A Validity Framework for the Use and Development of Exported Assessments
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The Validities of Pep and Some Characteristic Formulas in Modal Logic
THE VALIDITIES OF PEP AND SOME CHARACTERISTIC FORMULAS IN MODAL LOGIC Hong Zhang, Huacan He School of Computer Science, Northwestern Polytechnical University,Xi'an, Shaanxi 710032, P.R. China Abstract: In this paper, we discuss the relationships between some characteristic formulas in normal modal logic with their frames. We find that the validities of the modal formulas are conditional even though some of them are intuitively valid. Finally, we prove the validities of two formulas of Position- Exchange-Principle (PEP) proposed by papers 1 to 3 by using of modal logic and Kripke's semantics of possible worlds. Key words: Characteristic Formulas, Validity, PEP, Frame 1. INTRODUCTION Reasoning about knowledge and belief is an important issue in the research of multi-agent systems, and reasoning, which is set up from the main ideas of situational calculus, about other agents' states and actions is one of the most important directions in Al^^'^\ Papers [1-3] put forward an axiom scheme in reasoning about others(RAO) in multi-agent systems, and a rule called Position-Exchange-Principle (PEP), which is shown as following, was abstracted . C{(p^y/)^{C(p->Cy/) (Ce{5f',...,5f-}) When the length of C is at least 2, it reflects the mechanism of an agent reasoning about knowledge of others. For example, if C is replaced with Bf B^/, then it should be B':'B]'{(p^y/)-^iBf'B]'(p^Bf'B]'y/) 872 Hong Zhang, Huacan He The axiom schema plays a useful role as that of modus ponens and axiom K when simplifying proof. Though examples were demonstrated as proofs in papers [1-3], from the perspectives both in semantics and in syntax, the rule is not so compellent. -
The Substitutional Analysis of Logical Consequence
THE SUBSTITUTIONAL ANALYSIS OF LOGICAL CONSEQUENCE Volker Halbach∗ dra version ⋅ please don’t quote ónd June óþÕä Consequentia ‘formalis’ vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [...] Iohannes Buridanus, Tractatus de Consequentiis (Hubien ÕÉßä, .ì, p.óóf) Zf«±§Zh± A substitutional account of logical truth and consequence is developed and defended. Roughly, a substitution instance of a sentence is dened to be the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category. In particular atomic formulae can be replaced with any formulae containing. e denition of logical truth is then as follows: A sentence is logically true i all its substitution instances are always satised. Logical consequence is dened analogously. e substitutional denition of validity is put forward as a conceptual analysis of logical validity at least for suciently rich rst-order settings. In Kreisel’s squeezing argument the formal notion of substitutional validity naturally slots in to the place of informal intuitive validity. ∗I am grateful to Beau Mount, Albert Visser, and Timothy Williamson for discussions about the themes of this paper. Õ §Z êZo±í At the origin of logic is the observation that arguments sharing certain forms never have true premisses and a false conclusion. Similarly, all sentences of certain forms are always true. Arguments and sentences of this kind are for- mally valid. From the outset logicians have been concerned with the study and systematization of these arguments, sentences and their forms. -
Propositional Logic (PDF)
Mathematics for Computer Science Proving Validity 6.042J/18.062J Instead of truth tables, The Logic of can try to prove valid formulas symbolically using Propositions axioms and deduction rules Albert R Meyer February 14, 2014 propositional logic.1 Albert R Meyer February 14, 2014 propositional logic.2 Proving Validity Algebra for Equivalence The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R) Albert R Meyer February 14, 2014 propositional logic.3 Albert R Meyer February 14, 2014 propositional logic.4 1 Algebra for Equivalence Algebra for Equivalence for example, The set of rules for ≡ in DeMorgan’s law the text are complete: ≡ NOT(P AND Q) ≡ if two formulas are , these rules can prove it. NOT(P) OR NOT(Q) Albert R Meyer February 14, 2014 propositional logic.5 Albert R Meyer February 14, 2014 propositional logic.6 A Proof System A Proof System Another approach is to Lukasiewicz’ proof system is a start with some valid particularly elegant example of this idea. formulas (axioms) and deduce more valid formulas using proof rules Albert R Meyer February 14, 2014 propositional logic.7 Albert R Meyer February 14, 2014 propositional logic.8 2 A Proof System Lukasiewicz’ Proof System Lukasiewicz’ proof system is a Axioms: particularly elegant example of 1) (¬P → P) → P this idea. It covers formulas 2) P → (¬P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES (→) and NOT. The only rule: modus ponens Albert R Meyer February 14, 2014 propositional logic.9 Albert R Meyer February 14, 2014 propositional logic.10 Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the inference rule. -
Validity and Soundness
1.4 Validity and Soundness A deductive argument proves its conclusion ONLY if it is both valid and sound. Validity: An argument is valid when, IF all of it’s premises were true, then the conclusion would also HAVE to be true. In other words, a “valid” argument is one where the conclusion necessarily follows from the premises. It is IMPOSSIBLE for the conclusion to be false if the premises are true. Here’s an example of a valid argument: 1. All philosophy courses are courses that are super exciting. 2. All logic courses are philosophy courses. 3. Therefore, all logic courses are courses that are super exciting. Note #1: IF (1) and (2) WERE true, then (3) would also HAVE to be true. Note #2: Validity says nothing about whether or not any of the premises ARE true. It only says that IF they are true, then the conclusion must follow. So, validity is more about the FORM of an argument, rather than the TRUTH of an argument. So, an argument is valid if it has the proper form. An argument can have the right form, but be totally false, however. For example: 1. Daffy Duck is a duck. 2. All ducks are mammals. 3. Therefore, Daffy Duck is a mammal. The argument just given is valid. But, premise 2 as well as the conclusion are both false. Notice however that, IF the premises WERE true, then the conclusion would also have to be true. This is all that is required for validity. A valid argument need not have true premises or a true conclusion. -
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem Yannick Forster Dominik Kirst Gert Smolka Saarland University Saarland University Saarland University Saarbrücken, Germany Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] [email protected] Abstract like decidability, enumerability, and reductions are avail- We formalise the computational undecidability of validity, able without reference to a concrete model of computation satisfiability, and provability of first-order formulas follow- such as Turing machines, general recursive functions, or ing a synthetic approach based on the computation native the λ-calculus. For instance, representing a given decision to Coq’s constructive type theory. Concretely, we consider problem by a predicate p on a type X, a function f : X ! B Tarski and Kripke semantics as well as classical and intu- with 8x: p x $ f x = tt is a decision procedure, a function itionistic natural deduction systems and provide compact д : N ! X with 8x: p x $ ¹9n: д n = xº is an enumer- many-one reductions from the Post correspondence prob- ation, and a function h : X ! Y with 8x: p x $ q ¹h xº lem (PCP). Moreover, developing a basic framework for syn- for a predicate q on a type Y is a many-one reduction from thetic computability theory in Coq, we formalise standard p to q. Working formally with concrete models instead is results concerning decidability, enumerability, and reducibil- cumbersome, given that every defined procedure needs to ity without reference to a concrete model of computation. be shown representable by a concrete entity of the model. -
Sufficient Condition for Validity of Quantum Adiabatic Theorem
Sufficient condition for validity of quantum adiabatic theorem Yong Tao † School of Economics and Business Administration, Chongqing University, Chongqing 400044, China Abstract: In this paper, we attempt to give a sufficient condition of guaranteeing the validity of the proof of the quantum adiabatic theorem. The new sufficient condition can clearly remove the inconsistency and the counterexample of the quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93, 160408, (2004)]. Keywords: Quantum adiabatic theorem; Integral formalism; Differential formalism; Berry phase. PACS: 03.65.-w; 03.65.Ca; 03.65.Vf. 1. Introduction The quantum adiabatic theorem (QAT) [1-4] is one of the basic results in quantum physics. The role of the QAT in the study of slowly varying quantum mechanical system spans a vast array of fields and applications, such as quantum field [5], Berry phase [6], and adiabatic quantum computation [7]. However, recently, the validity of the application of the QAT had been doubted in reference [8], where Marzlin and Sanders (MS) pointed out an inconsistency. This inconsistency led to extensive discussions from physical circles [9-18]. Although there are many different viewpoints as for the inconsistency, there seems to be a general agreement that the origin is due to the insufficient conditions for the QAT. More recently, we have summarized these different viewpoints of studying the inconsistency of the QAT through the reference [19] where we have noticed that there are essentially two different types of inconsistencies of the QAT in reference [8], that is, MS inconsistency and MS counterexample. Most importantly, these two types are often confused as one [19]. -
5.1 Standard Form, Mood, and Figure
An Example of a Categorical Syllogism All soldiers are patriots. No traitors are patriots. Therefore no traitors are soldiers. The Four Conditions of Standard Form ◦ All three statements are standard-form categorical propositions. ◦ The two occurrences of each term are identical. ◦ Each term is used in the same sense throughout the argument. ◦ The major premise is listed first, the minor premise second, and the conclusion last. The Mood of a Categorical Syllogism consists of the letter names that make it up. ◦ S = subject of the conclusion (minor term) ◦ P = predicate of the conclusion (minor term) ◦ M = middle term The Figure of a Categorical Syllogism Unconditional Validity Figure 1 Figure 2 Figure 3 Figure 4 AAA EAE IAI AEE EAE AEE AII AIA AII EIO OAO EIO EIO AOO EIO Conditional Validity Figure 1 Figure 2 Figure 3 Figure 4 Required Conditio n AAI AEO AEO S exists EAO EAO AAI EAO M exists EAO AAI P exists Constructing Venn Diagrams for Categorical Syllogisms: Seven “Pointers” ◦ Most intuitive and easiest-to-remember technique for testing the validity of categorical syllogisms. Testing for Validity from the Boolean Standpoint ◦ Do shading first ◦ Never enter the conclusion ◦ If the conclusion is already represented on the diagram the syllogism is valid Testing for Validity from the Aristotelian Standpoint: 1. Reduce the syllogism to its form and test from the Boolean standpoint. 2. If invalid from the Boolean standpoint, and there is a circle completely shaded except for one region, enter a circled “X” in that region and retest the form. 3. If the form is syllogistically valid and the circled “X” represents something that exists, the syllogism is valid from the Aristotelian standpoint. -
Introduction to Validity
Slide 1 Introduction to Validity Presentation to the National Assessment Governing Board This presentation addresses the topic of validity. It begins with some first principles of validity. Slide 2 Validity is important. * “one of the major deities in the pantheon of the psychometrician...” (Ebel, 1961, p. 640) * “the most fundamental consideration in developing and evaluating tests” (AERA, APA, NCME, 1999, p. 9) 2 These two quotes highlight the importance of validity in measurement and assessment. The first of these quotes is by a renowned psychometrician, Robert Ebel. The second quote is from the Standards for Educational and Psychological Testing. Slide 3 Two Important Concepts 1) Construct * a label used to describe behavior * refers to an unobserved (latent) characteristic of interest * Examples: creativity, intelligence, reading comprehension, preparedness 3 There are two important concepts to validity. The first one is the notion of a construct. Constructs can be defined as labels used to describe an unobserved behavior, such as honesty. Such behaviors vary across individuals; although they may leave an impression, they cannot be directly measured. In the social sciences virtually everything is a construct. Slide 4 Construct (continued) * don’t exist -- “the product of informed scientific imagination” * operationalized via a measurement process 4 Linda Crocker and James Algina described constructs as “the product of informed scientific imagination.” Constructs represent something that is abstract, but can be operationalized through the measurement process. Measurement can involve a paper and pencil test, a performance, or some other activity through observation which would then produce variables that can be quantified, manipulated, and analyzed. Slide 5 Two Important Concepts 2) Inference * “Informed leap” from an observed, measured value to an estimate of underlying standing on a construct * Short vs. -
Propositional Logic
Propositional Logic LX 502 - Semantics September 19, 2008 1. Semantics and Propositions Natural language is used to communicate information about the world, typically between a speaker and an addressee. This information is conveyed using linguistic expressions. What makes natural language a suitable tool for communication is that linguistic expressions are not just grammatical forms: they also have content. Competent speakers of a language know the forms and the contents of that language, and how the two are systematically related. In other words, you can know the forms of a language but you are not a competent speaker unless you can systematically associate each form with its appropriate content. Syntax, phonology, and morphology are concerned with the form of linguistic expressions. In particular, Syntax is the study of the grammatical arrangement of words—the form of phrases but not their content. From Chomsky we get the sentence in (1), which is grammatical but meaningless. (1) Colorless green ideas sleep furiously Semantics is the study of the content of linguistic expressions. It is defined as the study of the meaning expressed by elements in a language and combinations thereof. Elements that contribute to the content of a linguistic expression can be syntactic (e.g., verbs, nouns, adjectives), morphological (e.g., tense, number, person), and phonological (e.g., intonation, focus). We’re going to restrict our attention to the syntactic elements for now. So for us, semantics will be the study of the meaning expressed by words and combinations thereof. Our goals are listed in (2). (2) i. To determine the conditions under which a sentence is true of the world (and those under which it is false) ii. -
Semantics for Classical Predicate Logic (Part I)∗
Semantics for Classical Predicate Logic (Part I)∗ Hans Halvorson Formal logic begins with the assumption that the validity of an argu- ment depends only on its logical form, and not on its content. In particu- lar, if two arguments have the same logical form, then either they are both valid, or they are both invalid. But what is logical form? Our first ap- proach (before the midsemester break) was to identify logical form with sentence connecting words, such as “and”, “or”, and “if . then . ”. But we found that there are some arguments whose validity is not due to sen- tence connecting words; instead, these arguments are valid because of cer- tain quantifier phrases, such as “all . ” or “some . ”. We have now proposed four rules of inference for quantifier phrases: U-Elimination, U-Introduction, E-Introduction, and E-Elimination. It should seem reasonably clear that these rules are acceptable, i.e. will not lead you from true premises to a false conclusion. But since the rules are somewhat subtle, we should provide an independent method for checking their va- lidity. Furthermore, it is not at all clear whether or not these rules form a “complete” system — i.e. if there is a sense in which they can be used to reproduce any valid argument. Let’s consider first the question of the safety of these rules, and com- pare with the case of the sentence connective rules. How do we know that the rule &E is safe? Roughly speaking, we know that in every situation where P & Q is true, P is also true. -
Lecture 11: the Syntax of Propositional Logic
Lecture 11: Propositional Logic Statements The Syntax of Propositional Logic The Syntax of . Module Home Page Aims: Title Page • To discuss what we mean by a logic; JJ II • To look at what is meant by propositional logic; J I • To look at the syntax of propositional logic. Page 1 of 11 Back Full Screen Close Quit 11.1. Propositional Logic • In this section of the course, we introduce logic in general, and then look at one particular logic, propositional logic. We use this to introduce many of the main ideas Propositional Logic that are to be found in every logic. We will look at the syntax and semantics of Statements this logic, how to translate simple sentences of English into syntactically well-formed The Syntax of . expressions of this logic, and how to show the validity of arguments in a number of ways. Module Home Page • You are warned that logic terminology and notation is quite variable between different books. I have been consistent within these notes, and have also tried to point out cases where books use different terminology and notation. Title Page • Logic is the study of reasoning. JJ II • It is used to study arguments, comprising premisses and conclusions, such as the following: J I 1. Every elephant is grey. 2. Clyde is an elephant. Page 2 of 11 3. Therefore, Clyde is grey. This a valid argument. Back Why? Because: If all the premisses are true, then the conclusion must be true. This definition of a valid argument says nothing about whether the premisses are in fact Full Screen true: their truth is not necessary for the validity of the argument, as witnessed by the validity of the following argument (despite the falsity of its first premiss): Close 1. -
Basic Concepts of Logic
BASIC CONCEPTS OF LOGIC 1. What is Logic? ..................................................................................................2 2. Inferences and Arguments ................................................................................2 3. Deductive Logic versus Inductive Logic ..........................................................5 4. Statements versus Propositions.........................................................................6 5. Form versus Content .........................................................................................7 6. Preliminary Definitions.....................................................................................9 7. Form and Content in Syllogistic Logic ...........................................................11 8. Demonstrating Invalidity Using the Method of Counterexamples .................13 9. Examples of Valid Arguments in Syllogistic Logic .......................................19 10. Exercises for Chapter 1...................................................................................22 11. Answers to Exercises for Chapter 1................................................................25 2 Hardegree, Symbolic Logic 1. WHAT IS LOGIC? Logic may be defined as the science of reasoning. However, this is not to suggest that logic is an empirical (i.e., experimental or observational) science like physics, biology, or psychology. Rather, logic is a non-empirical science like mathematics. Also, in saying that logic is the science of reasoning, we do not mean that