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17.09 Questions, Answers, and Evidence Scientific Method and Research Ethics 17.09 Questions, Answers, and Evidence Dr. C. D. McCoy Plan for Part 1: Deduction 1. Logic, Arguments, and Inference 1. Questions and Answers 2. Truth, Validity, and Soundness 3. Inference Rules and Formal Fallacies 4. Hypothetico-Deductivism What is Logic? • “What is logic?” is a question that philosophers and logicians have debated for millennia. We won’t attempt to give a complete answer to it here… Historical Perspectives on Logic • Aristotle held that logic is the instrument by which we come to know: important to coming to know is understanding the nature of things, especially the necessity of things. • Immanuel Kant held that logic is a “canon of reasoning”: a catalog of the correct forms of judgment. Logic is formal; it abstracts completely from semantic content. • Gottlob Frege held that logic is a “body of truths”: it has its own unique subject matter, which includes concepts and objects (like numbers!). • Many authors also hold that “generality” and “truth” are central notions in logic. Reasoning and Argument • Reasoning is a cognitive process that • Begins from facts, assumptions, principles, examples, instances, etc. • And ends with a solution, a conclusion, a judgment, a decision, etc. • In logic we call an item of reasoning an argument. • An argument consists in a series of statements, called the premisses, and a single statement, called the conclusion. Assessing Arguments • One reason to study logic is to understand why some instances of reasoning are good and why some are bad. • Logic neatly separates the ways arguments can be good or bad into two kinds: • The truth of the premisses. • What relation the premisses have to the conclusion. • Logic largely concerns the analysis of the latter (although the nature of truth is important to logic too!). Logic cannot tell you whether some premisses are true or not (unless they are logically true). Examples • I was very hungry last night. • Therefore I went to sleep. • It is either Tuesday or Wednesday. • It is not Tuesday. • It is not Wednesday. • Therefore… • In the first example there is no relation between premiss and conclusion. • In the second example the premisses are inconsistent. Example • “If Congress has the authority to compel us to purchase health insurance, it must also have the authority to force us to buy products and services related to transportation, food, housing and the like.” (J. Norman, Huffington Post, 11 Feb 2011) • Congress does have the authority to compel us to purchase health insurance. • Therefore… • This argument can easily be made valid. But it commits an informal fallacy: it is a “slippery slope” argument. Formal Logic We will focus on formal deductive logic in this part of the lecture. The “right” relation between premisses and conclusion in deductive logic is called validity. One way an argument can be logically defective, then, is to be invalid: the conclusion does not follow deductively from the premisses. Assertions, Statements, Propositions • In an argument, the premisses and conclusion are considered assertions. • An assertion is represented as a declarative statement uttered with “assertoric force”: the truth of the statement is being asserted in the utterance. • It is common to think of the referent of a declarative sentence as an abstract proposition. A proposition is something that possesses a truth value, that is, it can be true or false. • Thus we say that a statement is true if it refers to a true proposition, and we say that a statement is false if it refers to a false proposition. If it is not clear to which proposition a statement refers, then that does not mean that the statement is neither true nor false, rather it is merely ambiguous. Examples • “We are, at the moment, in Vienna.” -> [We are, at the moment, in Vienna] -> [False] • “We are, at the moment, in Stockholm.” -> [We are, at the moment, in Stockholm] -> [True] • “Ok.” -> [?] • “Where are we at the moment?” -> [?] Interlude: Questions • Interrogative sentences are normally not part of deductive logic. But some philosophers and logicians have developed a logic of questions, which is called “erotetic logic”. • If we are committed to representing sentences propositionally, how can we represent an interrogative? • “Where are we at the moment?” • -> “We are in _______ at the moment.” • -> [We are in Amsterdam, Bangkok, Copenhagen, …, Stockholm, …at the moment]. Validity and Logical Consequence • An argument, recall, is a structured set of assertions: a collection of assertions called the premisses and an assertion called the conclusion. • An argument is valid just when it is not possible for the conclusion to be false when the premisses are true. • It is worth remarking that arguments may be valid or invalid; there are neither true arguments nor false arguments. • If an argument is valid, then we say that the conclusion is a logical consequence of the premisses, or that the premisses entail the conclusion. • To show that an argument is valid one must prove it; to show that it is invalid one must find a counterexample. Examples • If it is raining, then the sidewalks will be wet. • It is in fact raining. • Therefore the sidewalks will be wet. • I always become hungry at noon. • I am in fact hungry at the moment. • It must be noon. • I was really hungry last night. • I went to sleep. Why is Logic Formal and General? • All humans are mortal. • All Greeks are humans. • All Greeks are mortal. • All felines are mammals. • All tigers are felines. • All tigers are mammals. • All Y are Z. • All X are Y. • All X are Z. Why is Logic Formal and General? • As the preceding examples illustrate, the precise meaning of a sentence does not matter for determining the validity or invalidity of an argument. • Thus it makes sense to abstract away from the “non-logical” content of sentences when assessing its logical properties. • One does this by replacing the content by symbols which stand in for the abstracted content. This is what makes logic formal (and general). • Example: • “If it is raining, then the sidewalks will get wet.” : R -> W • “It is raining.” : R • “Therefore the sidewalks will get wet.” : W • This pattern of inference is called modus ponens. Valid Inference Rules • Modus ponens • Disjunctive • Conjunction • A -> B Syllogism Simplification • A • A v B • A & B • B • ¬A • A • B • Modus Tollens • Disjunction • A -> B • Hypothetical Introduction • ¬B Syllogism • A • ¬A • A -> B • A v B • B -> C • A -> C Fallacious Inferences • Affirming the • Affirming a Consequent Disjunct • A -> B • A v B • B • A • A • ¬B • Denying the • Denying a Antecedent Conjunct • A -> B • A & B • ¬A • ¬B • ¬B • ¬A The Deductive Method of Testing • A scientist conceives of or invents a theory. • The scientist then determines various consequences of the theory through logical deduction. • He or she assesses these consequences for consistency, compatibility with other theories, etc. • The scientist then determines certain singular statements (predictions) which are at odds with all known theories. • These statements are then put to empirical test, the results of which generate new singular statements that are either consistent with the prediction (verification) or inconsistent with it (falsification). Hypothetico-Deductivism • Let H be some hypothesis. • We determine that H entails some prediction P by logical deduction. • We gather some evidence E. • If P & E are consistent, then H is verified. • If P & E are inconsistent, then H is falsified. Limitations of Deductivism • Suppose we have two hypotheses H1 and H2 that entail P, and P and evidence E are consistent. Are they equally corroborated by the evidence E? • Suppose hypothesis H is corroborated. Why make use of it in practical contexts? There are presumably any number of other hypotheses that would be corroborated by the same evidence, but which make divergent predictions. • Why choose a hypothesis H1 for testing? Does deductivism presume that the method of hypothesis selection is no better than guessing? Plan for Part 1: Deduction 1. Logic, Arguments, and Inference 1. Questions and Answers 2. Truth, Validity, and Soundness 3. Inference Rules and Formal Fallacies 4. Hypothetico-Deductivism Plan for Part 2: Induction 1. Inductive Logic 1. Kinds of Inductive Inference 2. Evidence and Degree of Support 2. The Problem of Induction 1. Hume on the Problem of Induction 2. Goodman on the New Riddle of Induction 3. Inductive Probability 4. Bayesianism 5. Hypothesis Confirmation Inductive Logic • Deductive logic is concerned with provability and truth- preserving inference. A deductively valid argument is one where the conclusion necessarily follows from the premisses. A deductively valid argument with true premisses (a sound argument) therefore has a true conclusion. • Inductive logic is concerned with certain arguments which are deductively invalid; these arguments are thought to be good arguments, even though the conclusion does not follow from the premisses. Such arguments are also called ampliative, for the conclusion goes beyond what is contained in the premisses. Examples • The sun has risen every day in human memory. • Therefore, the sun will (probably) rise again tomorrow. • I always become hungry at noon. • I am in fact hungry at the moment. • Therefore it is probably be noon. • A random survey of 34,525 Americans indicated that 18% of them smoke. • 18% of Americans smoke. Kinds of Inductive Argument • Inductive generalization or enumerative induction: An inference from some set of individuals to a larger set of individuals. • I observed numerous black ravens, therefore all ravens must be black. • Statistical syllogism: An inference from a larger set of individuals to a smaller set of individuals. • 18% of Americans smoke, so probably 18% of Chicagoans smoke. • Analogical argument: An inference from one set of individuals to another based on a similarity between them. • John and Ann both like comic books. John also likes superhero movies, so Ann probably does too.
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