© 2016 by Pearson Education, Inc. Chapter 8 Valid Inferences © 2016 by Pearson Education, Inc.

Learning Outcomes Evaluate the logical strength of inferences presented to imply or entail that their conclusions must certainly be true if we take their premises to be true Recognize reasoning fallacies masquerading as valid inferences Learning Outcomes The chapter first evaluates the logical strength of inferences presented to imply or entail that their conclusions must certainly be true if one takes their premises to be true. It concludes by recognizing reasoning fallacies masquerading as valid inferences.

Chapter Opening Video Chapter Opening Video The video explains the how valid inferences are drawn. It shows how just a few valid inferences can be drawn given the limited constraints of the context.

Structure of the Reasoning Inferences offered as certain Reasoning with declarative statements Reasoning about classes of objects Reasoning about relationships Structure of the Reasoning Valid: Describes an argument or inference. Truth of the premises implies that conclusion is true. False conclusion implies that one or more of the premises is false. Inferences offered as certain People make arguments believing that truth of the premises guarantees the truth of the conclusion. Reasoning with declarative statements Include valid argument templates that derive their structures from the way simple statements interact grammatically when one uses logical prepositions or adverbs. Reasoning about classes of objects Words like some and all are used to express one’s ideas about how individual objects and groups of objects relate. Reasoning about relationships Terms that describe relationships help make valid inferences about the objects to which the terms apply.

Structure of the Organ Transplant Scenario The organ transplant scenario is represented in the form of a map.

Inferences Offered as Certain People make arguments believing that truth of the premises guarantees truth of the conclusion Thoughtful and fair-minded interpretation is essential Context and intention of the argument maker are to be considered Inferences Offered as Certain People make arguments believing that truth of the premises guarantees truth of the conclusion. Aristarchus deduced that the Sun was much larger than the Earth. Carl Sagan explains that there was no other inference Aristarchus could draw except that the Sun was large and far away. Thoughtful and fair-minded interpretation is essential. People do not always speak with logical precision. Context and the intention of the argument maker should be considered.

Inferences Offered as Certain Laws of nature or logic cannot be suspended Logical strength of an inference can be evaluated by a counterexample Inferences Offered as Certain Laws of nature or logic cannot be suspended. Strong critical thinking requires people not to attribute more than the argument maker intends. Logical strength of an inference can be evaluated by a counterexample. Argument is not valid if all the premises are true but the conclusion is false in a counterexample.

Reasoning with Declarative Statements Denying the consequent Premise #1 - If A, then B Premise #2 - Not B Conclusion - Therefore, not A Affirming the antecedent Premise #2 - A Conclusion - Therefore, B Reasoning with Declarative Statements Denying the consequent Premise #1 - If A, then B. Premise #2 - Not B. Conclusion - Therefore, not A. Logical strength is not the only consideration when evaluating real life arguments. Facts and logic are different things. Affirming the antecedent - Argument template relies on the meaning and grammatical power of “if then” expressions and the second premise confirms that the “if” portion is true. Premise #2 – A. Conclusion - Therefore, B.

Reasoning with Declarative Statements Disjunctive syllogism Premise #1 - Either A or B Premise #2 - Not A Conclusion - Therefore, B Reasoning with Declarative Statements Disjunctive syllogism - People presented with various alternatives logically reduce options. Premise #1 - Either A or B. Premise #2 - Not A. Conclusion - Therefore, B.

Grammatically Equivalent Structures Certain grammatically equivalent structures provide multiple ways to express the same thing.

Simulation Simulation This simulation helps practice with “neither, unless and only” arguments.

Reasoning About Classes of Objects Applying a generalization Premise #1 - Every member of group F is a member of group G Premise #2 - Individual object X is a member of F Conclusion - Object X is a member of G Reasoning About Classes of Objects Applying a generalization - When a generalization asserts that a given characteristic to each of the members of a class of objects, it is logical to conclude that a given individual or subgroup of individuals that are members of the class possess that characteristic. Premise #1 - Every member of group F is a member of group G. Premise #2 - Individual object X is a member of F. Conclusion - Object X is a member of G.

Reasoning About Classes of Objects Applying an exception Premise #1 - Every member of group F is a member of group G Premise #2 - Object X is not a member of G Conclusion - Object X is not a member of F Power of only Only has the ability to change the meaning of a sentence depending on where it is placed Reasoning About Classes of Objects Applying an exception - If every member of a given class of objects has a certain characteristic, and one or more specific objects do not have that characteristic, it is logical to infer that specific objects are not members of the class. Premise #1 - Every member of group F is a member of group G. Premise #2 - Object X is not a member of G. Conclusion - Object X is not a member of F. Power of only Only has the ability to change the meaning of a sentence depending on where it is placed.

Simulation Simulation This simulation helps practice with arguments pertaining to classes and objects.

Power of Only Power of Only Meanings change depending on where only is positioned in the example.

Power of Only Power of Only Meanings change depending on where only is positioned in the example.

Reasoning About Relationships Terms that describe relationships help make valid inferences about the objects to which the terms apply Transitivity relationship If x has a transitive relationship to y, and y has the same transitive relationship to z x has the same transitive relationship to z Reasoning About Relationships Terms that describe relationships help make valid inferences about the objects to which the terms apply. Example - John is Susan’s younger brother. So, they must have the same mother or the same father. Understanding of the logical implications of relational terms is part of one’s comprehension of language. Transitivity relationship If x has a transitive relationship to y, and y has the same transitive relationship to z. x has the same transitive relationship to z. Example - David is Stanley’s neighbor. So, Stanley is David’s neighbor.

Reasoning About Relationships Reflexivity relationship If x has a reflexive relationship to y, then y has the same reflexive relationship to x Identity relationship If x is y, then y is x Reasoning About Relationships Reflexivity relationship If x has a reflexive relationship to y, then y has the same reflexive relationship to x. Example - Leonardo DiCaprio played Jordan Belfort in the 2013 film The Wolf of Wall Street. Actor who played Jordan Belfort in that film was nominated for an Oscar. So, Leonardo DiCaprio received an Oscar nomination for his performance in that film. Identity relationship If x is y, then y is x.

Discussion Question “Eliminate all other factors, and the one which remains must be the truth” - Sherlock Holmes Is it possible to follow Sherlock Holmes’ advice on how to figure out the one right answer to a problem? Discussion Question Sir Arthur Conan Doyle’s character calls the arguments he put forth with certainty deductions. "Eliminate all other factors, and the one which remains must be the truth“ - Sherlock Holmes Is it possible to follow Sherlock Holmes’ advice on how to figure out the one right answer to a problem?

Fallacies Masquerading as Valid Arguments Fallacies when reasoning with declarative statements Fallacies when reasoning about classes of objects Fallacies of false reference Personal infallibility?—We don't think so Fallacies Masquerading as Valid Arguments - Precision of thought and expression helps avoid mistakes when making or evaluating arguments offered as valid inferences. A counterexample has the ability to: Reveal the illogical structure. Expose the fallacy. Suppress the argument’s apparent persuasiveness. Fallacies when reasoning with declarative statements Familiar fallacies can mimic logically correct declarative statements. Fallacies when reasoning about classes of objects False classification, and fallacies of composition and division are errors committed while reasoning about classes of objects. Fallacies of false reference Occur when reasoning about relationships like identity, reflexivity, or transitivity. Personal infallibility? —We don't think so A healthy sense of skepticism, the study of human history, and life experience suggests that the probability of personal infallibility approximates zero.

Fallacies When Reasoning with Declarative Statements Affirming the consequent Premise #1 - If A, then B Premise #2 - B Conclusion - Therefore, A Not logical to believe that A must be true because B is true Fallacies When Reasoning with Declarative Statements Affirming the consequent Premise #1 - If A, then B Premise #2 - B Conclusion - Therefore, A A may not be the only condition that brings about B. Not logical to believe that A must be true because B is true.

Fallacies When Reasoning with Declarative Statements Denying the antecedent Premise #1 - If A, then B Premise #2 - Not A Conclusion - Therefore not B Not logical to think that B cannot happen because A does not happen Fallacies When Reasoning with Declarative Statements Denying the antecedent Premise #1 - If A, then B Premise #2 - Not A Conclusion - Therefore not B A may not be the only condition that brings about B. Not logical to think that B cannot happen because A does not happen.

Fallacies When Reasoning About Classes of Objects False classification Example - Criminals enjoy mafia movies and Cassandra enjoys mafia movies Does not apply that Cassandra is a criminal Fallacies When Reasoning About Classes of Objects False classification Example - Criminals enjoy mafia movies and Cassandra enjoys mafia movies. Does not apply that Cassandra is a criminal. A feature or attribute can be true of two groups without requiring that one group must be classified as part of the other group.

Fallacies When Reasoning About Classes of Objects Fallacies of composition A group and each of its members may not have the same attributes Fallacies of division Attribute to each individual member of a group a characteristic that is true of the group as a whole Fallacies When Reasoning About Classes of Objects Attribute that applies to the parts may not apply to the whole, or vice versa. Fallacies of composition A group and each of its members may not have the same attributes. Example - It is in each person’s financial interest to cheat a little on his or her income tax return. So, it is financially good for the nation if people cheat on their taxes. Fallacies of division Attribute to each individual member of a group a characteristic that is true of the group as a whole. Example - The United States of America has the right to enter into treaties and to declare war. Therefore each of the 50 states has the right individually to declare war or to enter into treaties.

Fallacies of False Reference Occur when reasoning about relationships Identity, reflexivity, or transitivity Comparable reasoning mistake Occurs when people are not aware that same object, person, or event can be identified using multiple descriptions Fallacies of False Reference Occur when reasoning about relationships. Identity, reflexivity, or transitivity. The ambiguity of certain expressions can be the source of a mistaken inference. Comparable reasoning mistake Occurs when people are not aware that same object, person, or event can be identified using multiple descriptions. Knowing, believing, wanting, or intending something when it is described or named in one way does not imply that the person knows, believes, wants, or intends that same thing as described or named in another way. Example - Tyler at age 10 has often told his Mom that in college he wants to learn how big buildings and bridges are built. These are subjects addressed in civil engineering. Therefore, Tyler has announced that he plans to major in civil engineering in college. Tyler is 10 years old. He has no idea what civil engineering is.

Personal Infallibility? The human species is capable of inferring with certainty the implications of rules, laws, principles, and regulations Capacity for certainty drives people toward wrongheaded conclusions Probability of personal infallibility approximates to zero Personal Infallibility? The human species is capable of inferring with certainty the implications of rules, laws, principles, and regulations. Power to reason with certainty can lead to ominous results. Capacity for certainty drives people toward wrongheaded conclusions. Probability of personal infallibility approximates to zero. Not all beliefs, values, assumptions, and interpretations are true.

Sketchnote Video Sketchnote Video The video summarizes how valid inferences can be made using the limited number of constraints and rules of the context.