Topic
Contraposition
About: Contraposition is a research topic. Over the lifetime, 20 publications have been published within this topic receiving 460 citations.
Papers
Book•
30 Sep 1974
TL;DR: The history of logic can be traced back to the early sixteenth century, when the first formal logic was proposed as discussed by the authors. But it is difficult to trace the evolution of formal logic to the present day.
Abstract: I/Historical Introduction.- 1. The Publication of Medieval Works.- 2. Scholasticism in Italy and Germany.- 3. Scholasticism in France and Spain.- 4. Humanism.- 5. Rudolph Agricola and His Influence.- 6. Petrus Ramus and His Influence.- 7. Seventeenth Century Logic: Eclecticism.- 8. Humanism and Late Scholasticism in Spain.- 9. Other Schools of Logic.- 10. A Note on Terminology.- II/Meaning and Reference.- I. The Nature of Logic.- 1. The Contents of Logical Text-books.- 2. The Definition of Logic.- 3. The Object of Logic.- II. Problems of Language.- 1. Terms: Their Definition and Their Main Divisions.- 2. The Relationship between Mental, Spoken and Written Terms.- 3. Other Divisions of Terms.- 4. Sense and Reference.- 5. Propositions and Their Parts.- 6. Sentence-Types and Sentence-Tokens.- 7. Complex Signifiables and Truth.- 8. Other Approaches to Truth.- 9. Possibility and Necessity.- II. Supposition Theory.- 1. Supposition, Acceptance and Verification.- 2. Proper, Improper, Relative and Absolute Supposition.- 3. Material Supposition.- 4. Simple Supposition.- 5. Natural Personal Supposition.- 6. Ampliation.- 7. Appellation.- III. Semantic Paradoxes.- 1. Problems Arising from Self-Reference.- 2. Solution One: Self-Reference Is Illegitimate.- 3. Solution Two: All Propositions Imply Their Own Truth.- 4. Solution Three: Insolubles Assert Their Own Falsity.- 5. Solution Four: Two Kinds of Meaning.- 6. Solution Five: Two Truth-Conditions.- 7. Later Writing on Insolubles.- III/Formal Logic. Part One: Unanalyzed Propositions.- I. The Theory of Consequence.- 1. The Definition of Consequence.- 2. The Definition of Valid Consequence.- 3. Formal and Material Consequence.- 4. 'Ut Nunc' Consequence.- 5. The Paradoxes of Strict Implication.- 6. Rules of Valid Consequence.- II. Propositional Connectives.- 1. Compound Propositions in General.- 2. Conditional Propositions.- 3A. Rules for Illative Conditionals.- 3B. Rules for Promissory Conditionals.- 4. Biconditionals.- 5. Conjunctions.- 6. Disjunctions.- 7. De Morgan's Laws.- 8. Other Propositional Connectives.- III. An Analysis of the Rules Found in Some Individual Authors.- 1. Paris in the Early Sixteenth Century.- 2. Oxford in the Early Sixteenth Century.- 3. Germany in the Early Sixteenth Century.- 4. Spain in the Third Decade of the Sixteenth Century.- 5. Spain in the Second Part of the Sixteenth Century.- 6. Germany in the Early Seventeenth Century.- IV/ Formal Logic. Part Two: The Logic of Analyzed Propositions.- I. The Relationships Between Propositions.- 1. The Quality and Quantity of Propositions.- 2. Opposition.- 3. Equipollence.- 4. Simple and Accidental Conversion.- 5. Conversion by Contraposition.- II. Supposition Theory and Quantification.- 1. The Divisions of Personal Supposition.- 2. Descent and Ascent.- III. Categorical Syllogisms.- 1. Figures and Modes.- 2. How to Test the Validity of a Syllogism.- 3. Proof by Reduction.- 4. Syllogisms with Singular Terms.- Appendix/Latin Texts.- 1. Primary Sources.- 2. Secondary Sources on the History of Logic 1400-1650.- Index of names.
227 citations
1 Jan 2001
TL;DR: This paper gives a brief introduction to a particular machine learning method known as inductive logic programming, and it is argued that this method, unlike many current statistically based machine learning methods, implies a view of grammar learning that bears close affinity to the views linguists have of the logical problem of language acquisition.
Abstract: This paper gives a brief introduction to a particular machine learning method known as inductive logic programming. It is argued that this method, unlike many current statistically based machine learning methods, implies a view of grammar learning that bears close affinity to the views linguists have of the logical problem of language acquisition. Two experiments in grammar learning using this technique are described, using a unification grammar formalism, and positive-only data. What is Inductive Logic Programming? Inductive Logic Programming (Muggleton & DeRaedt 1994:629-679) is a machine learning technique that builds logical theories here, (full) first order logic to explain observations. ‘Explain’ here means that it is possible to deduce the evidence from the axioms of the theory (and not be able to deduce negative evidence). ILP is best introduced via the following schema and consequent derivation: (1) Background & Hypothesis Evidence We do not assume a tabula rasa: for reasons that every linguist will be familiar with, it is necessary to assume a fairly rich set of background assumptions to constrain the space of possible hypotheses. Given this background, and the evidence, the task is to come up with a hypothesis such that when it is conjoined with the background, the evidence can be deduced from it. 32 James Cussens & Stephen Pulman Each of the components in the above schema is represented as a set of logical statements. Notice that schema 1 is logically equivalent to 2, since if P Q then P → Q (the deduction theorem), and P → Q ≡ ¬Q → ¬P (contraposition): (2) Background & Evidence Hypothesis (where the overline indicates negation.) Since Background & Evidence is by hypothesis, consistent, it will be the case by Herbrand's theorem (provided that we restrict the form of H and E) that there is some finite set of ground clauses that are true in every model of that expression. Step 3 of the derivation is: (3) Find set of clauses C true in every model of: Background & Evidence. Notice that we represent this set of clauses as a negation, to make succeeding steps tidier. Since this set of clauses is true in every model of Background & Evidence, then the following step of the derivation holds: (4) Background & Evidence C Hypothesis Note that Hypothesiswill be a subset of C . The remaining two steps of the derivation follow simply:
72 citations
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TL;DR: In this paper, the authors define des different interpretations du principe de preference formelle defini par Aristote dans les "Topiques" and du role qu'elles jouent dans le cadre des theories of la preference, des theories de l'utilite and des theories DE la decision developpees par S. Hallden, L. J. Savage and G. E. Moore
Abstract: Etude des differentes interpretations du principe de preference formelle defini par Aristote dans les «Topiques» et du role qu'elles jouent dans le cadre des theories de la preference, des theories de l'utilite et des theories de la decision developpees par S. Hallden, L. J. Savage et G. E. Moore
16 citations
TL;DR: It is argued that contraposition is valid for a class of natural language conditionals, if some modifications are allowed to preserve the meaning of the original conditional, if what is implicit in the original consequent is made explicit in the contrapositive antecedent.
16 citations
Journal Article•
TL;DR: In this article, a qualitative case study was adopted to reveal the preservice mathematics teachers' ability to determine the techniques of proofs on integers using the Proof Techniques Determination Form (PTDF).
Abstract: The aim of the study is to reveal the preservice mathematics teachers’ ability to determine the techniques of proofs on integers. A qualitative case study approach was adopted in this study. The participants of the study consisted of 172 preservice teachers enrolled in an elementary mathematics teaching program in their second and third years at a state university in Turkey. The data of the study were obtained from the Proof Techniques Determination Form (PTDF) which consists of six proofs on integers proven by different techniques and semi-structured interviews with five preservice teachers who were successful in different achievement levels of PTDF. The preservice teachers were asked to determine the proof techniques presented to them and express their warrants. The proof techniques used in PTDF are; direct proof, proof by induction, proof by contradiction, proof by contraposition, proof by counterexample and proof by confliction. At the end of the study, the preservice teachers were successful in determining proof by induction and direct proof technique. However, they were unsuccessful in determining proof by contraposition and proof by contradiction technique. While there was no difficulty in determining proof by counterexample technique, most of the preservice teachers had struggled in determining proof by confliction used to show that the proposition was false. The preservice teachers mostly used direct proof instead of proof by contraposition, proof by contraposition instead of proof by contradiction, and direct proof instead of proof by confliction. It was determined that the preservice teachers tended to evaluate the technique of any proof as direct proof. Sources of these difficulties were: deficiencies in understanding the differences between proof by contradiction and proof by contraposition, usage of same warrants of both techniques, and accepting general warrants which are valid for every proof as the property of direct proof.
11 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2019 | 3 |
| 2018 | 1 |
| 2017 | 2 |
| 2013 | 2 |
| 2008 | 1 |