The Operations of Conversion, Obversion and Contraposition Are Applied to Categorical Propositions to Yield New Categorical Propositions

Total Page:16

File Type:pdf, Size:1020Kb

The Operations of Conversion, Obversion and Contraposition Are Applied to Categorical Propositions to Yield New Categorical Propositions 4.4 CONVERSION, OBVERSION AND CONTRAPOSITION The operations of conversion, obversion and contraposition are applied to categorical propositions to yield new categorical propositions. Conversion consists of simply switching the subject term with the predicate term while leaving the quality and quantity of the proposition unaltered. The result of applying conversion to a categorical proposition is called the converse of the proposition. Thus, for example, the converse of “All dogs are mammals” is “All mammals are dogs.” The converses of E and I propositions are logically equivalent to them (that is, they necessarily have the same truth values.) The converse of A and O propositions are not, in general, logically equivalent to them. Similarly, if we form an argument whose premise is a categorical proposition and whose conclusion is the converse of it, then the argument is valid if the premise is an E and I proposition and, in general, is invalid if the premise is an A or an O proposition. Obversion consists of both (1) changing the quality of the proposition (leaving the quantity the same), and (2) complementing the predicate term. To complement the predicate term, one typically attaches the prefix “non-“ to it. The result of obversion is called obverse of the proposition to which it is applied. The obverse of any categorical proposition, A, E, I or O, is logically equivalent to it. Any argument whose premise is a categorical proposition and whose conclusion is its obverse is a valid argument. Contraposition consist of both (1) switching the subject with the predicate term (while leaving the quality and quantity of the proposition unaltered), and (2) complementing both terms. Thus, for example, the contrapositive of “All dogs are mammals” is “All non-mammals are non-dogs.” The result of contraposition is called contrapositive of the proposition to which it is applied. The contrapositive of A propositions and O propositions are logically equivalent to the originals, while the contrapositives of E and I propositions are not, in general, logically equivalent to the originals. If we form an argument whose premises is a categorical proposition and whose conclusion is the contrapositive of it, then the argument is valid if the premise is an A or an O proposition and, in general is invalid if the premise is an E or an I proposition. Conversion, obversion and contraposition may be used in sequence to prove certain arguments valid. There are two basic points to remember: The first is that doubly complementing a term (e.g. non non-P) yields the equivalent of the term that is doubly complemented. The second is that the operations of conversion and contraposition must be correctly applied: conversion must be applied only to E and I propositions and contraposition only to A and O propositions. Here is an example of using the operations to show that “All A are non-B; therefore, no B are A” is a valid argument: All A are non-B. All non-non-B are non-A. Contraposition of an A proposition All B are non-A. Replacing “non-non-B” with “B.” No B are non-non-A Obversion No B are A. Replacing “non-non-A” with “A.”.
Recommended publications
  • Categorize" a Person, Place, Time, Thing, Or Situation Is to Characterize It As a Member of a Class of Similar Things
    CHAPTER 2 CATEGORICAL PROPOSITIONS A. The Structure of Categorical Propositions To "categorize" a person, place, time, thing, or situation is to characterize it as a member of a class of similar things. One does not consider the thing in question from its purely individual point of view, that is, in terms of the qualities it has without relationship to any other things. Upon being categorized, an individual thing is known by properties that it has by virtue of its being a member of the class of things referred to by that category. All propositions expressed in the Aristotelian system of logic are called categorical propositions because they are constructed using two categories of things: the subject category (or class) and the predicate category (or class). The copula, or connector between the subject and predicate, of a categorical proposition indicates how its subject category is related to its predicate category. The copula specifies whether members of the subject category are also members of the predicate category or whether members of the subject category are not members of the predicate category. Illustration: Subject Copula Predicate Quality of Copula (1) Strawberries are red. Affirmative (2) Strawberries are not red. Negative However, this is not enough. (1) declares that members of the class of strawberries are also members of the class of red things, but (1) is ambiguous because it does not specify the quantity of the subject class that is included in the predicate class. Thus, Ms. A might take (1) to mean “all strawberries are red,” while Mr. B might take (1) to mean “some strawberries are red.” Ms.
    [Show full text]
  • CSE Yet, Please Do Well! Logical Connectives
    administrivia Course web: http://www.cs.washington.edu/311 Office hours: 12 office hours each week Me/James: MW 10:30-11:30/2:30-3:30pm or by appointment TA Section: Start next week Call me: Shayan Don’t: Actually call me. Homework #1: Will be posted today, due next Friday by midnight (Oct 9th) Gradescope! (stay tuned) Extra credit: Not required to get a 4.0. Counts separately. In total, may raise grade by ~0.1 Don’t be shy (raise your hand in the back)! Do space out your participation. If you are not CSE yet, please do well! logical connectives p q p q p p T T T T F T F F F T F T F NOT F F F AND p q p q p q p q T T T T T F T F T T F T F T T F T T F F F F F F OR XOR 푝 → 푞 • “If p, then q” is a promise: p q p q F F T • Whenever p is true, then q is true F T T • Ask “has the promise been broken” T F F T T T If it’s raining, then I have my umbrella. related implications • Implication: p q • Converse: q p • Contrapositive: q p • Inverse: p q How do these relate to each other? How to see this? 푝 ↔ 푞 • p iff q • p is equivalent to q • p implies q and q implies p p q p q Let’s think about fruits A fruit is an apple only if it is either red or green and a fruit is not red and green.
    [Show full text]
  • Conditional Statement/Implication Converse and Contrapositive
    Conditional Statement/Implication CSCI 1900 Discrete Structures •"ifp then q" • Denoted p ⇒ q – p is called the antecedent or hypothesis Conditional Statements – q is called the consequent or conclusion Reading: Kolman, Section 2.2 • Example: – p: I am hungry q: I will eat – p: It is snowing q: 3+5 = 8 CSCI 1900 – Discrete Structures Conditional Statements – Page 1 CSCI 1900 – Discrete Structures Conditional Statements – Page 2 Conditional Statement/Implication Truth Table Representing Implication (continued) • In English, we would assume a cause- • If viewed as a logic operation, p ⇒ q can only be and-effect relationship, i.e., the fact that p evaluated as false if p is true and q is false is true would force q to be true. • This does not say that p causes q • If “it is snowing,” then “3+5=8” is • Truth table meaningless in this regard since p has no p q p ⇒ q effect at all on q T T T • At this point it may be easiest to view the T F F operator “⇒” as a logic operationsimilar to F T T AND or OR (conjunction or disjunction). F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 3 CSCI 1900 – Discrete Structures Conditional Statements – Page 4 Examples where p ⇒ q is viewed Converse and contrapositive as a logic operation •Ifp is false, then any q supports p ⇒ q is • The converse of p ⇒ q is the implication true. that q ⇒ p – False ⇒ True = True • The contrapositive of p ⇒ q is the –False⇒ False = True implication that ~q ⇒ ~p • If “2+2=5” then “I am the king of England” is true CSCI 1900 – Discrete Structures Conditional Statements – Page 5 CSCI 1900 – Discrete Structures Conditional Statements – Page 6 1 Converse and Contrapositive Equivalence or biconditional Example Example: What is the converse and •Ifp and q are statements, the compound contrapositive of p: "it is raining" and q: I statement p if and only if q is called an get wet? equivalence or biconditional – Implication: If it is raining, then I get wet.
    [Show full text]
  • Logical Connectives Good Problems: March 25, 2008
    Logical Connectives Good Problems: March 25, 2008 Mathematics has its own language. As with any language, effective communication depends on logically connecting components. Even the simplest “real” mathematical problems require at least a small amount of reasoning, so it is very important that you develop a feeling for formal (mathematical) logic. Consider, for example, the two sentences “There are 10 people waiting for the bus” and “The bus is late.” What, if anything, is the logical connection between these two sentences? Does one logically imply the other? Similarly, the two mathematical statements “r2 + r − 2 = 0” and “r = 1 or r = −2” need to be connected, otherwise they are merely two random statements that convey no useful information. Warning: when mathematicians talk about implication, it means that one thing must be true as a consequence of another; not that it can be true, or might be true sometimes. Words and symbols that tie statements together logically are called logical connectives. They allow you to communicate the reasoning that has led you to your conclusion. Possibly the most important of these is implication — the idea that the next statement is a logical consequence of the previous one. This concept can be conveyed by the use of words such as: therefore, hence, and so, thus, since, if . then . , this implies, etc. In the middle of mathematical calculations, we can represent these by the implication symbol (⇒). For example 3 − x x + 7y2 = 3 ⇒ y = ± ; (1) r 7 x ∈ (0, ∞) ⇒ cos(x) ∈ [−1, 1]. (2) Converse Note that “statement A ⇒ statement B” does not necessarily mean that the logical converse — “statement B ⇒ statement A” — is also true.
    [Show full text]
  • 2-2 Statements, Conditionals, and Biconditionals
    2-2 Statements, Conditionals, and Biconditionals Use the following statements to write a 3. compound statement for each conjunction or SOLUTION: disjunction. Then find its truth value. Explain your reasoning. is a disjunction. A disjunction is true if at least p: A week has seven days. one of the statements is true. q is false, since there q: There are 20 hours in a day. are 24 hours in a day, not 20 hours. r is true since r: There are 60 minutes in an hour. there are 60 minutes in an hour. Thus, is true, 1. p and r because r is true. SOLUTION: ANSWER: p and r is a conjunction. A conjunction is true only There are 20 hours in a day, or there are 60 minutes when both statements that form it are true. p is true in an hour. since a week has seven day. r is true since there are is true, because r is true. 60 minutes in an hour. Then p and r is true, 4. because both p and r are true. SOLUTION: ANSWER: ~p is a negation of statement p, or the opposite of A week has seven days, and there are 60 minutes in statement p. The or in ~p or q indicates a an hour. p and r is true, because p is true and r is disjunction. A disjunction is true if at least one of the true. statements is true. ~p would be : A week does not have seven days, 2. which is false. q is false since there are 24 hours in SOLUTION: a day, not 20 hours in a day.
    [Show full text]
  • 1 UNIT 4 PROPOSITIONS Contents 4.0 Objectives 4.1 Introduction 4.2
    UNIT 4 PROPOSITIONS Contents 4.0 Objectives 4.1 Introduction 4.2 History of Logic and Proposition 4.3 Propositions and Sentences 4.4 Propositions and Judgments 4.5 Types of Proposition 4.6 Quality and Quantity 4.7 Let Us Sum Up 4.8 Key Words 4.9 Further Readings and References 4.10 Answers to Check Your Progress 4.0 OBJECTIVES As we know inference is the main subject matter of logic. The term refers to the argument in which a proposition is arrived at and affirmed or denied on the basis of one or more other propositions accepted as the starting point of the process. To determine whether or not an inference is correct the logician examines the propositions that are the initial and end points of that argument and the relationships between them. This clearly denotes the significance of propositions in the study of logic. In this unit you are expected to study: • the nature • the definition • the types and forms of propositions • the difference between propositions and sentences and judgments • the description of various types of propositions viewed from different standpoints like, composition, generality, relation, quantity, quality, and modality. 4.1 INTRODUCTION Classical logic concerns itself with forms and classifications of propositions. We shall begin with the standard definition of proposition. A proposition is a declarative sentence which is either true or false but not both. Also a proposition cannot be neither true nor false. A proposition is always expressed with the help of a sentence. For example - the same proposition “It is raining” can be expressed in English, Hindi, and Sanskrit and so on.
    [Show full text]
  • New Approaches for Memristive Logic Computations
    Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 6-6-2018 New Approaches for Memristive Logic Computations Muayad Jaafar Aljafar Portland State University Let us know how access to this document benefits ouy . Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Electrical and Computer Engineering Commons Recommended Citation Aljafar, Muayad Jaafar, "New Approaches for Memristive Logic Computations" (2018). Dissertations and Theses. Paper 4372. 10.15760/etd.6256 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. For more information, please contact [email protected]. New Approaches for Memristive Logic Computations by Muayad Jaafar Aljafar A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering Dissertation Committee: Marek A. Perkowski, Chair John M. Acken Xiaoyu Song Steven Bleiler Portland State University 2018 © 2018 Muayad Jaafar Aljafar Abstract Over the past five decades, exponential advances in device integration in microelectronics for memory and computation applications have been observed. These advances are closely related to miniaturization in integrated circuit technologies. However, this miniaturization is reaching the physical limit (i.e., the end of Moore’s Law). This miniaturization is also causing a dramatic problem of heat dissipation in integrated circuits. Additionally, approaching the physical limit of semiconductor devices in fabrication process increases the delay of moving data between computing and memory units hence decreasing the performance. The market requirements for faster computers with lower power consumption can be addressed by new emerging technologies such as memristors.
    [Show full text]
  • Immediate Inference
    Immediate Inference Dr Desh Raj Sirswal, Assistant Professor (Philosophy) P.G. Govt. College for Girls, Sector-11, Chandigarh http://drsirswal.webs.com . Inference Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. Immediate Inference Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition Eduction Here we will know about Eduction in details. Eduction The second form of Immediate Inference is Eduction. It has three types – Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence. Details are given below: Conversion An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid. Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false. Steps for Conversion Reversing the subject and the predicate terms in the premise. Valid Conversions Convertend Converse A: All S is P.
    [Show full text]
  • Introduction to Logic CIS008-2 Logic and Foundations of Mathematics
    Introduction to Logic CIS008-2 Logic and Foundations of Mathematics David Goodwin [email protected] 11:00, Tuesday 15th Novemeber 2011 Outline 1 Propositions 2 Conditional Propositions 3 Logical Equivalence Propositions Conditional Propositions Logical Equivalence The Wire \If you play with dirt you get dirty." Propositions Conditional Propositions Logical Equivalence True or False? 1 The only positive integers that divide 7 are 1 and 7 itself. 2 For every positive integer n, there is a prime number larger than n. 3 x + 4 = 6. 4 Write a pseudo-code to solve a linear diophantine equation. Propositions Conditional Propositions Logical Equivalence True or False? 1 The only positive integers that divide 7 are 1 and 7 itself. True. 2 For every positive integer n, there is a prime number larger than n. True 3 x + 4 = 6. The truth depends on the value of x. 4 Write a pseudo-code to solve a linear diophantine equation. Neither true or false. Propositions Conditional Propositions Logical Equivalence Propositions A sentence that is either true or false, but not both, is called a proposition. The following two are propositions 1 The only positive integers that divide 7 are 1 and 7 itself. True. 2 For every positive integer n, there is a prime number larger than n. True whereas the following are not propositions 3 x + 4 = 6. The truth depends on the value of x. 4 Write a pseudo-code to solve a linear diophantine equation. Neither true or false. Propositions Conditional Propositions Logical Equivalence Notation We will use the notation p : 2 + 2 = 5 to define p to be the proposition 2 + 2 = 5.
    [Show full text]
  • Logic, Proofs
    CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true.
    [Show full text]
  • The Peripatetic Program in Categorical Logic: Leibniz on Propositional Terms
    THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 65 THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS MARKO MALINK Department of Philosophy, New York University and ANUBAV VASUDEVAN Department of Philosophy, University of Chicago Abstract. Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such as reductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titled Specimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule of reductio ad absurdum in a purely categor- ical calculus in which every proposition is of the form AisB. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule of reductio ad absurdum,but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic ¬ known as RMI→ . From its beginnings in antiquity up until the late 19th century, the study of formal logic was shaped by two distinct approaches to the subject. The first approach was primarily concerned with simple propositions expressing predicative relations between terms.
    [Show full text]
  • Testing Validity Using Venn's Diagrams
    TESTING VALIDITY USING VENN'S DIAGRAMS To test the validity of a categorical syllogism, one can use the method of Venn diagrams. Since a categorical syllogism has three terms, we need a Venn diagram using three intersecting circles, one representing each of the three terms in a categorical syllogism. A three term diagram has eight regions (the number of regions being 2n where n is the number of terms). The following chart gives the extension of the predicates in the various regions of the diagram. Region S (Minor Term) P (Major Term) M (Middle Term) 1 yes no no 2 yes yes no 3 no yes no 4 yes no yes 5 yes yes yes 6 no yes yes 7 no no yes 8 no no no In order to use a Venn diagram to test a syllogism, the diagram must be filled in to reflect the contents of the premises. Remember, shading an area means that that area is empty, the term represented has no extension in that area. What one is looking for in a Venn diagram test for validity is an accurate diagram of the conclusion of the argument that logically follows from a diagram of the premises. Since each of the premises of a categorical syllogism is a categorical proposition, diagram the premise sentences independently and then see whether the conclusion has already been diagramed. If so, the argument is valid. If not, then it is not. Remember, one definition of validity is that the propositional (informational) content of the conclusion is already expressed in the premises.
    [Show full text]