA B TA FB TA B FA TB → → F A TA TA T AEveryFA combinationFA of top-level operator and sign occurs in one of the ¬ ¬ For every formula of a class α with a top level operator and sign (T or above cases. Tableau proof rules by those classes are shown in Figure 3.A For the followingConjunctive cases of formulas we defineDisjunctive one successor formulaUniversal Existential F for‣ trueExample and false) 1 4 7 as9 is indicated, accessible wefrom define 1 4 7 twowhich successor is accessble formulas from 1α 41 etc.and tableau is closed if every branch contains some pair of formulas of the form α2: ‣ These tables essentiallyσTAand encodeσFA.A theproof semanticsfor modal of the logic. logic formula consists of a closed tableau LECTURE NOTES FEBRURARY 18, 2010 α α1 α2 β β1 β2 starting with the root 1FA. TA B TA TB TA B TA TB ∧ ∨ FA B FA FB FA B FA FB σα σβ σν 1 σπ 2 ∨ ∧ (α) (β) (ν∗) (π) FA B TA FB TA B FA TB σα σβ σβ σ ν σ π → → 1 1 2 0 0 F A TA TA T A FA FA σα ¬ ¬ 2 1 For the following cases of formulas we define one successor formula σ accessible from σ and σ occurs on the branch already 2 σ is a simple unrestricted extension of σ, i.e., σ is accessible from σ and no other prefix on the branch starts with σ LECTURE NOTES FEBRURARY 18, 2010 Figure 3: Tableau proof rules for QML The tableau rules can also be used to analyze F A A as follows: → ♦ 1 F A A (1) → ♦ 1 T A (2) from 1 1 F ♦A (3) from 1 stop No more proof rules can be used because the modal formulas are ν rules, which are only applicable for accessible prefixes that already occur on the branch.
Load more

Recommended publications
  • DRAFT: Final Version in Journal of Philosophical Logic Paul Hovda
    Tensed mereology DRAFT: final version in Journal of Philosophical Logic Paul Hovda There are at least three main approaches to thinking about the way parthood logically interacts with time. Roughly: the eternalist perdurantist approach, on which the primary parthood relation is eternal and two-placed, and objects that persist persist by having temporal parts; the parameterist approach, on which the primary parthood relation is eternal and logically three-placed (x is part of y at t) and objects may or may not have temporal parts; and the tensed approach, on which the primary parthood relation is two-placed but (in many cases) temporary, in the sense that it may be that x is part of y, though x was not part of y. (These characterizations are brief; too brief, in fact, as our discussion will eventually show.) Orthogonally, there are different approaches to questions like Peter van Inwa- gen's \Special Composition Question" (SCQ): under what conditions do some objects compose something?1 (Let us, for the moment, work with an undefined notion of \compose;" we will get more precise later.) One central divide is be- tween those who answer the SCQ with \Under any conditions!" and those who disagree. In general, we can distinguish \plenitudinous" conceptions of compo- sition, that accept this answer, from \sparse" conceptions, that do not. (van Inwagen uses the term \universalist" where we use \plenitudinous.") A question closely related to the SCQ is: under what conditions do some objects compose more than one thing? We may distinguish “flat” conceptions of composition, on which the answer is \Under no conditions!" from others.
    [Show full text]
  • The Rise and Fall of LTL
    The Rise and Fall of LTL Moshe Y. Vardi Rice University Monadic Logic Monadic Class: First-order logic with = and monadic predicates – captures syllogisms. • (∀x)P (x), (∀x)(P (x) → Q(x)) |=(∀x)Q(x) [Lowenheim¨ , 1915]: The Monadic Class is decidable. • Proof: Bounded-model property – if a sentence is satisfiable, it is satisfiable in a structure of bounded size. • Proof technique: quantifier elimination. Monadic Second-Order Logic: Allow second- order quantification on monadic predicates. [Skolem, 1919]: Monadic Second-Order Logic is decidable – via bounded-model property and quantifier elimination. Question: What about <? 1 Nondeterministic Finite Automata A = (Σ,S,S0,ρ,F ) • Alphabet: Σ • States: S • Initial states: S0 ⊆ S • Nondeterministic transition function: ρ : S × Σ → 2S • Accepting states: F ⊆ S Input word: a0, a1,...,an−1 Run: s0,s1,...,sn • s0 ∈ S0 • si+1 ∈ ρ(si, ai) for i ≥ 0 Acceptance: sn ∈ F Recognition: L(A) – words accepted by A. 1 - - Example: • • – ends with 1’s 6 0 6 ¢0¡ ¢1¡ Fact: NFAs define the class Reg of regular languages. 2 Logic of Finite Words View finite word w = a0,...,an−1 over alphabet Σ as a mathematical structure: • Domain: 0,...,n − 1 • Binary relation: < • Unary relations: {Pa : a ∈ Σ} First-Order Logic (FO): • Unary atomic formulas: Pa(x) (a ∈ Σ) • Binary atomic formulas: x < y Example: (∃x)((∀y)(¬(x < y)) ∧ Pa(x)) – last letter is a. Monadic Second-Order Logic (MSO): • Monadic second-order quantifier: ∃Q • New unary atomic formulas: Q(x) 3 NFA vs. MSO Theorem [B¨uchi, Elgot, Trakhtenbrot, 1957-8 (independently)]: MSO ≡ NFA • Both MSO and NFA define the class Reg.
    [Show full text]
  • Probabilistic Semantics for Modal Logic
    Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators).
    [Show full text]
  • On Constructive Linear-Time Temporal Logic
    Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. On Constructive Linear-Time Temporal Logic Kensuke Kojima1 and Atsushi Igarashi2 Graduate School of Informatics Kyoto University Kyoto, Japan Abstract In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corre- sponds to a type system for binding-time analysis via the Curry-Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction and Kripke semantics for constructive LTL with conjunction and disjunction, and prove soundness and completeness. Distributivity of the “next” operator over disjunction “ (A ∨ B) ⊃ A ∨ B” is rejected from a computational viewpoint. We also give a formalization by sequent calculus and its cut-elimination procedure. Keywords: constructive linear-time temporal logic, Kripke semantics, sequent calculus, cut elimination 1 Introduction Temporal logic is a family of (modal) logics in which the truth of propositions may depend on time, and is useful to describe various properties of state transition systems. Linear-time temporal logic (LTL, for short), which is used to reason about properties of a fixed execution path of a system, is temporal logic in which each time has a unique time that follows it.
    [Show full text]
  • A Unifying Field in Logics: Neutrosophic Logic
    University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2007 A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS - 6th ed. Florentin Smarandache University of New Mexico, [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Logic and Foundations Commons, Other Applied Mathematics Commons, and the Set Theory Commons Recommended Citation Florentin Smarandache. A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS - 6th ed. USA: InfoLearnQuest, 2007 This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (sixth edition) InfoLearnQuest 2007 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (sixth edition) This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/basic Copyright 2007 by InfoLearnQuest. Plenty of books can be downloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm Peer Reviewers: Prof. M. Bencze, College of Brasov, Romania.
    [Show full text]
  • A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics
    FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 1 Contents: Preface by C. Le: 3 0. Introduction: 9 1. Neutrosophy - a new branch of philosophy: 15 2. Neutrosophic Logic - a unifying field in logics: 90 3. Neutrosophic Set - a unifying field in sets: 125 4. Neutrosophic Probability - a generalization of classical and imprecise probabilities - and Neutrosophic Statistics: 129 5. Addenda: Definitions derived from Neutrosophics: 133 2 Preface to Neutrosophy and Neutrosophic Logic by C.
    [Show full text]
  • Aristotelian Temporal Logic: the Sea Battle
    Aristotelian temporal logic: the sea battle. According to the square of oppositions, exactly one of “it is the case that p” and “it is not the case that p” is true. Either “it is the case that there will be a sea battle tomorrow” or “it is not the case that there will be a sea battle tomorrow”. Problematic for existence of free will, and for Aristotelian metaphysics. Core Logic – 2005/06-1ab – p. 3/34 The Master argument. Diodorus Cronus (IVth century BC). Assume that p is not the case. In the past, “It will be the case that p is not the case” was true. In the past, “It will be the case that p is not the case” was necessarily true. Therefore, in the past, “It will be the case that p” was impossible. Therefore, p is not possible. Ergo: Everything that is possible is true. Core Logic – 2005/06-1ab – p. 4/34 Megarians and Stoics. Socrates (469-399 BC) WWW WWWWW WWWWW WWWWW WW+ Euclides (c.430-c.360 BC) Plato (c.427-347 BC) WWW WWWWW WWWWW WWWWW WW+ Eubulides (IVth century) Stilpo (c.380-c.300 BC) Apollonius Cronus Zeno of Citium (c.335-263 BC) gg3 ggggg ggggg ggggg ggg Diodorus Cronus (IVth century) Cleanthes of Assos (301-232 BC) Chrysippus of Soli (c.280-207 BC) Core Logic – 2005/06-1ab – p. 5/34 Eubulides. Strongly opposed to Aristotle. Source of the “seven Megarian paradoxes”, among them the Liar. The Liar is attributed to Epimenides the Cretan (VIIth century BC); (Titus 1:12).
    [Show full text]
  • Artificial Intelligence Dynamic Reasoning with Qualified Syllogisms
    Artificial Intelligence ELSEVIER Artificial Intelligence 93 ( 1997 ) 105 167 Dynamic reasoning with qualified syllogisms Daniel G. Schwartz’ Department of Computer Science, Florida State University, Tallahassee. FL 32306-4019, USA Received January 1996: revised January 1997 Abstract A gualij%e~ syllogism is a classical Aristoteiean syllogism that has been “qualified” through the use of fuzzy quantifiers, likelihood modifiers, and usuality modifiers, e.g., “Most birds can Ry; Tweety is a bird; therefore, it is likely that Tweety can fly.” This paper introduces a formal logic Q of such syllogisms and shows how this may be employed in a system of nonmonotonj~ reasoning. In process are defined the notions of path logic and dynamic reasoning system (DRS) The former is an adaptation of the conventional formal system which explicitly portrays reasoning as an activity that takes place in time. The latter consists of a path logic together with a multiple- inhe~tance hierarchy. The hierarchy duplicates some of the info~ation recorded in the path logic, but additionally provides an extralogical spec#icity relation. The system uses typed predicates to formally distinguish between properties and kinds of things. The effectiveness of the approach is demonstrated through analysis of several “puzzles” that have appeared previously in the literature, e.g., Tweety the Bird, Clyde the Elephant, and the Nixon Diamond. It is also outlined how the DRS framework accommodates other reasoning techniques-in particular, predicate circumscription, a “localized” version of default logic, a variant of nonmonotonic logic, and reason maintenance. Furthermore it is seen that the same framework accomodates a new formulation of the notion of unless.
    [Show full text]
  • Boxes and Diamonds: an Open Introduction to Modal Logic
    Boxes and Diamonds An Open Introduction to Modal Logic F19 Boxes and Diamonds The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université de Lorraine Jeremy Avigad, Carnegie Mellon University Tim Button, University College London Walter Dean, University of Warwick Gillian Russell, Dianoia Institute of Philosophy Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, Columbia University Dana Hägg, University of Calgary Zesen Qian, Carnegie Mellon University Boxes and Diamonds An Open Introduction to Modal Logic Remixed by Richard Zach Fall 2019 The Open Logic Project would like to acknowledge the gener- ous support of the Taylor Institute of Teaching and Learning of the University of Calgary, and the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Cover illustrations by Matthew Leadbeater, used under a Cre- ative Commons Attribution-NonCommercial 4.0 International Li- cense. Typeset in Baskervald X and Nimbus Sans by LATEX. This version of Boxes and Diamonds is revision ed40131 (2021-07- 11), with content generated from Open Logic Text revision a36bf42 (2021-09-21). Free download at: https://bd.openlogicproject.org/ Boxes and Diamonds by Richard Zach is licensed under a Creative Commons At- tribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Cre- ative Commons Attribution 4.0 Interna- tional License. Contents Preface xi Introduction xii I Normal Modal Logics1 1 Syntax and Semantics2 1.1 Introduction....................
    [Show full text]
  • Explaining Multi-Stage Tasks by Learning Temporal Logic Formulas from Suboptimal Demonstrations
    Robotics: Science and Systems 2020 Corvalis, Oregon, USA, July 12-16, 2020 Explaining Multi-stage Tasks by Learning Temporal Logic Formulas from Suboptimal Demonstrations Glen Chou, Necmiye Ozay, and Dmitry Berenson Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 Email: gchou, necmiye, dmitryb @umich.edu f g Abstract—We present a method for learning to perform multi- stage tasks from demonstrations by learning the logical structure and atomic propositions of a consistent linear temporal logic (LTL) formula. The learner is given successful but potentially suboptimal demonstrations, where the demonstrator is optimizing a cost function while satisfying the LTL formula, and the cost function is uncertain to the learner. Our algorithm uses Fig. 1. Multi-stage manipulation: first fill the cup, then grasp it, and then the Karush-Kuhn-Tucker (KKT) optimality conditions of the deliver it. To avoid spills, a pose constraint is enforced after the cup is grasped. demonstrations together with a counterexample-guided falsifi- In this paper, our insight is that by assuming that demonstra- cation strategy to learn the atomic proposition parameters and tors are goal-directed (i.e. approximately optimize an objective logical structure of the LTL formula, respectively. We provide function that may be uncertain to the learner), we can regu- theoretical guarantees on the conservativeness of the recovered atomic proposition sets, as well as completeness in the search larize the LTL learning problem without being given
    [Show full text]
  • Post-Defense Draft with Corrections
    THE A-THEORY: A THEORY BY MEGHAN SULLIVAN A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Philosophy Written under the direction of Dean Zimmerman and approved by New Brunswick, New Jersey October, 2011 © 2011 Meghan Sullivan ALL RIGHTS RESERVED ABSTRACT OF THE DISSERTATION The A-Theory: A Theory by Meghan Sullivan Dissertation Director: Dean Zimmerman A-theories of time postulate a fundamental distinction between the present and other times. This distinction manifests in what A-theorists take to exist, their accounts of property change, and their views about the appropriate temporal logic. In this dissertation, I argue for a particular formulation of the A-theory that dispenses with change in existence and makes tense operators an optional formal tool for expressing the key theses. I call my view the minimal A-theory. The first chapter introduces the debate. The second chapter offers an extended, logic-based argument against more traditional A-theories. The third and fourth chapters develop my alternative proposal. The final chapter considers a problem for A-theorists who think the contents of our attitudes reflect changes in the world. ii Acknowledgements I’ve loved working on this project over the past few years, largely because of the ex- citing philosophers I’ve been fortunate to work alongside. Tim Williamson first intro- duced me to tense logic while I was at Oxford. He helped me through some confused early stages, and his work on modality inspires the minimal A-theory.
    [Show full text]
  • Tense Or Temporal Logic Gives a Glimpse of a Number of Related F-Urther Topics
    TI1is introduction provi'des some motivational and historical background. Basic fonnal issues of propositional tense logic are treated in Section 2. Section 3 12 Tense or Temporal Logic gives a glimpse of a number of related f-urther topics. 1.1 Motivation and Terminology Thomas Miiller Arthur Prior, who initiated the fonnal study of the logic of lime in the 1950s, called lus logic tense logic to reflect his underlying motivation: to advance the philosophy of time by establishing formal systems that treat the tenses of nat­ \lrallanguages like English as sentence-modifying operators. While many other Chapter Overview formal approaches to temporal issues have been proposed h1 the meantime, Pri­ orean tense logic is still the natural starting point. Prior introduced four tense operatm·s that are traditionally v.rritten as follows: 1. Introduction 324 1.1 Motiv<Jtion and Terminology 325 1.2 Tense Logic and the History of Formal Modal Logic 326 P it ·was the case that (Past) 2. Basic Formal Tense Logic 327 H it has always been the case that 2.1 Priorean (Past, Future) Tense Logic 328 F it will be the case that (Future) G it is always going to be the case that 2.1.1 The Minimal Tense Logic K1 330 2.1 .2 Frame Conditions 332 2.1.3 The Semantics of the Future Operator 337 Tims, a sentence in the past tense like 'Jolm was happy' is analysed as a temporal 2.1.4 The lndexicai'Now' 342 modification of the sentence 'John is happy' via the operator 'it was the case thai:', 2.1.5 Metric Operators and At1 343 and can be formally expressed as 'P HappyO'olm)'.
    [Show full text]