Study Guides Chapter 1 - Basic Training Argument: A group of propositions is an argument when one or more of the propositions in the group is/are used to give evidence (or if you like, reasons, or grounds) in support of one of the other propositions in the group that is receiving support. Conclusion: A proposition that is the part of an argument that is being supported by the premises. Deductive Argument: An argument whose premises are claimed to guarantee the truth of its conclusion. Deductive arguments can be either valid or invalid and either sound or unsound. Inductive Argument: An argument whose premises are claimed to increase the likelihood that its conclusion is true. Inductive arguments can be weak or strong. Logic: The basic principles and techniques that are used in distinguishing correct (good) reasoning from incorrect (bad) reasoning. Premise: A proposition that is part of an argument and is used to offer support for the conclusion. Proposition: An assertion that is either true or false. (Synonymous with Statement. ) Sentence: A unit of connected speech or writing that can have various uses, including that of asserting propositions. Sound Argument: A valid deductive argument all of whose premises are true. Valid Argument: A deductive argument whose conclusion would have to be true if its premises were true. 1
Chapter 2 - Recognizing and Analyzing Arguments Language Uses Expressive: An expressive use of language is one in which the speaker reveals feeling, attitudes, and such, but does not assert any matter of fact. Directive: A directive use of language is one in which the speaker commands or encourages listeners to perform some action. Informative: An informative use of language is one in which the speaker states a matter of fact. Ritual: A ritual use of language is one in which words are spoken as part of a formal or informal ceremony. Performative: A performative use of language is one in which the speaker accomplishes something by the speaking. Kinds of Arguments Deduction: A deduction is an argument in which the premises are intended to guarantee the truth of the conclusion. Induction: An induction is an argument whose premises are intended to establish that the conclusion is likely true. Sorites: A chain of arguments in which the conclusions of earlier arguments serve as the premises of later arguments. Enthymeme: An argument with a missing premise or conclusion. Groups of propositions often mistaken for arguments Descriptions: A description consists of one or more propositions that is (are) intended merely to convey information about something. Explanations: An explanation is intended to account for why something is true that is already known to be true. An explanation is distinguished from an argument in this respect since an argument is intended to establish that something that is not known to be true is in fact true. Good Arguments Hypothetical Syllogism: If A implies B and if B implies C, then A must imply C. Modus Ponens: If A implies B and A is true, then B must be true. Modus Tollens: If A implies B and B is false, then A must be false. Disjunctive Syllogism: If either A or B is true and A is false, then B must be true; or if either A or B is true and B is false, then A must be true. Bad Arguments The Fallacy of Affirming the Consequent: If A implies B, and B is true, it does not follow that A must be true. The Fallacy of Denying the Antecedent: If A implies B and A is false, it does not follow that B must be false. 2
Chapter 3 Disputes and Definitions Disputes I: Attitudes and Beliefs Attitude: A difference in attitude is one in which the parties do not disagree about matters of fact, but differ in their feelings or attitudes. Belief: A disagreement in belief in one in which one person takes something to be true and the other denies it. Both: Some disputes involve both attitudes and beliefs. Neither: Some apparent disputes are neither actual disagreements nor actual differences. Disputes II: Genuine and Verbal Genuine: Dispute remains even when terms are clearly defined. Verbal: Dispute is a function of terms that are not clearly defined and dissolves when terms are clarified. Non-Lexical Definitions Emotive: The assignment of meaning to a term that is designed to arouse emotions, pro or con. Precising: The assignment of meaning to a term designed to clarify its application and to eliminate vagueness. Stipulative: The assignment of a meaning to a term by mutual agreement. Such definitions are neither true nor false. Theoretical: The assignment of meaning to a term on the basis of a theoretical framework. Lexical Definitions Definitions with synonyms: The assignment of meaning to a term by transferring to it the meaning of another term presumed to be more readily known and roughly equivalent to it. Operations definitions: The assignment of meaning to a term on the basis of some observational effect that the term is supposed to produce. Genus/Species: The assignment of meaning to a general term on the basis of its affiliation with a general category and on the basis of its specific difference from other members of that category. Lexical Definition Defects Accidental: A definition is accidental if it fails to state the essential characteristics of the term being defined. Circular: A definition is circular if the term being defined is used in the definition itself. Emotive: An emotive definition is one which attempts to arouse emotions rather than clarify the concept. Figurative: A figurative definition is one which uses a metaphor or image rather than making an attempt to state the essential characteristics of the concept being defined. Negative: A negative definition is one which says what a term does not mean rather than what it does mean. Obscure: An obscure definition is one which uses language even less well known than the term being defined. Too Broad: A definition is too broad if it includes more than actually falls within the domain of the term being defined. Too Narrow: A definition is too narrow if it excludes some things that are included within the domain of the term being defined. Too Broad and Too Narrow: A definition can be both too exclusive and too inclusive. 3
Chapter 4 - Identifying Fallacies 1. Ignorance: Committed when an attempt is made to justify a conclusion on the basis of a lack of evidence. 2. Pity: Committed when an attempt is made to justify a conclusion on the basis of pity. 3. Desire: Committed when an attempt is made to justify a conclusion on the basis of a desire that it is true. 4. Authority: Committed when an attempt is made to justify a conclusion by an inappropriate appeal to authority. 5. Begging the Question: Committed when the conclusion is assumed as a premise 6. Character-Abusive: Discrediting an argument on the basis of a person s questionable character. 7. Character-Circumstantial: Discrediting an argument on the basis of a person s obvious bias. 8. Complex Question: Assumes an answer to an unasked question embedded in the question asked. 9. False Cause: Assumes that one event is the cause of another when is it actually not the cause. 10. Popularity: An appeal to popular opinion as a supporting reason. 11. Force: An appeal to force as a supporting reason. 12. Hasty Generalization: (Converse Accident) Generalization on the basis of exceptions 13. Accident: Use of a general rule that does not fit an exceptional case. 14. Irrelevant Conclusion: Jumping to conclusions on the basis of unwarranted assumptions. 15. Equivocation: Invalid conclusion based on an ambiguous use of a term. 16. Amphiboly: Invalid conclusion based on a faulty grammatical construction. 17. Accent: Invalid conclusion based on misleading emphasis or de-emphasis of words or phrases. 18. Composition: An inference that a whole or collection has the same properties its parts or members have 19. Division: An inference that the parts of a whole or members of a collection have the same properties as the whole. 4
Chapter 5 - Categorical Propositions Standard-Form Categorical Propositions: E Proposition: No S is P Universal Quantity; Negative Quality A Proposition: All S is P Universal Quantity; Affirmative Quality I Proposition: Some S is P Particular Quantity; Affirmative Quality O Proposition: Some S is not P Particular Quantity; Negative Quality Distribution A: S-term distributed; P-term undistributed (DU) E: both terms distributed (DD) I: both terms undistributed (UU) O: S-term undistributed; P-term distributed (UD) Immediate Inferences A: All S is P: Some P are S (valid by limit) E: No S is P: No P are S (valid) I: Some S is P: Some P are S (valid) O: Some S is not P (not valid) Obversion: A: All S is P: No S are non-p (valid) E: No S is P: All S are non-p (valid) I: Some S is P: Some S are not non-p (valid) O: Some S is not P: Some S are non-p (valid) Contraposition: A: All S is P: All non-p are non-s (valid) E: No S is P: Some non-p are not non- S (valid by limit) I: Some S is P: (not valid) O: Some S is not P: Some non-p are not non-s (valid) 5
Chapter 6 - Categorical Arguments Categorical Fallacies: 1. Illicit Terms: A valid standard-form categorical syllogism must contain exactly, and only, three class terms. 2. Illicit Minor: In a valid standard-form categorical syllogism, if the minor term is distributed in the conclusion, it must be distributed in the minor premise. 3. Illicit Major: In a valid standard-form categorical syllogism, if the major term is distributed in the conclusion, it must be distributed in the major premise. 4. Illicit Middle: In a valid standard-form categorical syllogism, the middle term must be distributed in at least one premise. 5. Illicit Quality: In a valid standard-form categorical syllogism, the conclusion must be negative if there is a negative premise. 6. The Existential Fallacy: In a valid standard-form categorical syllogism, the conclusion cannot be particular if both premises are universal. 7. Two Negatives: A valid standard-form categorical syllogism can't have two negative premises. Venn Diagrams Test for Validity Step One: Draw three interlocking circles and label each one with an uppercase letter designating one of the classes in the argument being tested. Step Two: Diagram the major and the minor premises but never diagram the conclusion. It does not matter which premise you diagram first, unless one of the premises is a particular proposition and the other one is a universal proposition. In this case, always diagram the universal proposition first. Step Three: Examine the diagram to see if it contains a diagram of the conclusion of the syllogism you are testing. If it does, then the argument is valid; if it doesn't, the argument is invalid. Valid Forms: Figure 1 AAA EAE AII EIO; Figure 2 EAE AEE EIO AOO Figure 3 IAI AII OAO EIO Figure 4 AEE IAI EIO Traditionally Valid Forms Figure 1: AAI; EAO Figure 2: AEO; EAO Figure 3: AAI; EAO Figure 4: AEO; EAO; AAI 6
Chapter 7 - Sentential Propositions Kinds of Sentential Propositions: The Conjunction: p and q. The conjunction is symbolized as p q. The parts of the conjunction are called conjuncts. The conjunction is true if and only if both of its conjuncts are true. The Disjunction: either p or q. The disjunction is symbolized as p v q. The parts of the disjunction are called disjuncts. The disjunction is true if and only if either or both of its disjuncts is (are) true. The Negation: not p. The negation is symbolized as ~ p. The negation is true if and only if what it negates is false and false if and only if what it negates is true. The Conditional: if p then q. The conditional is symbolized as p q. It has two parts: the if part is the antecedent and the then part is the consequent. The conditional is false only if the antecedent is true and the consequent is false. The Biconditional: p if and only if q. The biconditional is symbolized as p q. The biconditional is true if and only if p and q have the same truth-value and false if and only if p and q have different truth-values. Chapter 8 - Sentential Truth Tables and Argument Forms Tautology: A proposition that is true under every possible interpretation. Contradiction: A proposition that is false under every possible interpretation. Contingent Proposition: A proposition that is true under some interpretations and false under others. Rules of Inference (So far) 1. Modus Ponens (MP) p q; p; Therefore q 2. Modus Tollens (MT) p q; q; Therefore p 3. Hypothetical Syllogism (HS) p q; q r; Therefore p r 4. Disjunctive Syllogism (DS (p v q; q, Therefore p ( p v q; p; Therefore, q 7
Chapter 9- Sentential Proofs Rules of Inference 1. Modus Ponens (MP) p q; p; Therefore q 2. Modus Tollens (MT) p q; q; Therefore p 3. Hypothetical Syllogism (HS) p q; q r; Therefore p r 4. Constructive Dilemma (CD) (p q) (r s); p v r; Therefore q v s 5. Disjunctive Syllogism (DS (p v q; q, Therefore p ( p v q; p; Therefore, q) 6. Simplification (Simp) p q Therefore p (p q Therefore q) 7. Addition (Add) p Therefore p v q 8. Conjunction (Conj) p; q Therefore p q 9. Absorption (Abs) p q, Therefore p (p q) Standard Logical Equivalences 1. DeMorgan's (DM) (p q) ( p v q); (p v q) ( p q) 2. Commutation (Comm) (p q) (q p); (p v q) (q v p) 3. Association (Assn) [p v (q v r)] [(p v q) v r]; [p (q r)] [(p q) r] 4. Distribution (Dist) [p (q v r)] [(p q) v (p r)]; [p v (q r)] [(p v q) (p v r)] 5. Double Negation (DN) p p 6. Transposition (Trans) (p q) ( q p) 7. Material Implication (Imp) (p q) ( p v q) 8. Exportation (Exp) [(p q) r] [p (q r)] 9. Material Equivalence (EQ) (p q) [(p q) (q p)]; [(p q) v ( p q)] 10. Tautology (Taut) p (p v p); p (p p) Assumption Rules: You may assume any proposition, at any time, in any deduction, as long as the assumption is discharged according to the rules IP or CP before concluding the deduction. Assumed Premise (AP): The abbreviation for the justification when the Assumption Rule is employed Conditional Proof (CP): Allows you to infer a conditional proposition from an assumed premise. The conditional proposition inferred must have the assumed premise as its antecedent and the immediately preceding line as its consequent. Step I: Assume a premise (AP). Step II: Make deductions from this premise. Step III: Discharge the Assumed Premise with a conditional proposition (AP q) Indirect Proof (IP): Allows you to infer the negation of an assumed premise only if a contradiction is derived from the assumed premise. Step I: Assume a premise (AP). Step II: Derive a contradiction (p ~p). Step III: Discharge the Assumed Premise with a negation of the Assumed Premise (~AP) 8
Chapter 10 - Predicate Logic Quantification Rules of Inference 1. Universal Instantiation (UI) (x)(gx. Ga) (x)gx Gu ( a is an individual constant and u is a free variable) 2. Universal Generalization (UG) Gu (x)gx ( u is a free variable. Note this rule does not allow the inference: Ga (x) Gx, where a is an individual constant) 3. Existential Generalization (EG) Ga ( x)gx Gu ( x)gx a is an individual constant and u is a free variable 4. Existential Instantiation (EI) ( x)gx Ga a is an individual constant that has not occurred previously in the deduction; note that it is not a valid application of EI to infer ( x)gx Gu, where u is a free variable Quantifier Negation (QN) ~( x)~gx (x)gx ~(x)~gx ( x)gx ~( x)gx (x)~gx ~(x)gx ( x)~gx Translating A, E, I, O Categorical Propositions into Predicate Logic A Proposition: All S are P: (x) (Sx Px) E Proposition: No S are P: (x)(sx ~Px) I Proposition: Some S are P: ( x)(sx Px) O Proposition: Some S are not P: ( x)(sx ~Px) 9