# There are two common forms of deductively valid conditional argument: modus ponens and modus tollens.

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1 INTRODUCTION TO LOGICAL THINKING Lecture 6: Two types of argument and their role in science: Deduction and induction 1. Deductive arguments Arguments that claim to provide logically conclusive grounds for their conclusion are deductive arguments (here the conclusion is claimed to lead down from the premises). Where the premises do indeed guarantee the conclusion, the argument is held to be deductively valid. 1.1 Deductive validity Where the premises guarantee with certainty the conclusion, the relation between premises and conclusion is referred to as deductive validity. Consider this argument: All jumblies have two moggs Sam is a jumblie So Sam has two moggs Notice the conclusion follows logically from (i.e., to be entailed by) the premises here. 1.2 The different forms of deductive argument Typically, a deductive argument comes in the form of a syllogism (an argument with two premises and a conclusion, and there is a single term which appears in both premises (as in all the examples we have used so far): All A are B X is A Therefore X is B One common type of syllogism is the hypothetical syllogisms. This includes a hypothetical or conditional statement in one of the premises "If... then...". They are not asserting two statements they are only asserting a single statement. The proposition which follows the "If" is called the "antecedent", and the proposition which follows the "then" is called the consequent. There are two common forms of deductively valid conditional argument: modus ponens and modus tollens. Modus ponens ( putting or asserting the antecedent ) Premise 1: If you are a mother, then you are female Premise 2: A You are a mother Conclusion: Therefore B Therefore you are female antecedent = A = you are a mother consequent = B = you are female Modus tollens ( taking away or denying the consequent ): Not B You are not female Therefore not A Therefore you are not a mother antecedent = A = you are a mother consequent = B = you are female However, if you assert the consequent or deny the antecedent, your argument will be not valid. We have an error or fallacy in the argument: Fallacy of asserting the consequent B Therefore A You are female Therefore you are a mother

2 Fallacy of denying the antecedent: Not A Therefore not B You are not a mother Therefore you are not female Another common type of syllogism is what is called argument by elimination. Here the syllogism involves a disjunctive statement in the premises. A disjunctive statement is an "Either... or..." statement: Either A or B Not A Therefore B Either he is sleeping or he is watching TV He is not sleeping Therefore he is watching TV We can also have deductively valid forms in which the premises are a mixture of hypothetical and disjunctive statements: Either A or B If A then C If B then C Therefore C Either he's sleeping or he's watching TV If he's sleeping then he's breathing If he's watching TV then he's breathing Therefore he's breathing 2. Inductive arguments "Inductive means "to bring forward". In inductive arguments, the premises support the conclusion to a greater or less extent, so we are looking at degrees of likeliness or probability. For example: Many smokers die of lung cancer. John is a smoker. Therefore John will die of lung cancer. Here, even if the premises are true, the conclusion is not guaranteed, because the first premise, by saying "many", allows that some smokers won't die of lung cancer. John could be one of those. There may be something about John which puts him in the low-risk category, such that it is not even probable that he will die of lung cancer. Now consider this argument: I have 1000 balls in this bag. 999 are red and 1 is black. Therefore, when I put my hand in and draw out one ball, it will be red Here, the relationship between premises and conclusion is very strong indeed, meaning that the conclusion is very likely. However, it is still not guaranteed, because there is an outside chance that I'll draw out the single black ball. With inductive arguments, we never have logical validity in the deductive sense; the conclusion is never entailed by the premises. So all inductive arguments are logically invalid. Instead, what we have is degrees of strength or plausibility. Varieties of inductive argument Inductive arguments come in different forms, but they all involve stating the premises and then moving beyond them. - Predictive argument: we make a prediction based on the past or present (e.g., The logic exam has always been 2 hours long. Therefore, it will be 2 hours long next year ). - Causal reasoning: we infer a causal relation from past experience (e.g., I can t log on. Therefore, the network must be down ). - Statistical generalisation: we observe statistical trends in a sample and then project these onto the population. Eg: if 70% of our sample first year students are female, then we might conclude inductively that 70% of all first years are female. - Argument from analogy: we conclude something based on a comparison/similarity between two things. Eg: John and Mary are registered psychologists, and they both practice. Lee also a registered psychologist. Therefore, Lee is likely to be in practice.

4 4.2 Deduction in science Evidence will also play a role in the way deductive arguments are used in science. Here, we need to know about background events in the history of science. Early last century, philosophers of science were arguing that science proceeds inductively by verification (i.e., showing that the theory is true). That is, we propose a theory, and then find instances and evidence which support that theory. But Karl Popper, said that this method is flawed, because no matter how many supporting instances we find, we can never prove true a general theory. For example, imagine that our theory is: "All crows are black". This theory is based on our past observations. Instead, according to Popper, we attempt to falsify the theory, because we can falsify a general theory. A single observation of a nonblack crow would falsify our theory. So Popper introduced falsificationism as a scientific approach, and falsifiability (instead of verifiability) as the criterion for a scientific theory. According to Popper, whatever your theory is, you must be able to specify what observation would in principle make it false. If your theory is consistent with any actual or possible state of affairs, then your theory is unfalsifiable, and so it is not scientific. This is where deduction comes in, because falsification is done via a deductive argument. It uses the familiar form of modus tollens (denying the consequent): Not B Therefore not A If the theory is true, we shall observe X (e.g., If all crows are black, then the crow around the corner will be black) We do not observe X (The crow around the corner is not black) Therefore the theory is not true (Therefore, not all crows are black) But notice that the first premise in the above crow argument is itself a deduction. All crows are black. There is a crow around the corner. The crow around the corner will be black In other words, our hypothesis, the observation which we propose to make to support the theory is logically entailed by the theory. This is part of the hypothetico-deductive method, and is essential to sound theory testing. In Step 1 we begin with our theory (e.g., All crows are black ) and in Step 2 we logically/deductively derive a hypothesis from the theory (e.g., The crow around the corner will be black ). It is this hypothesis that we empirically test in Step 3. Because of the logical validity of modus tollens, disconfirming evidence is stronger than confirming evidence (just as Popper argued). Obviously, if a theory is true, it will never be falsified. But Popper was not claiming that we should actually falsify our scientific theory. He was insisting that we make clear what observation would falsify our theory if our theory were false. A common problem is that it s not clear sometimes how the researcher's hypothesis is derived from the theory that is supposedly being tested. Also, when we put a hypothesis to empirical test in psychology, we typically use a sample drawn from a larger population of interest. Therefore, we deal with probabilities in our statistical test, and these probabilities are included in our argument s premises. So even if we use modus tollens, our conditional first premise has reference to likeliness in the consequent. As we ll see, this means that we never actually falsify the hypothesis that we are testing. (These issues are crucial for the way in which we go about doing scientific psychology, and we are just flagging them here; we ll return to them later) Conclusion In conclusion, evidence can be used to either support (via induction) or disconfirm (via deduction) a theory. And we use deduction to derive a testable hypothesis from our theory. In the next lecture, we begin to look at this process of evaluating arguments. References Cohen, M. R. & Nagel, E. (1957). An introduction to logic and scientific method. London: Routledge & Kegan Paul.

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