called the Art of Reasoning.” Mill (1858) writes: “Logic is not the science of Belief,
but the science of Proof, or. Evidence. So far forth as belief professes to be ...
A Philosophical Introduction to Formal Logic Chapter 0: What is Logic? Logic is the normative study of reasoning. In other words, logic is the study of what makes reasoning good or bad. Since it is a normative study of reasoning, logic is different from psychology. The psychologist wants to know (among other things) how we actually reason. By contrast, the logician wants to know how we ought to reason, even if we never actually reason as we should. And yet, logic has a psychological subject insofar as reasoning is regarded as a mental process. Now, the process of reasoning is a kind of movement from some collection of beliefs that the reasoner accepts at some time to some possibly different collection of beliefs that the reasoner accepts at a later time. But in order to facilitate the critical evaluation of reasoning, the logician abstracts away from the messy details of how the reasoning process is implemented by humans and represents instances of reasoning as static, linguistic objects, called arguments.1
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Not all logicians have taken the view of logic taken here; however, the view here is a traditional one. For example, Whately (1811) writes: “Logic, in the most extensive sense which the name can with propriety be made to bear, may be considered as the Science, and also as the Art, of Reasoning. It investigates the principles on which argumentation is conducted, and furnishes rules to secure the mind from error in its deductions. Its most appropriate office, however, is that of instituting an analysis of the process of the mind in Reasoning; and in this point of view it is, as I have said, strictly a Science: while, considered in reference to the practical rules above mentioned, it may be called the Art of Reasoning.” Mill (1858) writes: “Logic is not the science of Belief, but the science of Proof, or Evidence. So far forth as belief professes to be founded upon proof, the office of logic is to supply a test for ascertaining whether or not the belief is well grounded.” Venn (1876) writes: “It is impossible to direct attention too prominently to the fact that logic (and therefore Probability as a branch of logic) is not concerned with what men do believe, but with what they ought to believe, if they are to believe correctly.” Peirce (1877) writes: “The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. Consequently, reasoning is good if it be such as to give a true conclusion from true premises, and not otherwise. Thus, the question of its validity is purely one of fact and not of thinking. A being the premises and B the conclusion, the question is, whether these facts are really so related that if A is B is. If so, the inference is valid; if not, not. It is not in the least the question whether, when the premises are accepted by the mind, we feel an impulse to accept the conclusion also. It is true that we do generally reason correctly by nature. But that is an accident; the true conclusion would remain true if we had no impulse to accept it; and the false one would remain false, though we could not resist the tendency to believe in it.” Moreover, this tradition is still very much alive today, for an example of which, see Priest (2000), who writes: “Logic is the study of what counts as a good reason for what, and why.”
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An argument is a collection of sentences that are related to one another in a special way: some of the sentences, called the premisses of the argument, are intended to evidentially support another one of the sentences, called the conclusion of the argument. When an argument is good, the premisses succeed in supporting the conclusion. If the premisses of an argument fail to support its conclusion, then the argument is bad. In many cases, we have an intuitive, pretheoretical sense for when an argument is good or bad. For example, consider the following argument:2
(1)
Every politician is a liar. Frank is a politician. Therefore, Frank is a liar.
Even if you have never taken logic before, you probably recognize argument (1) as a good argument. And even if you cannot articulate exactly why argument (1) is a good argument, you still probably recognize that it is a good argument. If it is not obvious to you that argument (1) is a good argument, that’s okay for now. Regardless of whether or not you find the argument obviously good, keep the following question in mind: What is it about a good argument that makes it a good argument? Please stop and reflect on this question for a few minutes before continuing on. When asking yourself what it is about a good argument that makes it a good argument, you might find it helpful to think about some examples of bad arguments as well. Compare argument (1) above with the following argument:
(2)
Every politician is a liar. Suzy is not a politician. Therefore, Suzy is a liar.
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In writing out arguments, I will follow this stylistic convention: I will write the premisses of the argument above a solid line, and I will write the conclusion of the argument below the line. Sometimes, I will give a separate name to each line in an argument. For example, in an argument with two premisses and a conclusion, I might label the first premiss (P1), the second premiss (P2), and the conclusion (C).
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I claim that argument (2) is not a good argument. The premisses of argument (2) do not evidentially support the conclusion of argument (2). Are you convinced? If so, ask yourself why argument (2) is a bad argument. If you are not immediately convinced that argument (2) is a bad argument, that’s okay for now. Learning about what makes an argument good or bad is the main point of this text, after all! We are now in a position to restate what it is that logic is all about. Logic is the study of the relation of evidential support. The logician wants to know when some premisses are evidence for the truth of some conclusion. But knowing that some premisses evidentially support some conclusion does not necessarily tell you that the conclusion is true. The premisses may be false, after all! Moreover, in an important sense, the logician does not care whether the conclusion of an argument is true or false. Rather, the logician wants to know something conditional: if the premisses were true, would they evidentially support the conclusion? Most logicians today are interested in a special case of the relation of evidential support: the relation of logical consequence.3 The relation of logical consequence is a deductive relation that holds between the premisses of an argument and the conclusion of that argument when the truth of the premisses guarantees the truth of the conclusion. Argument (1) above has this feature. If the premisses of the argument are both true, then the conclusion cannot be false. If the premisses and the conclusion of an argument stand in the relation of logical consequence, then we say that the conclusion follows logically from the premisses. And if the conclusion of an argument follows logically from its premisses, then the argument is a good one: the premisses evidentially support the conclusion.
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See Etchemendy (1990) for a very detailed discussion of the relation of logical consequence.
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The premisses of an argument might evidentially support some conclusion despite the fact that the conclusion does not follow logically from the premisses. In fact, many arguments that we care about are such that even if the premisses of the argument are true, the conclusion is not guaranteed to be true. For example, we are often interested in arguments like the following:
(3)
Only one out of several million tickets in lottery L is a winning ticket. Alistair has exactly one ticket in lottery L. Therefore, Alistair does not have a winning ticket.
In argument (3), if the premisses are both true, then the conclusion is very, very likely to be true. And so, it seems that the premisses evidentially support the conclusion. But the truth of the premisses does not guarantee the truth of the conclusion. Hence, the relation of logical consequence – understood as a deductive relation – only represents a species of evidential support. Therefore, even after we have a complete account of logical consequence, we will not have a complete account of evidential support. Later in this text, we will spend some time thinking about probability in order to better understand what makes arguments like (3) examples of good reasoning. We have already idealized reasoning in an important way by replacing beliefs with sentences and processes of thought with static arguments. In order to better study what makes reasoning good or bad, we will use special symbols to construct a formal language within which we may describe various kinds of evidential relation with mathematical precision. We will think of the sentences of our formal language as translations of sentences that occur in ordinary language, and we will construct a well-defined relation within our formal language to represent the relation of evidential support – or some species of that relation – that holds for sentences in ordinary language. Hence, we are studying reasoning at two removes. In place of the messy details about how brains represent the world and modify their representations over time, we 4
substitute details about the structure of some linguistic objects. And instead of studying the linguistic objects directly, we study mathematical representations of them. As Whately observed, logic is both a science and an art. We are mostly interested in the science of logic: our aim is to understand what makes reasoning good or bad. But we do not want to lose sight of the art of logic. We do not just want a formal account of good and bad reasoning. Ultimately, we want to be able to reason better as a result of studying logic. How can we learn to reason better? In this book, we will learn to reason better by studying formal models of good reasoning and by thinking about what makes reasoning good. We then try to make our own reasoning good by emulating our formal models, and in some cases, translating our own reasoning into a formal model to check it. In order to improve our reasoning, we need to imagine that our reasoning really can be improved. And that brings us to the first rule of practical logic: In order to learn, we must desire to learn and not rest content with what we already think.4 So, ask questions. Challenge your own beliefs and the beliefs of the people around you. When you find that you do not understand something, inquire harder. Talk to your friends and classmates. Interrogate your teachers. Consult books and websites. Make careful observations of the world around you. Make guesses and test them. Run experiments. Never give up until Nature has relinquished her secrets to you. Nothing else you learn will ever be as important as an unquenchable thirst to know.
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Paraphrased from Peirce’s lecture, “The First Rule of Logic,” reprinted in The Essential Peirce, Volume 2, 48.
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References Etchemendy, J. (1990) The Concept of Logical Consequence. Cambridge: Harvard University Press. Mill, J. (1858) A System of Logic. New York: Harper & Brothers. Available at http://books.google.com/books?id=she97TwKwkC&printsec=frontcover&dq=%22system+of+logic%22+inauthor:mill&source=bl&ot s=rFh05BWP_N&sig=zyOIo2g_Wz9LqMwbVPLYBJeDtM&hl=en&sa=X&ei=ImwgUPfWDI6xqAHmoICADQ&ved=0CDQ Q6AEwAA#v=snippet&q=%22logic%20is%20not%20the%20science%20of%20belief%22&f=f alse Peirce, C. (1877) “The Fixation of Belief,” Illustrations of the Logic of Science, serialized in Popular Science Monthly. Available at http://books.google.com/books?id=dywDAAAAMBAJ&pg=PA115&dq=popular+science+mont hly+1877+volume+11&hl=en&sa=X&ei=KuuiT5SrBIicgwePoKDXCA&ved=0CFIQ6AEwBQ #v=onepage&q&f=false Peirce, C. (1898) “The First Rule of Logic,” The Essential Peirce, Vol. 2. Bloomington: Indiana University Press. Priest, G. (2000) Logic: A Very Short Introduction. Oxford: Oxford University Press. Venn, J. (1876) The Logic of Chance, 2nd Edition. London: MacMillan and Co. Available at http://books.google.com/books?id=BbNVAAAAYAAJ&printsec=frontcover&dq=editions:8An mNMKe618C&source=bl&ots=4W9Kf7ehSv&sig=TDf-Kc-pmE9daGzJGltRZQOd_I&hl=en&sa=X&ei=tlwgUNCqA8XcqgG034GQDw&ved=0CFIQ6AEwBzg K#v=onepage&q=Logic%20(and%20therefore%20probability%20as%20a%20branch%20of%2 0logic)&f=false Whately, R. (1840) Elements of Logic, 7th Edition. London: B. Fellowes. Available at http://books.google.com/books?id=xJI9AAAAYAAJ&printsec=frontcover&dq=editions:DEK_ MjVNky8C&source=bl&ots=JFp3h403Uz&sig=TrPuq3b_3RCuesMl6e9u0IzYYzo&hl=en&sa= X&ei=xFMgUIqnLsrArQGA2YHABA&ved=0CDQQ6AEwAQ#v=onepage&q&f=false
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