1 Scientific Method and Research Ethics Questions, Answers, and Evidence Dr. C. D. McCoy
2 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
3 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
4 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.
5 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.
6 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).
7 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.
8 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.
9 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.
10 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.
11 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? -> [?]
12 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].
13 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.
14 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.
15 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.
16 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.
17 Valid Inference Rules Modus ponens A -> B A B Modus Tollens A -> B B A Disjunctive Syllogism A v B A B Hypothetical Syllogism A -> B B -> C A -> C Conjunction Simplification A & B A Disjunction Introduction A A v B
18 Fallacious Inferences Affirming the Consequent A -> B B A Affirming a Disjunct A v B A B Denying the Antecedent A -> B A B Denying a Conjunct A & B B A
19 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).
20 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.
21 Limitations of Deductivism Suppose we have two hypotheses H 1 and H 2 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 H 1 for testing? Does deductivism presume that the method of hypothesis selection is no better than guessing?
22 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
23 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
24 Inductive Logic Deductive logic is concerned with provability and truthpreserving 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.
25 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.
26 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.
27 Inductive Support In valid deductive arguments, the support for the conclusion is water-tight : the conclusion cannot be false if the premisses are true. In strong inductive arguments, the support for the conclusion is less than water-tight, but still good enough to consider the conclusion reasonable. A weak inductive argument is one that falls short. This leads to the concept of inductive support: how well the premisses support the conclusion is a matter of degree, one which can be assessed.
28 The Problem of Induction Scottish philosopher David Hume is famous for, among other things, his version of the problem of induction. Consider some inductive inference, for example that the sun will rise tomorrow. Hume argues that it is conceivable, it is possible, that sun will not rise tomorrow. What justifies the inference that it will? It cannot be that such an inference has always worked in the past, for that is an inductive inference and begs the question. It cannot some other fact, such as the uniformity of nature, for that fact is also obtained by an inductive inference, hence the justification offered is circular.
29 The New Riddle of Induction The problem of induction continues to be discussed. The most well-known response to it is by Nelson Goodman. Goodman wonders why induction must be justified in the way Hume maintains. What, after all, justifies deduction? Any justification one offers will be just as circular as the ones Hume considers for induction! He argues that rules of induction (and deduction) are the outcome of a process of reflective equilibrium: certain inferences seem good, some bad. Rules are suggested to capture the distinction. Evidence may lead the rules to be revised, but equally the rules may leads us to reconsider the original division of inferences into good and bad. Eventually a conceptual equilibrium is reached.
30 The New Riddle of Induction The problem of induction continues to be discussed. The most well-known response to it is by Nelson Goodman. Goodman thus rejects the need for a justification of induction. He introduces a new problem, however, the new riddle of induction. Consider the predicate grue. An object is grue just when it is observed before time t and is green or it is blue. We observe a number of emeralds, which are all green. But they are also grue! Why should we expect that the emeralds we observe after time t are green and not grue? Goodman thus demands an explanation for why some predicates are projectible to the future and others are not.
31 Inductive Probability Progress in developing an inductive logic came from distancing it from deductive logic conceptually. Recall that a crucial difference between deduction and induction is in terms of degree of support. It is quite natural to adopt a probabilistic way of describing this inductive support. A strong inductive argument warrants the conclusion to a high degree: the conclusion has a high probability of being true. A weak inductive argument provides little warrant for the conclusion: the conclusion is unlikely. Rudolf Carnap, logical empiricist and one of the most influential philosophers of science of the 20 th century.
32 Objective and Subjective Probabilities There are two ways of thinking of probability in this context. They are objective: inductive support is simply a relation between propositions, just as validity in deductive arguments is a relation between propositions. They are subjective: inductive support depends partly on one s degree of belief in the propositions involved; how one s degrees of beliefs should change based on new evidence, however, is entirely objective.
33 Bayesianism The Bayesian approach is founded on two ideas: An agent s degrees of belief can be represented formally as probabilities. A rational agent updates his or her degrees of belief in light of evidence according to Bayes s formula. Consider a simple proposition p. If I fully believe p, then my degree of belief (or credence) is 1: cr(p)=1. Bayesians usually hold that only tautologies (propositions that are logically true) can have a credence of 1. So let us suppose that my credence in p is some non-trivial probability, for example 0,7. Then my credence in p is 0,3. Now suppose I obtain some evidence e that is relevant to p. My credence in this evidence, let s suppose, is high: cr(e)=0,9.
34 Bayesianism How should I revise my credence in p in light of this evidence e? The Bayesian updating rule is the Bayes rule: cr p e = cr(e p) cr(e) cr p. posterior probability = likelihood evidence prior probability My new degree of belief, after updating, is set equal to the posterior probability: the probability of the proposition conditional on the evidence. Let s suppose that the likelihood of e given p is fairly low: cr(e p)=0,1. Then my updated credence is 0,78; this unlikely evidence should increase my credence in p from 0,7 to 0,78.
35 Questions about Bayesianism Where do the prior probabilities come from? From previous conditionalizations! But how far back does one go? And where do the original priors come from? For practical purposes it is enough to recognize that we do have degrees of belief and we can approximate them with numbers. Doesn t the intrusion of subjectivity undermine Bayesianism? No. The Bayesian approach separates the subjectivity of belief from the objectivity of evidence s impact on belief. It cannot tell you what to believe, but it can tell you how your beliefs should change in light of evidence. Dutch book arguments aim to show that the Bayesian conditionalization rule is uniquely rational, and convergence arguments aim to show that regardless of differing priors, sufficient evidence will always lead agents to similar conclusions.
36 Hypothesis Confirmation One desirable application of inductive logic is to hypothesis confirmation: favorable evidence should make a hypothesis more credible, whereas unfavorable evidence should make a hypothesis less credible. Carl Hempel, a logical empiricist, investigated confirmation in the 1940s and raised some paradoxes which arise in the logic of confirmation, including the famous raven paradox.
37 The Raven Paradox Nicod s Criterion: A universal generalization of the form All As are Bs is confirmed by x is both A and B. Equivalence Condition: A statement and any logically equivalent formulation of it are equally confirmed by instances. Consider the statement All ravens are black. This statement is confirmed by the statement This is a black raven. All ravens are black is logically equivalent to All non-black things are non-ravens. By Nicod s criterion, this second statement is confirmed by This is a white shoe.
38 The Raven Paradox How can looking in my closet tell me anything about the colors of ravens? Is indoor ornithology (as Goodman quips) really possible? There are many responses to the raven paradox One is that there is no paradox. Observing white shoes does confirm that all ravens are black, but the degree of confirmation is negligible. This is the answer given by Bayesians, who can make quantitative the difference between observing a white shoe and observing a black raven on the posterior probability of the generalization. (Some do accept the paradox though and choose to, for example, reject Nicod s criterion.)
39 Bayesian Confirmation How should a Bayesian understand hypothesis confirmation? One approach is to introduce a probability threshold: if one s credence in a hypothesis is above this threshold, then one considers the hypothesis confirmed. The usual interpretation of confirmation in the Bayesian approach, however, is to consider a hypothesis confirmed if conditionalization on evidence simply raises one s degree of belief, that is, just when cr(h e)>cr(h).
40 Bayesian Confirmation Does Bayesian Confirmation Theory solve the Problem of Induction? Does Bayesian Confirmation Theory solve the Raven Paradox?
41 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
42 Plan for Part 3: Abduction 1. Inference to the Best Explanation 1. What is an explanation? 2. What makes an explanation best? 3. Objections to IBE 2. Peircean Abduction vs IBE 3. Hypothesis Formation and Scientific Discovery
43 Inference to the Best Explanation Discussions on scientific method often raise a special inference pattern, distinct from deductive and inductive inference: the inference to the best explanation. In the latter 20 th century it became particularly prominent in philosophical discussions thanks to the work of Gilbert Harman and Peter Lipton.
44 Inference to the Best Explanation Suppose that you are walking in woods in winter, and there is snow everywhere on the ground. You see distinctive tracks in the snow like these: Who or what made them? It seems obvious that one should infer that a person wearing snowshoes recently passed this way. Why? Because that is the hypothesis which best explains what you see.
45 Inference to the Best Explanation Clearly inference to the best explanation (IBE) is not deductively valid. But is it just induction? Perhaps. It is certainly an ampliative inference. Proponents of IBE argue that explanatory considerations are not evidence in the sense that it is used in inductive inferences however. An inference to the best explanation therefore treats explanatory goodness as adequate justification for drawing a conclusion. IBE is often also called abduction, having a logic of its own distinct from induction and deduction. The term abduction was coined by Charles Peirce in the early 20 th Century. He used it to refer to a process of forming explanatory hypotheses, not an inference to a specific explanatory hypothesis. More on Peircean abduction later.
46 Inference to the Best Explanation IBE is often glossed in the following way: Given evidence E and candidate hypotheses H 1, H 2, which explain E, infer the truth of the hypothesis which best explains E.
47 What is an Explanation? It seems as if one should have a good idea of what an explanation is in order to ascertain the best one. If one looks at the philosophical literature on explanation, however, one sees little consensus. As Lipton puts it, it seems that IBE is therefore attempting to account for the obscure in terms of the equally obscure. The original account of explanation is due to Hempel and Oppenheim, who characterize an explanation as a sound deductive argument which features a law of nature essentially. This is known as the Deductive Nomological account. Other philosophers have argued that an explanation must articulate the causes responsible for the phenomenon being explained.
48 What is the Best Explanation? Is the best explanation the one that is most likely to be true? If that were the case, then IBE would be a species of inductive inference. Moreover, it would not give any guidance about hypothesis selection, since one does not know in advance which hypothesis is the most likely one to be true among the candidate hypotheses. Lipton argues that the best explanation is not the likeliest but the loveliest. The loveliest explanation is the one that, if correct, would provide the most understanding. Nevertheless, he insists that the loveliest explanation and the likeliest explanation will usually be the same.
49 Best of a Bad Lot? The most well-known criticism of IBE comes from Bas van Fraassen. He points out that inferring the truth of the best explanation depends on the true explanation being among the candidate hypotheses. Van Fraassen also argues that any probabilistic IBE rule must match Bayesian conditionalization or else be probabilistically incoherent. In other words, IBE is either induction or irrational.
50 Peirce on Abduction Contemporary discussion of IBE/abduction characterizes it as a justified inference to the truth of a hypothesis. As said, Peirce did not conceive of it in this way. For him it was a method for generating explanatory hypotheses. Abduction is the process of forming explanatory hypotheses. It is the only logical operation which introduces any new idea.
51 Peirce on Abduction Peirce recognized three distinct logics in scientific inquiry: abduction, deduction, and induction. Abduction concerns all operations by which hypotheses are engendered. Deduction concerns the derivation of predictions and comparison of these predictions with observation. Induction concerns the assessment of the hypothesis in light of evidence. What is logical, though, about the invention of hypotheses? Is abduction better understood as concerning the adoption of hypotheses as worthy candidates for further investigation?
52 Discovery and Justification Recall the distinction Popper made between the psychology of knowledge and the logic of knowledge. This distinction is similar to a wellknown one made by Hans Reichenbach in 1938: The context of discovery, which concerns the actual processes of thought. The context of justification, which concerns a rational reconstruction of processes of thought. The descriptive task of epistemology is concerned with this.
53 Discovery and Justification Reichenbach s terminology has come to mark a distinction between phases of research, splitting the scientific process into discovery and justification. Such a distinction has been heavily criticized since at least Kuhn, as history shows that there is no clear division along these lines. In any case, this is clearly not what Reichenbach meant, although logical empiricists, like Popper, did not think that the process of discovery could be rationally reconstructed. It always involved some creativity. Peirce, then, can be understood as claiming that there was a logic of some kind to the discovery process, one which somehow involved explanatory considerations.
54 Logic(s) of Discovery So, to repeat, what is logical about the invention of hypotheses? The invention of hypotheses is clearly an ampliative inference, so it is not deductive. Can it be characterized as a kind of induction? Does it fit within the Bayesian framework? Perhaps. There are other approaches which are worth considering however
55 The Learner s Paradox Recall the Value Problem from Plato s dialogue Meno? There is another famous problem in it too: the Learner s Paradox. How can one learn something? Either we know that thing or we do not. If we know it, then there is no point in inquiring about it. If we do not know it, then there is no point in inquiring it either, for we would not recognize it even were we to stumble across it. The usual response to the paradox is to say that we must know something about the thing in advance of inquiring about it, so that we can recognize it.
56 Scientific Problems A scientific problem, then, can be posed as a question. Sometimes this is an explanatory question ( Why x? ) and sometimes a descriptive question ( What is x? ). Recall the brief discussion of the logic of questions, erotetic logic. There it was suggested that a question is a set of answers. What delimits the set of answers are constraints on the question. In terms of problems and solutions, a problem is defined by its constraints, which fix its possible solutions.
57 Method What makes something a method? One way to think of a method is a procedure which, if followed, reliably leads to success at the relevant goal. An epistemic method, we might say, is one that, if followed, reliably leads to knowledge, understanding, truth, etc. A minimum threshold for something to be methodological (reliable) is that it is better than guessing. By this standard, solving empirical problems ( recalcitrant data ) is a reliable means for making progress. Is solving explanatory problems a reliable means for making progress too?
58 Analogies in Science Besides problem solving, another form of reasoning which is often seen as relevant to discovery is analogy. But it seems that there can be no universal inference rule that captures analogy. The success of an analogical inference is context-dependent. For this reason they are considered heuristic: suggestive rather than demonstrative. Mary Hesse, one of the foremost philosophers of science in the 20 th century writing on models, analogy, and inference.
59 Non-Empirical Theory Confirmation Richard Dawid (of SU) and collaborators have attempted to broaden the application of Bayesian confirmation to cases where empirical evidence is not easily available (for example, in string theory). Why should the E in Bayes Formula be empirical evidence? If evidence is that which justifies, then perhaps meta-empirical facts, for example facts about the process of research or inquiry (in some context), are evidentially relevant to the evaluation of hypotheses. This idea, while fully in the context of justification, is relevant to the process of discovery, at least in the sense that confirmation here is more suggestive than demonstrative.
60 Plan for Part 3: Abduction 1. Inference to the Best Explanation 1. What is an explanation? 2. What makes an explanation best? 3. Objections to IBE 2. Peircean Abduction vs IBE 3. Hypothesis Formation and Scientific Discovery
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