Deductive Reasoning
Deductive reasoning in mathematics involves drawing specific conclusions from general principles or premises. It follows a top-down approach, starting with a general theory and applying it to specific cases to reach a logical conclusion. This method is fundamental to mathematical proofs and problem-solving, providing a systematic way to establish the truth of mathematical statements.
8 Key excerpts on "Deductive Reasoning"
eBook - ePubThought and Knowledge
An Introduction to Critical Thinking
- Diane F. Halpern(Author)
- 2013(Publication Date)
- Psychology Press(Publisher)
...75) A distinction is often made between inductive and Deductive Reasoning. (See the Thinking as Hypothesis Testing chapter for a related discussion of this topic.) In inductive reasoning, observations are collected that support or suggest a conclusion. It is a way of projecting information from known examples to the unknown (Heit, 2000). For example, if every person you have ever seen has only one head, you would use this evidence to support the conclusion (or suggest the hypothesis) that everyone in the world has only one head. Of course, you cannot be absolutely certain of this fact. It is always possible that someone you have never met has two heads. If you met just one person with two heads, your conclusion must be wrong. Thus, with inductive reasoning you can never prove that your conclusion or hypothesis is correct, but you can disprove it. With inductive reasoning, if the premises are true, the conclusion is probably true. In Deductive Reasoning, we begin with statements known or believed to be true, like “everyone has only one head,” and then conclude or infer that La Tisha, a woman you have never met, will have only one head. This conclusion follows logically from the earlier statement. If we know that it is true that everyone has only one head, then it MUST be true that any specific person will have only one head. Similarly, if I show you a rectangle that is 2’ by 3’, then the area of the rectangle must be 6 square feet. Deductive Reasoning is sometimes described as reasoning “down” from beliefs about the nature of the world to particular instances. Rips (1988) argues that deduction is a general purpose mechanism for cognitive tasks: deduction “enables us to answer questions from information stored in memory, to plan actions according to goals, and to solve certain kinds of puzzles” (p...
eBook - ePubThought and Knowledge
An Introduction to Critical Thinking
- Diane F. Halpern, Dana S. Dunn(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...75) A distinction is often made between inductive and Deductive Reasoning. (See Chapter 6 “Thinking as Hypothesis Testing” for a related discussion of this topic.) In inductive reasoning observations are collected that support or suggest a conclusion. It is a way of projecting information from known examples to the unknown (Heit, 2000). For example, if every person you have ever seen has only one head, you would use this evidence to support the conclusion (or suggest the hypothesis) that everyone in the world has only one head. Of course, you cannot be absolutely certain of this fact. It is always possible that someone you have never met has two heads. If you met just one person with two heads, your conclusion must be wrong. Thus, with inductive reasoning you can never prove that your conclusion or hypothesis is correct, but you can disprove it. With inductive reasoning, if the premises are true, the conclusion is probably true. In Deductive Reasoning, we begin with statements known or believed to be true, like “everyone has only one head,” and then conclude or infer that La Tisha, a woman you have never met, will have only one head. This conclusion follows logically from the earlier statement. If we know that it is true that everyone has only one head, then it MUST be true that any specific person will have only one head. Similarly, if we show you a rectangle that is 2’ by 3’, then the area of the rectangle must be 6 square feet. Deductive Reasoning is sometimes described as reasoning “down” from beliefs about the nature of the world to particular instances. Rips (1988) argues that deduction is a general-purpose mechanism for cognitive tasks—deduction “enables us to answer questions from information stored in memory, to plan actions according to goals, and to solve certain kinds of puzzles” (p...
eBook - ePub- W. James Potter(Author)
- 2019(Publication Date)
- SAGE Publications, Inc(Publisher)
...Chapter 7 Deducing: Reasoning with Logic from General Principles I. The Deduction Algorithm Step 1: Begin With an Observation Step 2: Identify a Relevant General Principle Step 3: Use the Two Premises to Reason Logically to a Conclusion II. Heuristics Heuristic 1: Probability Premise Heuristic 2: Conditional Reasoning III. Avoiding Traps Trap 1: Faulty Major Premise Trap 2: Irrelevant Major Premise Trap 3: Too Complex a Major Premise Is Needed Trap 4: Conditional Reasoning Trap 5: Irrational Reasoning Trap 6: Unwillingness to Build Knowledge Structures IV. Chapter Review Exercises Deduction is the skill of using a few premises to reason logically toward a conclusion. The basic procedure of deduction follows a reasoning process in the form of a syllogism, which is a set of three statements. The first statement in the set is called the major premise ; it is usually a general principle or rule. The second statement is called the minor premise ; it is usually an observation. The third statement is the reasoned conclusion. We use logic to see if the observation fits the rule and then derive the conclusion. Perhaps the most familiar example of a syllogism is the one that uses the following two premises: (1) All men are mortal and (2) Socrates is a man. From this we can conclude that Socrates is mortal. The first premise is the major one; that is, it states a general proposition. The second premise is the minor one; that is, it provides information about something specific (in this case, a specific person) in a way that relates it to the major premise. Using logic, we see that the observation in the second premise fits the rule in the first premise and we conclude that Socrates is mortal. Deduction is the skill that the fictional detective Sherlock Holmes employed so successfully to make sense of clues and solve crimes. He knew a great deal about the physical world and about human behavior; this knowledge was his bank of major premises...
eBook - ePub- Robert Gass, John Seiter(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...As Leighton (2006) commented: Deductive Reasoning is a cornerstone of scientific research in which claims to new knowledge are evaluated rigorously. Scientists, however, are not the only ones who should profit from systematic thinking. Students stand to become better decision makers, problem solvers, and thinkers when they learn to reason systematically. (pp. 109–110) In this chapter we examined Deductive Reasoning. Unlike inductive reasoning, which deals with probabilities, deduction reasons from known premises, or premises taken to be true, to reach certain conclusions. Ordinary people are not particularly adept at distinguishing valid from invalid arguments. However, their ability to use deduction increases with training and practice. Deduction has practical applications to everyday life, though we may not always realize when we are using it. The syllogism is one of the most common ways of learning to assess validity and invalidity. Three types of syllogisms were examined; categorical, conditional, and disjunctive. Rules and strategies for evaluating each type were also provided. With practice, the rules become almost second nature. Notes 1 One could just as easily argue that people are naturally given to mystical or magical thinking. 2 For the record, Frank Zappa is listed as #22 on Rolling Stone ’s greatest guitarists of all-time list. Brian May is #26 and Jerry Garcia #46. Other guitarists such as Ry Cooder, Steve Vai (Whitesnake), Alice Cooper, and Joe Satriani (Deep Purple), also read music. 3 Some pregnancies might include vaginal bleeding, which could be mistaken for a period, but that is not the same as menstruation. 4 The reasoning is as follows: If Neanderthals and humans diverged onto separate evolutionary paths long ago (say, 400 thousand years) and remained separated, then humans in different parts of the world should have roughly the same amount of Neanderthal DNA. Such is not the case, however...
eBook - ePubMathematical Reasoning
Patterns, Problems, Conjectures, and Proofs
- Raymond Nickerson(Author)
- 2011(Publication Date)
- Psychology Press(Publisher)
...6 CHAPTER Informal Reasoning in Mathematics A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks. (Rota, 1981) The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue. That the process of mathematical discovery is not revealed in the finished proofs that mathematicians publish was pointed out by Evariste Galois, the brilliant French mathematician who, after inventing group theory, died in a duel at the age of 21. It has been convincingly documented by Polya (1954a, 1954b) and Lakatos (1976). In addition to deducing the implications of axioms, mathematicians also invent new axiomatic systems, and this cannot be done by Deductive Reasoning alone. As Polish-American mathematician Stanislav Ulam (1976) puts it, “In mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing” (p. 154). Rucker (1982) makes essentially the same point: “In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it” (p...
eBook - ePubThe Psychology of Thinking
Reasoning, Decision-Making and Problem-Solving
- John Paul Minda(Author)
- 2020(Publication Date)
- SAGE Publications Ltd(Publisher)
...For that reason, many of the examples in the literature were framed as statements of confidence in a prediction or a generalization. Induction can also be described as going beyond the given evidence to discover something new via thinking. With deductive logic, on the other hand, we are attempting to explain how people make specific conclusions and to determine if that conclusion is valid. Deduction often starts with a general statement (“My jacket is either in my backpack or at school”), and then proceeds to more specific statements (“It's not in my backpack”). Rather than going beyond the given evidence to discover something new via thinking, deduction often involves verifying that which is already known. The two kinds of reasoning are related in terms of how we use them in everyday thinking. Although the psychology of induction and deduction differ, we tend to use them together and it can be difficult to tell if you are using inductive reasoning or Deductive Reasoning. For example, if I buy a coffee from McDonald's, take a sip, and discover that it is very hot, I conclude that McDonald's coffee is very hot. That is a generalization, and I can infer that other McDonald's will serve hot coffee. Categorical induction might even allow me to project the property of “hot” to coffee purchased at other, similar restaurants. In both cases, I am relying on inductive reasoning and past experience to make some conclusions about the future. Suppose, then, that I form a general premise about McDonald's coffee. This premise can be stated as: Premise: McDonald's coffee is hot. A premise is a statement of facts about something. In this case, it's a statement of fact about McDonald's coffee. A premise like this can be used to make precise conclusions...
eBook - ePubCritical Thinking Across the Curriculum
A Brief Edition of Thought & Knowledge
- Diane F. Halpern(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...This conclusion follows logically from the earlier statement. If we know that it is true that everyone has only one head, then it must also be true that any specific person will have only one head. This conclusion necessarily follows from the belief; if the belief is true, the conclusion must be true. Deductive Reasoning is sometimes described as reasoning “down” from beliefs about the nature of the world to particular instances. Rips (1988) argued that deduction is a general-purpose mechanism for cognitive tasks. According to Rips, deduction “enables us to answer questions from information stored in memory, to plan actions according to goals, and to solve certain kinds of puzzles” (p. 117). The notion of reasoning up from observations and reasoning down from hypotheses is schematically shown in Fig. 4.1. Although it is common to make a distinction between inductive and Deductive Reasoning (Neubert & Binko, 1992), the distinction may not be a particularly useful description of how people reason in real life. In everyday contexts, we switch from inductive to Deductive Reasoning in the course of thinking. Our hypotheses and beliefs guide the observations we make, whereas our observations, in turn, modify our hypotheses and beliefs. Often, this process will involve a continuous interplay of inductive and Deductive Reasoning. Thinking in real-world contexts almost always involves the use of multiple types of thinking skills. FIG. 4.1 A pictorial distinction between deductive and inductive reasoning. In most real-world settings, we use both types of reasoning recursively LINEAR ORDERING Reasoning is simply a matter of getting your facts straight. —B. F. Anderson (1980, p. 62) Joel is stronger than Bill, but not as strong as Richard. Richard is stronger than Joel, but not as strong as Donald. Who is strongest and who is second strongest? Although I'm sure that you've never met Joel, Donald, Richard, and Bill, I'm also sure you could answer this question...
eBook - ePubThinking Critically About Critical Thinking
A Workbook to Accompany Halpern's Thought & Knowledge
- Diane F. Halpern, Heidi R. Riggio(Authors)
- 2013(Publication Date)
- Routledge(Publisher)
...The skills developed in this chapter are most similar to those used in college-level mathematics as they provide a single correct answer and require careful consideration and execution of all steps in the process. Deductive Reasoning is based on the assumptions that if certain information is true, then there are conclusions that must also be true. Both spatial and verbal strategies are used to help readers practice both modes of thinking. Common biases and errors in Deductive Reasoning are introduced. Many of these are discussed in several other chapters using different perspectives. The use of Deductive Reasoning skills in real world contexts is also highlighted so that readers can recognize when Deductive Reasoning skills are needed and when they are being persuaded with Deductive Reasoning techniques. Name: Date: Course/Section: Journal Entries Record your thoughts as you reflect on the material in the Deductive Reasoning chapter. The purpose of this journal is to let you step back and reflect on the material that you are learning. It is a place to record your discussions with yourself. Write about topics that are unclear or seem particularly useful to you. This is the time to make your own connections between and within chapters, from class, and the real-life that happens out of class. Use the next page, for your second entry. _______________________________ Date of first entry Name: Date: Course/Section: Journal - Second Entry for Chapter 4: _______________________________ Date of second entry Review of Deductive Reasoning Skills Category description: The skills presented in this chapter are used to determine if a conclusion is valid--that it, it must be true if the premises are true. These skills are used in many contexts including law, medicine, financial projections, and the science. Skill Description Example of Use a...
