Lesson 5: Inductive and Deductive Reasoning Conditional - a statement that tells if one thing happens, another will follow; stated as if p, then q written as p q; Contrapositive - a type of conditional that says if one thing doesn t happen, neither will another; stated as if not q, then not p written ~q ~p; Converse - a reversed conditional; if a conditional is p q, than its converse is q p There are two approaches to furthering knowledge: reasoning from known ideas and synthesizing observations. Inductive reasoning allows you to observe the world, and attempt to explain based on your observations. You start with no prior assumptions. Deductive reasoning consists of logical assertions from known facts. Terms to know Before one can start to understand logic, and thereby begin to prove geometric theorems, one must first know a few vocabulary words and symbols. Conditional: a conditional is something which states that one statement implies another. A conditional contains two parts: the condition and the conclusion, where the conditional statement implies the conclusion. A conditional is always in the form "If statement 1, then statement 2." In most mathematical notation, a conditional is often written in the form p q, which is read as "If p, then q" where p and q are both statements. the converse of a logical statement is when the conclusion becomes the condition and vice versa; i.e., p q becomes q p. For example, the converse of the statement "If someone is a woman, then they are a human" would be "If someone is a human, then they are a woman." The converse of a conditional does not necessarily have the same truth value as the original, though it sometimes does, as will become apparent later.
AND: And is a logical operator which is true only when both statements are true. For example, the statement "Diamond is the hardest substance known to man AND a diamond is a metal" is false. While the former statement is true, the latter is not. However, the statement "Diamond is the hardest substance known to man AND diamonds are made of carbon" would be true, because both parts are true. OR: If two statements are joined together by "or," then the truth of the "or" statement is dependant upon whether one or both of the statements from which it is composed is true. For example, the statement "Tuesday is the day after Monday OR Thursday is the day after Saturday" would have a truth value of "true," because even though the latter statement is false, the former is true. NOT: If a statement is preceded by "NOT," then it is evaluating the opposite truth value of that statement. The symbol for "NOT" is. For example, if the statement p is "Elvis is dead," then p would be "Elvis is not dead." The concept of "NOT" can cause some confusion when it relates to statements which contain the word "all." For example, if r is " ". "All men have hair," then r would be "All men do not have hair" or "No men have hair." Do not confuse this with "Not all men have hair" or "Some men have hair." The "NOT" should apply to the verb in the statement: in this case, "have." p can also be written as NOT p or ~p. NOT p may also be referred to as the "negation of p." The inverse of a conditional says that the negation of the condition implies the negation of the conclusion. For example, the inverse of p q is p q. Like a converse, an inverse does not necessarily have the same truth value as the original conditional. Bi-conditional: A bi-conditional is conditional where the condition and the conclusion imply one another. A bi-conditional starts with the words "if and only if." For example, "If and only if p, then q" means both that p implies q and that q implies p. Premise: A premise is a statement whose truth value is known initially. For example, if one were to say "If today is Thursday, then the cafeteria will serve burritos," and one knew that what day it was, then the premise would be "Today is Thursday" or "Today is not Thursday." : The symbol which denotes a conditional. p q is read as "if p, then q." Iff: Iff is a shortened form of "if and only if." It is read as "if and only if." : The symbol which denotes a bi-conditional. p q is read as "If and only if p, then q." : The symbol for "therefore." p q means that one knows that p is true (p is true is the premise), and has logically concluded that q must also be true.
: The symbol for "and." +: The symbol for "or." : The symbol for if and only if. Deductive Reasoning There are a few forms of deductive logic. One of the most common deductive logical arguments is modus ponens, which states that: p q p q (If p, then q) (p, therefore q) An example of modus ponens: If I stub my toe, then I will be in pain. I stubbed my toe. Therefore, I am in pain. Although there are other forms of deductive logic, these will suit our purposes for this course. Inductive Reasoning Inductive reasoning is a logical argument which does not definitely prove a statement, but rather assumes it. Inductive reasoning is used often in life. Polling is an example of the use of inductive reasoning. If one were to poll one thousand people, and 300 of those people selected choice A, then one would infer that 30% of any population might also select choice A. This would be using inductive logic, because it does not definitively prove that 30% of any population would select choice A. Because of this factor of uncertainty, inductive reasoning should be avoided when possible when attempting to prove geometric properties. Summarizing Logic for Informal Geometry: There are four basic Logic Statements that you need for Informal Geometry. This is a simplification of the Logic rules, but will help you sort out all of the verbiage.
Note: We use another sign here, which is a single arrow and not a double arrow, but it means the same thing. ( is equal in meaning to ) Conditional: Statement; if p, then q (If it rains, you use an umbrella) p q If q, then p (If you use an umbrella, it rains) q p If not p, then not q (If doesn t rain, you don t use an umbrella) ~ p ~q If not q, then not p (If you are not using an umbrella, then it is not raining) ~q ~p Lesson 5 Review Explain why each statement might be true or might be untrue. Tell if each is an example of inductive or deductive reasoning? Justify your answers. 1. All men are mortal. Joe is a man. Therefore Joe is mortal. If the first two statements are true, then the conclusion must be true. 2 2. To get a high school diploma from The Ogburn School, a student must have 24 credits. Cindy has more than 24 credits. Therefore, Cindy must have a high school diploma. 3. This cat is black. That cat is black A third cat is black. Therefore all cats are black. 4. This marble from the bag is black. That marble from the bag is black. A third marble from the bag is black. Therefore all the marbles in the bag are black. 5. For problems A-E, write the converse, inverse, and contrapositive statements based on the given conditional statement. A. If I own a dog, then I own an animal.
B. If I go to be early, then I sleep well. C. If this is Thursday, then I do not go to church. D. If today is Wednesday, the yesterday was Tuesday. E. If 5x = 10, then x = 2.