5.1 Standard Form, Mood, and Figure
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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. -
7.1 Rules of Implication I
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification. -
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. -
Saving the Square of Opposition∗
HISTORY AND PHILOSOPHY OF LOGIC, 2021 https://doi.org/10.1080/01445340.2020.1865782 Saving the Square of Opposition∗ PIETER A. M. SEUREN Max Planck Institute for Psycholinguistics, Nijmegen, the Netherlands Received 19 April 2020 Accepted 15 December 2020 Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cogni- tive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leak- ing O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute -
A Validity Framework for the Use and Development of Exported Assessments
A Validity Framework for the Use and Development of Exported Assessments By María Elena Oliveri, René Lawless, and John W. Young A Validity Framework for the Use and Development of Exported Assessments María Elena Oliveri, René Lawless, and John W. Young1 1 The authors would like to acknowledge Bob Mislevy, Michael Kane, Kadriye Ercikan, and Maurice Hauck for their valuable input and expertise in reviewing various iterations of the framework as well as Priya Kannan, Don Powers, and James Carlson for their input throughout the technical review process. Copyright © 2015 Educational Testing Service. All Rights Reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS, GRE, LISTENTING. LEARNING. LEADING, TOEIC, and TOEFL are registered trademarks of Educational Testing Service (ETS). EXADEP and EXAMEN DE ADMISIÓN A ESTUDIOS DE POSGRADO are trademarks of ETS. 1 Abstract In this document, we present a framework that outlines the key considerations relevant to the fair development and use of exported assessments. Exported assessments are developed in one country and are used in countries with a population that differs from the one for which the assessment was developed. Examples of these assessments include the Graduate Record Examinations® (GRE®) and el Examen de Admisión a Estudios de Posgrado™ (EXADEP™), among others. Exported assessments can be used to make inferences about performance in the exporting country or in the receiving country. To illustrate, the GRE can be administered in India and be used to predict success at a graduate school in the United States, or similarly it can be administered in India to predict success in graduate school at a graduate school in India. -
Traditional Logic II Text (2Nd Ed
Table of Contents A Note to the Teacher ............................................................................................... v Further Study of Simple Syllogisms Chapter 1: Figure in Syllogisms ............................................................................................... 1 Chapter 2: Mood in Syllogisms ................................................................................................. 5 Chapter 3: Reducing Syllogisms to the First Figure ............................................................... 11 Chapter 4: Indirect Reduction of Syllogisms .......................................................................... 19 Arguments in Ordinary Language Chapter 5: Translating Ordinary Sentences into Logical Statements ..................................... 25 Chapter 6: Enthymemes ......................................................................................................... 33 Hypothetical Syllogisms Chapter 7: Conditional Syllogisms ......................................................................................... 39 Chapter 8: Disjunctive Syllogisms ......................................................................................... 49 Chapter 9: Conjunctive Syllogisms ......................................................................................... 57 Complex Syllogisms Chapter 10: Polysyllogisms & Aristotelian Sorites .................................................................. 63 Chapter 11: Goclenian & Conditional Sorites ........................................................................ -
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. -
Common Sense for Concurrency and Strong Paraconsistency Using Unstratified Inference and Reflection
Published in ArXiv http://arxiv.org/abs/0812.4852 http://commonsense.carlhewitt.info Common sense for concurrency and strong paraconsistency using unstratified inference and reflection Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Unstratified Reflection is the Norm....................................... 11 Abstraction and Reification .............................................. 11 This paper develops a strongly paraconsistent formalism (called Direct Logic™) that incorporates the mathematics of Diagonal Argument .......................................................... 12 Computer Science and allows unstratified inference and Logical Fixed Point Theorem ........................................... 12 reflection using mathematical induction for almost all of Disadvantages of stratified metatheories ........................... 12 classical logic to be used. Direct Logic allows mutual Reification Reflection ....................................................... 13 reflection among the mutually chock full of inconsistencies Incompleteness Theorem for Theories of Direct Logic ..... 14 code, documentation, and use cases of large software systems Inconsistency Theorem for Theories of Direct Logic ........ 15 thereby overcoming the limitations of the traditional Tarskian Consequences of Logically Necessary Inconsistency ........ 16 framework of stratified metatheories. Concurrency is the Norm ...................................................... 16 Gödel first formalized and proved that it is not possible -
To Determine the Validity of Categorical Syllogisms
arXiv:1509.00926 [cs.LO] Using Inclusion Diagrams – as an Alternative to Venn Diagrams – to Determine the Validity of Categorical Syllogisms Osvaldo Skliar1 Ricardo E. Monge2 Sherry Gapper3 Abstract: Inclusion diagrams are introduced as an alternative to using Venn diagrams to determine the validity of categorical syllogisms, and are used here for the analysis of diverse categorical syllogisms. As a preliminary example of a possible generalization of the use of inclusion diagrams, consideration is given also to an argument that includes more than two premises and more than three terms, the classic major, middle and minor terms in categorical syllogisms. Keywords: inclusion diagrams, Venn diagrams, categorical syllogisms, set theory Mathematics Subject Classification: 03B05, 03B10, 97E30, 00A66 1. Introduction Venn diagrams [1] have been and continue to be used to introduce syllogistics to beginners in the field of logic. These diagrams are perhaps the main technical tool used for didactic purposes to determine the validity of categorical syllogisms. The objective of this article is to present a different type of diagram, inclusion diagrams, to make it possible to reach one of the same objectives for which Venn diagrams were developed. The use of these new diagrams is not only simple and systematic but also very intuitive. It is natural and quite obvious that if one set is included in another set, which in turn, is included in a third set, then the first set is part of the third set. This result, an expression of the transitivity of the inclusion relation existing between sets, is the primary basis of inclusion diagrams. -
Aristotle and Gautama on Logic and Physics
Aristotle and Gautama on Logic and Physics Subhash Kak Louisiana State University Baton Rouge, LA 70803-5901 February 2, 2008 Introduction Physics as part of natural science and philosophy explicitly or implicitly uses logic, and from this point of view all cultures that studied nature [e.g. 1-4] must have had logic. But the question of the origins of logic as a formal discipline is of special interest to the historian of physics since it represents a turning inward to examine the very nature of reasoning and the relationship between thought and reality. In the West, Aristotle (384-322 BCE) is gener- ally credited with the formalization of the tradition of logic and also with the development of early physics. In India, the R. gveda itself in the hymn 10.129 suggests the beginnings of the representation of reality in terms of various logical divisions that were later represented formally as the four circles of catus.kot.i: “A”, “not A”, “A and not A”, and “not A and not not A” [5]. Causality as the basis of change was enshrined in the early philosophical sys- tem of the S¯am. khya. According to Pur¯an. ic accounts, Medh¯atithi Gautama and Aks.ap¯ada Gautama (or Gotama), which are perhaps two variant names for the same author of the early formal text on Indian logic, belonged to about 550 BCE. The Greek and the Indian traditions seem to provide the earliest formal arXiv:physics/0505172v1 [physics.gen-ph] 25 May 2005 representations of logic, and in this article we ask if they influenced each other.