DOCSLIB.ORG

Propositions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Propositions Module 1: Basic Logic Theme 1: Propositions English sentences are either true or false or neither. Consider the following sentences: 1. Warsaw is the capital of Poland. 2. ¾·5 = ¿. 3. How are you? The first sentence is true, the second is false, while the last one is neither true nor false. A statement that is either true or false but not both is called a proposition. Propositional logic deals with such statements and compound propositions that combine together simple propositions (e.g., combining sentences (1) and (2) above we may say “Warsaw is the capital of Poland and ¾·5=¿”). In order to build compound propositions we need rules on how to combine propositions. We Õ Ö denote propositions by lowercase letters Ô, or . Let us define: Ô Õ Ô ^ Õ ¯ The conjunction of and , denoted as , is the proposition Ô aÒd Õ; Õ and it is true when both Ô and are true and false otherwise. Ô Õ Ô _ Õ ¯ The disjunction of and , denoted as , is the proposition Ô ÓÖ Õ; Õ and it is false when both Ô and are false and true otherwise. Ô :Ô Ô ¯ The negation of , denoted either as or , is the proposition It is not true that Ô. = Õ = Ô _ Õ Example 1:LetÔ “Hawks swoop” and “Gulls glide”. Then isthesameas“Hawksswoop or gulls glide”. We also can translate back. For example, the English sentence “it is not true that Ô hawks swoop” can be written as : . Exercise 1A: With the same notation as in the example above write the following propositions sym- bolically: ¯ It is not true that “Hawks swoop and gulls glide”. ¯ “Hawks do not swoop or gulls do not glide”. 1 Theme 2: Truth Tables We can express compound propositions using a truth table that displays the relationships between the truth values of the simple propositions and the compound proposition. In the next three tables we show the truth tables for the negation, conjunction, and disjunction. Observe that any proposition Ì F Ô can take only two values, namely true, denoted ,orfalse, denoted . Therefore, for a com- ^ Õ µ pound proposition consisting of two propositions (e.g., Ô we must consider only four possible F assignments of Ì and . Table 1: The truth table for the negation. Ô :Ô T F F T Table 2: The truth table for the conjunction. Ô Õ Ô ^ Õ TTT TFF FTF FFF Table 3: The truth table for the disjunction. Ô Õ Ô _ Õ TTT TFT FTT FFF In this module we will often use truth tables. To construct a truth table for a statement (e.g., Ô _ Õ Ô Õ : ) containing two propositions, say and , one first builds two columns with all possible vales Õ ´Ì; Ì µ; ´Ì; F µ; ´F; Ì µ; ´F; F µ of Ô and (i.e., ), and then follows already accepted rules of inference Ô _ Õ to determine the truth value of the compound statement (say : ). Exercise 1B: Construct truth tables for the following statements: Ô ^:Ô ¯ ; Ô _:Õ ¯ . 2 Theme 3: Implications ¾ =¾ Ü =4 In mathematics we often deal with conditional statements like: “if Ü , then .Theif–then ! Õ Ô Õ statement is called implication and it is denoted as Ô . It is false when is true and is false and ! Õ true otherwise. The reader may inspect the truth table of Ô in Table 4 below. Table 4: The truth table for the implication. Ô Õ Ô ! Õ TTT TFF FTT FFT ! Õ Ô Õ It is important to emphasize that Ô is false only when is true and is false. In words, truth cannot imply a false statement, but false can imply truth. For example, consider the following statement ¾ if Ü = ¾; ØheÒ Ü =4 =¾ which is true even if the first part of this compound statement is not true, say when Ü . ! Õ Ô In the implication Ô , the proposition is called hypothesis or antecedent and the proposition Ô Õ is known as conclusion or consequent. The conclusion expresses a necessary condition for , while the hypothesis expresses a sufficient condition for Õ to hold. Some other common ways of ! Õ expressing the implication Ô are: Ô Õ ¯ if ,then ; Ô Õ ¯ implies ; Ô Õ ¯ if , ; Ô Õ ¯ only if ; Ô Õ ¯ is sufficient for ; Õ Ô ¯ if ; Õ Ô ¯ whenever ; Õ Ô ¯ is necessary for . Exercise 1C: Make truth tables for the following statements: !:Õ 1. Ô ; Ô ^:Õ µ ! Ö 2. ´ . 3 ! Õ There are some important related implications following from Ô , namely: ! Ô 1. The proposition Õ is called the converse. ! Õ :Õ !:Ô 2. The contrapositive of Ô is ; Ô !:Õ 3. The inverse is : . In Table 5 we compare the truth values of these propositions. Table 5: The truth table for the implication, contrapositive, converse, and inverse. Ô Õ Ô ! Õ :Õ !:Ô Õ ! Ô :Ô !:Õ TTT T T T TFF F T T FTT T F F FFT T T T É We say that two compound propositions È and are logically equivalent if they have the same truth values. We shall write È É or È ¸ É: ! Õ It should be observed from Table 5 that the implication Ô has the same truth values as the Õ !:Ô contrapositive : , but not as the converse and the inverse. Thus we can write Ô ! Õ :Õ !:Ô; Ô ! Õ 6 :Ô !:Õ; Ô ! Õ 6 Õ ! Ô: Example 2: Prove that Ô ! Õ :Ô _ Õ: We use the truth table. Our computation is shown in Table 6. Comparing the second column with the ! Õ :Ô _ Õ last one, we see that the truth values are the same for Ô and , so the above two compound propositions are logically equivalent. Table 6: The truth table for Example 2. Ô Õ Ô ! Õ :Ô :Ô _ Õ TTT F T TFF F F FTT T T FFT T T 4 Exercise 1D: Using the truth table prove that the following propositions are logically equivalent: Ô _ ´Õ ^ Ö µ ´Ô _ Õ µ ^ ´Ô _ Ö µ: In Exercise 1D the reader was asked to prove logical equivalence that is known under the name distributive law. This is an example of many other logical equivalences that we list in Table 7 and prove in the sequel. Table 7: Logical Equivalences Equivalence Name ^ Ì Ô Ô Identity laws Ô _ F Ô _ Ì Ì Ô Domination laws Ô ^ F F _ Ô Ô Ô Idempotent laws Ô ^ Ô Ô ´:Ôµ Ô : Double negation law _ Õ Õ _ Ô Ô Commutative laws Ô ^ Õ Õ ^ Ô _ ´Õ _ Ö µ ´Ô _ Õ µ _ Ö Ô Associative laws Ô ^ ´Õ ^ Ö µ ´Ô ^ Õ µ ^ Ö _ ´Õ ^ Ö µ ´Ô _ Õ µ ^ ´Ô _ Ö µ Ô Distributive laws Ô ^ ´Õ _ Ö µ ´Ô ^ Õ µ _ ´Ô ^ Ö µ ´Ô ^ Õ µ :Ô _:Õ : De Morgan’s laws :´Ô _ Õ µ :Ô ^:Õ All laws listed above can be easily proved using the truth table. The reader is encouraged to try to work out all the truth tables. Having such laws under our belt, we can prove many new logical equivalences without using the truth table. Example 3: Prove that :´Ô _ ´:Ô ^ Õ µµ :Ô ^:Õ :´Ô _ Õ µ: We proceed as follows ´Ô _ ´:Ô ^ Õ µµ :Ô ^:´:Ô ^ Õ µ : De Morgan’s law :Ô ^ ´:´:Ôµ _:Õ µ De Morgan’s law :Ô ^ ´Ô _:Õ µ double negation law ´:Ô ^ Ôµ _ ´:Ô ^:Õ µ distributive law 5 F _ ´:Ô ^:Õ µ :Ô ^ Ô F since ´:Ô ^:Õ µ _ F commutative law ´:Ô ^:Õ µ identity law :´Ô _ Õ µ : De Morgan’s law Thus the above logical equivalence is proved. The above is largely self-explanatory, but a few words of additional information follows: In the first statement above we, naturally, apply De Morgan’s law ´È _ ɵ=:È ^:É É É = :Ô ^ Õ : . In our case, is a compound statement , thus another application É = Ô _:Õ Ô ^ ´Õ _ Ö µ=´Ô ^ Õ µ _ ´Ô ^ Ö µ of De Morgan’s law implies : . Then we “multiply out”, that is, . The rest is simple. A compound proposition is called a tautology if it is always true, no matter what the truth values _:Ô Ì Ô of the propositions (e.g., Ô no matter what is the value of . Why?). A compound proposition is called a contradiction if it is always false, no matter what the truth ^:Ô Ì Ô values of the propositions (e.g., Ô no matter what is the value of . Why?). Finally, a proposition that is neither a tautology nor a contradiction is called a contingency. 6 Theme 4: Predicates and Quantifiers In mathematics we often have to deal with sentences like ¾ : Ü ¾Ü ·½ = ¼ Õ : Ò ; Ô or is a prime number which are not propositions since their values are neither true nor false since the values of the variables Ò È ´Üµ É´Òµ Ü and are not specified. We shall denote such statements as or and call propositional Ò functions or predicates of Ü or . Ü D È More formally, let È be a statement involving the variable that belongs to the set .Then is Ü ¾D È ´Üµ called a propositional function or predicate with respect to D if for each the sentence È is a proposition. The domain D is often called the universe of discourse of . Example 4: The statement above ¾ È ´Üµ: Ü ¾Ü ·½ = ¼ =½ Ü 6=½ É´¿µ É´Òµ Ò is true when Ü and is false for any .
Load more

Recommended publications
  • 1 Lecture 13: Quantifier Laws1 1. Laws of Quantifier Negation There
    Ling 409 lecture notes, Lecture 13 Ling 409 lecture notes, Lecture 13 October 19, 2005 October 19, 2005 Lecture 13: Quantifier Laws1 true or there is at least one individual that makes φ(x) true (or both). Those are also equivalent statements. 1. Laws of Quantifier Negation Laws 2 and 3 suggest a fundamental connection between universal quantification and There are a number of equivalences based on the set-theoretic semantics of predicate logic. conjunction and existential quantification and disjunction: Take for instance a statement of the form (∃x)~ φ(x)2. It asserts that there is at least one individual d which makes ~φ(x) true. That means that d makes ϕ(x) false. That, in turn, (∀x) ψ(x) is true iff ψ(a) and ψ(b) and ψ(c) … (where a, b, c … are the names of the means that (∀x) φ(x) is false, which means that ~(∀x) φ(x) is true. The same reasoning individuals in the domain.) can be applied in the reverse direction and the result is the first quantifier law: ∃ ψ ψ ψ ψ ( x) (x) is true iff (a) or (b) or c) … (where a, b, c … are the names of the (1) Laws of Quantifier Negation individuals in the domain.) Law 1: ~(∀x) φ(x) ⇐⇒ (∃x)~ φ(x) From that point of view, Law 1 resembles a generalized DeMorgan’s Law: (Possible English correspondents: Not everyone passed the test and Someone did not pass (4) ~(ϕ(a) & ϕ(b) & ϕ(c) … ) ⇐⇒ ~ϕ(a) v ~ϕ(b) v ϕ(c) … the test.) (5) Law 4: (∀x) ψ(x) v (∀x)φ(x) ⇒ (∀x)( ψ(x) v φ(x)) ϕ ⇐ ϕ Because of the Law of Double Negation (~~ (x) ⇒ (x)), Law 1 could also be written (6) Law 5: (∃x)( ψ(x) & φ(x)) ⇒ (∃x) ψ(x) & (∃x)φ(x) as follows: These two laws are one-way entailments, NOT equivalences.
    [Show full text]
  • Lexical Scoping As Universal Quantification
    University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science March 1989 Lexical Scoping as Universal Quantification Dale Miller University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Dale Miller, "Lexical Scoping as Universal Quantification", . March 1989. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-89-23. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/785 For more information, please contact [email protected]. Lexical Scoping as Universal Quantification Abstract A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is "discharged" after the goal succeeds or fails. This interpretation is similar to the interpretation of implicational goals: the goal D ⊃ G should succeed if when D is assumed, the goal G succeeds. The assumption D is discharged after G succeeds or fails. An interpreter for a logic programming language containing both universal quantifiers and implications in goals and the body of clauses is described. In its non-deterministic form, this interpreter is sound and complete for intuitionistic logic. Universal quantification can provide lexical scoping of individual, function, and predicate constants. Several examples are presented to show how such scoping can be used to provide a Prolog-like language with facilities for local definition of programs, local declarations in modules, abstract data types, and encapsulation of state.
    [Show full text]
  • Sets, Propositional Logic, Predicates, and Quantifiers
    COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete.
    [Show full text]
  • CHAPTER 1 the Main Subject of Mathematical Logic Is Mathematical
    CHAPTER 1 Logic The main subject of Mathematical Logic is mathematical proof. In this chapter we deal with the basics of formalizing such proofs and analysing their structure. The system we pick for the representation of proofs is natu- ral deduction as in Gentzen (1935). Our reasons for this choice are twofold. First, as the name says this is a natural notion of formal proof, which means that the way proofs are represented corresponds very much to the way a careful mathematician writing out all details of an argument would go any- way. Second, formal proofs in natural deduction are closely related (via the Curry-Howard correspondence) to terms in typed lambda calculus. This provides us not only with a compact notation for logical derivations (which otherwise tend to become somewhat unmanageable tree-like structures), but also opens up a route to applying the computational techniques which un- derpin lambda calculus. An essential point for Mathematical Logic is to fix the formal language to be used. We take implication ! and the universal quantifier 8 as basic. Then the logic rules correspond precisely to lambda calculus. The additional connectives (i.e., the existential quantifier 9, disjunction _ and conjunction ^) will be added via axioms. Later we will see that these axioms are deter- mined by particular inductive definitions. In addition to the use of inductive definitions as a unifying concept, another reason for that change of empha- sis will be that it fits more readily with the more computational viewpoint adopted there. This chapter does not simply introduce basic proof theory, but in addi- tion there is an underlying theme: to bring out the constructive content of logic, particularly in regard to the relationship between minimal and classical logic.
    [Show full text]
  • Precomplete Negation and Universal Quantification
    PRECOMPLETE NEGATION AND UNIVERSAL QUANTIFICATION by Paul J. Vada Technical Report 86-9 April 1986 Precomplete Negation And Universal Quantification. Paul J. Voda Department of Computer Science, The University of British Columbia, Vancouver, B.C. V6T IW5, Canada. ABSTRACT This paper is concerned with negation in logic programs. We propose to extend negation as failure by a stronger form or negation called precomplete negation. In contrast to negation as failure, precomplete negation has a simple semantic charaterization given in terms of computational theories which deli­ berately abandon the law of the excluded middle (and thus classical negation) in order to attain computational efficiency. The computation with precomplete negation proceeds with the direct computation of negated formulas even in the presence of free variables. Negated formulas are computed in a mode which is dual to the standard positive mode of logic computations. With uegation as failure the formulas with free variables must be delayed until the latter obtain values. Consequently, in situations where delayed formulas are never sufficiently instantiated, precomplete negation can find solutions unattainable with negation as failure. As a consequence of delaying, negation as failure cannot compute unbounded universal quantifiers whereas precomplete negation can. Instead of concentrating on the model-theoretical side of precomplete negation this paper deals with questions of complete computations and efficient implementations. April 1986 Precomplete Negation And Universal Quanti.flcation. Paul J. Voda Department or Computer Science, The University or British Columbia, Vancouver, B.C. V6T 1W5, Canada. 1. Introduction. Logic programming languages ba.5ed on Prolog experience significant difficulties with negation. There are many Prolog implementations with unsound negation where the computation succeeds with wrong results.
    [Show full text]
  • Not All There: the Interactions of Negation and Universal Quantifier Pu
    Not all there: The interactions of negation and universal quantifier Puk'w in PayPaˇȷuT@m∗ Roger Yu-Hsiang Lo University of British Columbia Abstract: The current paper examines the ambiguity between negation and the univer- sal quantifier Puk'w in PayPaˇȷuT@m, a critically endangered Central Salish language. I ar- gue that the ambiguity in PayPaˇȷuT@m arises from the nonmaximal, exception-tolerating property of Salish all, instead of resorting to the scopal interaction between negation and the universal quantifier, as in English. Specifically, by assuming that negation in PayPaˇȷuT@m is always interpreted with the maximal force, the ambiguity can be under- stood as originating from exceptions to this canonical interpretation. Whether or not this ambiguity is only available in PayPaˇȷuT@m is still unclear, and further data elicitation and cross-Salish comparison are underway. Keywords: PayPaˇȷuT@m (Mainland Comox), semantics, ambiguities, negation, univer- sal quantifier 1 Introduction This paper presents a preliminary analysis of the semantic ambiguity involved in the combination of negation and the universal quantifier Puk'w in PayPaˇȷuT@m, a critically endangered Central Salish language. The ambiguity between a negative element and a universal quantifier is also found in English. For example, consider the English paradigm in (1) from Carden (1976), where (1a) has only one reading while (1b) is ambiguous. (1) a. Not all the boys will run. :(8x;boy(x);run(x)) b. [ All the boys ] won’t run. i. :(8x;boy(x);run(x)) ii. (8x;boy(x));:(run(x)) In the traditional account, with the readings in (1a) and (1b-i), negation takes higher scope than the quantified DP at LF.
    [Show full text]
  • On Operator N and Wittgenstein's Logical Philosophy
    JOURNAL FOR THE HISTORY OF ANALYTICAL PHILOSOPHY ON OPERATOR N AND WITTGENSTEIN’S LOGICAL VOLUME 5, NUMBER 4 PHILOSOPHY EDITOR IN CHIEF JAMES R. CONNELLY KEVIN C. KLEMENt, UnIVERSITY OF MASSACHUSETTS EDITORIAL BOARD In this paper, I provide a new reading of Wittgenstein’s N oper- ANNALISA COLIVA, UnIVERSITY OF MODENA AND UC IRVINE ator, and of its significance within his early logical philosophy. GaRY EBBS, INDIANA UnIVERSITY BLOOMINGTON I thereby aim to resolve a longstanding scholarly controversy GrEG FROSt-ARNOLD, HOBART AND WILLIAM SMITH COLLEGES concerning the expressive completeness of N. Within the debate HENRY JACKMAN, YORK UnIVERSITY between Fogelin and Geach in particular, an apparent dilemma SANDRA LaPOINte, MCMASTER UnIVERSITY emerged to the effect that we must either concede Fogelin’s claim CONSUELO PRETI, THE COLLEGE OF NEW JERSEY that N is expressively incomplete, or reject certain fundamental MARCUS ROSSBERG, UnIVERSITY OF CONNECTICUT tenets within Wittgenstein’s logical philosophy. Despite their ANTHONY SKELTON, WESTERN UnIVERSITY various points of disagreement, however, Fogelin and Geach MARK TEXTOR, KING’S COLLEGE LonDON nevertheless share several common and problematic assump- AUDREY YAP, UnIVERSITY OF VICTORIA tions regarding Wittgenstein’s logical philosophy, and it is these RICHARD ZACH, UnIVERSITY OF CALGARY mistaken assumptions which are the source of the dilemma. Once we recognize and correct these, and other, associated ex- REVIEW EDITORS pository errors, it will become clear how to reconcile the ex- JULIET FLOYD, BOSTON UnIVERSITY pressive completeness of Wittgenstein’s N operator, with sev- CHRIS PINCOCK, OHIO STATE UnIVERSITY eral commonly recognized features of, and fundamental theses ASSISTANT REVIEW EDITOR within, the Tractarian logical system.
    [Show full text]
  • Existential Quantification
    Quantification 1 From last time: Definition: Proposition A declarative sentence that is either true (T) or false (F), but not both. • Tucson summers never get above 100 degrees • If the doorbell rings, then my dog will bark. • You can take the flight if and only if you bought a ticket • But NOT x < 10 Why do we want use variables? • Propositional logic is not always expressive enough • Consider: “All students like summer vacation” • Should be able to conclude that “If Joe is a student, he likes summer vacation”. • Similarly, “If Rachel is a student, she likes summer vacation” and so on. • Propositional Logic does not support this! Predicates Definition: Predicate (a.k.a. Propositional Function) A statement that includes at least one variable and will evaluate to either true or false when the variables(s) are assigned value(s). • Example: S(x) : (−10 < x) ∧ (x < 10) E(a, b) : a eats b • These are not complete! Predicates Definition: Domain (a.ka. Universe) of Discourse The collection of values from which a variable’s value is drawn. • Example: S(x) : (−10 < x) ∧ (x < 10), x ∈ ℤ E(a, b) : a eats b, a ∈ People, b ∈ Vegetables • In This Class: Domains may NOT hide operators • OK: Vegetables • Not-OK: Raw Vegetables (Vegetable ∧ ¬Cooked) Predicates S(x) : (−10 < x) ∧ (x < 10), x ∈ ℤ E(a, b) : a eats b, a ∈ People, b ∈ Vegetables • Can evaluate predicates at specific values (making them propositions): • What is E(Joe, Asparagus)? • What is S(0)? Combining Predicates with Logical Operators • In E(a, b) : a eats b, a ∈ people, b ∈ vegetables, change the domain of b to “raw vegetables”.
    [Show full text]
  • 8 : : H Responses -> : : : S: to Disks, Displays, 8 : : ATM, Etc
    USOO5867649A United States Patent (19) 11 Patent Number: 5,867,649 Larson (45) Date of Patent: Feb. 2, 1999 54) DANCE/MULTITUDE CONCURRENT Algorithms for Scalable Synchronization on Shaved COMPUTATION Memory Multiprocessors, J. Mollor-Crumley, ACM Trans actions on Computer Systems, vol. 9, No. 1, Feb. 1991. 75 Inventor: Brian Ralph Larson, Mpls., Minn. Verification of Sequential and Concurrent Programs, Apt. & 73 Assignee: Multitude Corporation, Bayport, Olderog; Springer-Verlag, 1991. Minn. The Science of Programming, D. Gries Springer-Verlag, 1981. 21 Appl. No.: 589,933 Computer Architecture R. Karin Prentice-Hall (1989) (CPT. 5). 22 Filed: Jan. 23, 1996 The Design and Analysis of Parallel Algorithms, S. Akl, (51) Int. Cl." ...................................................... G06F 15/16 Prentice-Hall, 1989. 52 U.S. Cl. ......................................................... 395/200.31 Executing Temporal Logic Programs, B.C. Moszkowski, 58 Field of Search ....................... 364/DIG. 1 MS File, Cambridge University Press, Copyright 1986, pp. 1-125. 364/DIG. 2 MS File: 395/377, 376, 379, 382, 385, 390, 391,561, 800, 705, 706, Primary Examiner Robert B. Harrell 712, 200.31, 800.28, 800.3, 800.36; 340/825.8 Attorney, Agent, or Firm-Haugen and Nikolai, P.A. 56) References Cited 57 ABSTRACT U.S. PATENT DOCUMENTS This invention computes by constructing a lattice of States. 4,814,978 3/1989 Dennis .................................... 395/377 Every lattice of States corresponding to correct execution 4,833,468 5/1989 Larson et al. 340/825.8 Satisfies the temporal logic formula comprising a Definitive 5,029,080 7/1991 Otsuki ..................................... 395/377 Axiomatic Notation for Concurrent Execution (DANCE) program. This invention integrates into a State-lattice com OTHER PUBLICATIONS putational model: a polymorphic, Strong type System; Reference Manual for the ADA Programming Language, US Visibility-limiting domains, first-order assertions, and logic Dept.
    [Show full text]
  • An Introduction to Formal Logic
    forall x: Calgary An Introduction to Formal Logic By P. D. Magnus Tim Button with additions by J. Robert Loftis Robert Trueman remixed and revised by Aaron Thomas-Bolduc Richard Zach Fall 2021 This book is based on forallx: Cambridge, by Tim Button (University College Lon- don), used under a CC BY 4.0 license, which is based in turn on forallx, by P.D. Magnus (University at Albany, State University of New York), used under a CC BY 4.0 li- cense, and was remixed, revised, & expanded by Aaron Thomas-Bolduc & Richard Zach (University of Calgary). It includes additional material from forallx by P.D. Magnus and Metatheory by Tim Button, used under a CC BY 4.0 license, from forallx: Lorain County Remix, by Cathal Woods and J. Robert Loftis, and from A Modal Logic Primer by Robert Trueman, used with permission. This work is licensed under a Creative Commons Attribution 4.0 license. You are free to copy and redistribute the material in any medium or format, and remix, transform, and build upon the material for any purpose, even commercially, under the following terms: ⊲ You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. ⊲ You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. The LATEX source for this book is available on GitHub and PDFs at forallx.openlogicproject.org.
    [Show full text]
  • Universal Quantification in a Constraint-Based Planner
    Universal Quantification in a Constraint-Based Planner Keith Golden and Jeremy Frank NASA Ames Research Center Mail Stop 269-1 Moffett Field, CA 94035 {kgolden,frank}@ptolemy.arc.nasa.gov Abstract Incomplete information: It is common for softbots to • have only incomplete information about their environ- We present a general approach to planning with a restricted class of universally quantified constraints. These constraints ment. For example, a softbot is unlikely to know about stem from expressive action descriptions, coupled with large all the files on the local file system, much less all the files or infinite universes and incomplete information. The ap- available over the Internet. proach essentially consists of checking that the quantified Large or infinite universes: The size of the universe is constraint is satisfied for all members of the universe. We • generally very large or infinite. For example, there are present a general algorithm for proving that quantified con- straints are satisfied when the domains of all of the variables hundreds of thousands of files accessible on a typical file are finite. We then describe a class of quantified constraints system and billions of web pages publicly available over for which we can efficiently prove satisfiability even when the Internet. The number of possible files, file path names, the domains are infinite. These form the basis of constraint etc., is effectively infinite. Given the presence of incom- reasoning systems that can be used by a variety of planners. plete information and the ability to create new files, it is necessary to reason about these infinite sets. 1 Introduction Constraints: As noted in (Chien et al.
    [Show full text]
  • (Universal) Quantification: a Generalised Quantifier Theory Analysis of -Men∗
    1 Covert (Universal) Quantification: A Generalised Quantifier Theory Analysis of -men∗ Sherry Yong Chen University of Oxford Introduction This paper explores the hypothesis of covert quantification – the idea that some nominals can be treated as having a (universal) quantifier covertly – by examining the interaction between the suffix -men and the particle dou within the Generalised Quantifier Theory framework (Barwise & Cooper, 1981; henceforth GQT). I begin by introducing the puzzle posed by -men in Mandarin Chinese (henceforth Chinese), a suffix that has been widely analysed as a collective marker but demonstrates various features of a plural marker. I argue that neither analysis is satisfactory given the interaction of -men and dou, the latter of which can be analysed as the lexical representation of the Matching Function (Rothstein, 1995; henceforth the M-Function). Crucially, the collective analysis of -men fails to explain why it can co-occur with dou, which requires access to individual atoms, while the plural analysis of -men fails to predict many of its distributions and interpretations. In light of these puzzles, I observe that when there is no overt universal quantifier in the sentence, nominals with -men turn out to be ambiguous between a strong and a weak reading, but they must receive a definite interpretation and have a “significant subpart” requirement, analogous to the semantic denotation of most of the X. This motivates the GQT treatment of - men which assumes covert quantification in its semantic representation. Moreover, the two readings of -men disambiguate in the presence of dou, with only the strong reading left, which further suggests the existence of a covert universal quantifier in -men, given that dou as the M-Function seeks a universal quantifier in the semantic composition (Pan, 2005; Zhang, 2007).
    [Show full text]