# Logic/Signed Articles/Mark Sainsbury

### Logic

Mark Sainsbury
University of Texas at Austin, USA

Logic is a normative science: it provides criteria for assessing the quality of reasoning. The norms come from reflecting on our actual practice of reasoning, but the aim is not to provide a theory which merely describes how we reason. Logic must make room for the possibility that we sometimes make mistakes, sometimes of a quite general and systematic character. That they are mistakes is argued for in ways which ultimately depend upon other features of our reasoning practice (there is no practice-transcending logic), and it is the logician’s task to provide compelling criteria which reveal good reasoning as good, and bad reasoning as bad.

What makes for good reasoning? Sometimes what is at issue is demonstrative reasoning, which can be thought of as the process of extracting information which is already present in some form. That Socrates is mortal is already contained in the information that all men are mortal and Socrates is a man. Other times what is at issue is “inductive” or (better) “non-demonstrative” reasoning, where the task is to reach a conclusion that in some sense goes beyond the information available as evidence. We infer that there is a mouse in the house because the cheese disappears overnight, even though the larder door is locked. This kind of relation between evidence and theory is crucial to much reasoning in the empirical sciences, and is brought to full theoretical fruit in theories of probability and statistics.

“If Saddam Hussein had weapons of mass destruction, the war in Iraq was justified. But he had no such weapons. So the war was not justified.” This exemplifies a common fallacy. One could point up the error by saying the argument overlooks the possibility that links with Al Quaeda might also have justified the war, even in the absence of WMDs. A logician can reveal the bad quality of the argument by analogy. If the quoted reasoning is good, the following analogous reasoning should be good: “If 2 is greater than 3, it is greater than 1. But it is not greater than 3. So it is not greater than 1.” This kind of analogy is developed into the logician’s notion of a model: a good argument should hold in every model, and a model is a set-theoretic structure which is supposed to provide a general account of the logical features of a thought or argument. As part of this process, logicians typically require the reasoning to be represented in some special notation, like Venn diagrams, or the language of classical first order logic. The argument with which this paragraph opened might be represented “P ${\displaystyle \to }$ Q, ${\displaystyle \lnot }$ P, therefore ${\displaystyle \lnot }$ Q” (using “${\displaystyle \lnot }$” for “not” and “${\displaystyle \to }$” for “if … then”). Since “P ${\displaystyle \to }$ Q” can be true when “P” is false and “Q” is true (for example “George Bush jr is a twin ${\displaystyle \to }$ he has a sibling”), an argument of this form may have true premises and a false conclusion, and so is not deductively valid (as logicians use this term). By contrast, arguments of the form “P ${\displaystyle \to }$ Q, ${\displaystyle \lnot }$ Q, therefore ${\displaystyle \lnot }$ P” can never lead from truth to falsehood, and so are deductively valid.

Linda is single, very bright, and a philosophy major. She was active in student politics, and known for her concern with issues of discrimination and social justice. She has now got a job. On the basis of this information, 85% of those quizzed in a survey assigned a higher probability to her being a bank teller active in the feminist movement than simply to her being a bank teller. This is another example of a common kind of erroneous reasoning (reported by Tversky, A. and Kahneman, D. (1983) “Extension versus intuitive reasoning: The conjunction fallacy in probability judgment”. Psychological Review, 90, 293–315). Once pointed out, it is obvious that a person is less likely to possess both of two independent traits (being a bank teller, being a feminist) than just one. If logic is properly described as the normative science of reasoning, then logic, at least in the broadest sense, deals not only with the kind of demonstrative reasoning discussed in the previous paragraph, but also with probabilistic reasoning and, more generally, with reasoning designed to reach conclusions not already contained in the data. In practice, most people who are called logicians study deduction, not induction, and there are serious doubts about whether inductive logic could ever attain the precision and completeness of deductive logic.

How could a science be normative? Often we contrast the purely descriptive aims of science with the normative aspirations of preachers and ethicists. Some believe that descriptions can be objective, whereas norms are subjective, not a reflection of any external reality but only of personal feeling. Yet there is no getting away from it: logic is normative, because it aims to identify good reasoning; and it is scientific, being (roughly, and in part) a branch of mathematics. Thinking about logic should make us reject facile contrasts between norms and science.

Originally from Giandomenico Sica, Ed., The Language of Science. Monza: Polimetrica. [1]

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