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A short introduction to formal logic

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A pamphlet, designed to introduce students to some basic concepts and tools from formal logic.
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  A short introduction to formal logic Dan Hicksv0.3.2, May 2, 2012 Thanks to Tim Pawl and my Fall 2011 Intro to Phi-losophy students for feedback on earlier versions. Myapproach to teaching logic has benefited a great dealfrom numerous exchanges with John Milanese. Contents 1 The idea of logic1 1.1 Arguments and argument patterns. . . . . . . . . 21.2 Truth, validity, and soundness. . . . . . . . . . . . 41.3 Deductive validity. . . . . . . . . . . . . . . . . . . 81.4 Testing validity. . . . . . . . . . . . . . . . . . . . 9 2 Some common argument patterns12 2.1 Conditionals. . . . . . . . . . . . . . . . . . . . . . 122.2 Categories. . . . . . . . . . . . . . . . . . . . . . . 162.3 Other sentence logic arguments. . . . . . . . . . . 172.4 Induction. . . . . . . . . . . . . . . . . . . . . . . 182.5 Practical reasoning. . . . . . . . . . . . . . . . . . 202.6 Conversational logic. . . . . . . . . . . . . . . . . . 222.7 Modal logics. . . . . . . . . . . . . . . . . . . . . . 27 3 Working with arguments in the wild27 3.1 Formulating an argument. . . . . . . . . . . . . . 283.2 Reconstructing (and evaluating) an argument. . . 293.3 Enthymemes. . . . . . . . . . . . . . . . . . . . . . 313.4 Improving arguments. . . . . . . . . . . . . . . . . 313.5 Terminology. . . . . . . . . . . . . . . . . . . . . . 32 Additional resources34Note for instructors36Fig. 1: Categorical syllogisms37Fig. 2: Other argument patterns38 1 The idea of logic Logic is the branch of philosophy that deals with arguments. Weoften think of arguments in connection with disagreements, de-bates, and even shouting. Call this way of thinking about ar-guments argument as disagreement . For example, we mighttalk about the argument (meaning the disagreement) between Re-publicans and Democrats over taxes and the budget. But there’sanother way of thinking about arguments. This is less commonthan argument as disagreement, but it’s still common. For ex-ample, suppose the Republicans say that we should balance thebudget because it would help the economy. Here we would saythat the Republicans are giving an argument  to support their po-sition on the budget. Call this way of thinking about arguments argument as reason . An argument, on this way of thinking, is a reason why someone does (or should) believe something  : the Re-publicans, in giving their argument, are giving reasons why you (or1  whoever they’re talking to) should believe that we should balancethe budget.It may seem to you that philosophy is mostly about argument asdisagreement. You’re probably not wrong to think this. But logicdeals only with argument as reason. Logicians — philosophers whostudy logic — are interested in the connection between reasons andthe claims they support. Throughout this pamphlet, we’re goingto be concerned entirely with arguments as reasons (even whenthose reasons show up in the context of a disagreement). Our goalwill be to equip you with some basic concepts — a logical toolkit— for use in your philosophy class. 1.1 Arguments and argument patterns It’s not the case that all logic textbooks are obligated to discussthis argument. But it’s such a standard example that I can’t helpbut start with it.(1) All humans are mortal.(2) Socrates is a human.(3) ∴ Socrates is mortal. (1,2)(Socrateswas a philosopher in ancient Athens, and is often con-sidered the father of Western philosophy. He famously died bydrinking poison.) This argument is in premise-conclusion form .In premise-conclusion form, each distinct claim of the argument iswritten on its own line, and the lines are numbered to make themeasy to refer to. Line (3) is called the conclusion of the argument.(See what I did there, with the line number?) The conclusion isthe claim that the argument is meant to support. If I were givingyou this argument, I would be trying to convince you that Socratesis mortal. (It’s not a very interesting or surprising conclusion. Butsimple examples like this are easier to follow. You’ll encountermany arguments with more interesting conclusions in your philos-ophy class.) Lines (1) and (2) of this argument are its premises .The premises are the reasons given to support the conclusion: if you believe that all humans are mortal and that Socrates is a hu-man, then you have reason to believe that Socrates is mortal. Bothpremises and conclusions are steps of the argument. Premise (1)is the first step; the last step, (3), is the conclusion.This is a good time to mention two basic rules for writing argu-ments in premise-conclusion form. First, every step is a declar-ative sentence . You can think of a declarative sentence as a sen-tence that can be true or false. Here are some things that aren’tdeclarative sentences, and so can’t be steps of an argument: × Is a human.This one isn’t grammatically a sentence — it doesn’t have asubject. × Make tacos!This one is a sentence, but it’s an imperative sentence — acommand. There’s a special logic for imperative sentences,but it’s not one we’ll discuss in this pamphlet. × Is it raining?Another sentence, but this one is a question. × I like corn.This one is actually tricky. It can be read as a declarativesentence: a report (true or false) of how I feel about corn.But it can also be read as an optative , which is an expressionof a wish, desire, or preference. On this reading, this sen-tence is like saying ‘Hooray for corn!’ — which can’t be trueor false.2  Our second rule is that premises come first, conclusionscome last . This rule is just about order: the conclusion of theargument should be the very last step, and the first step (at least)should be a premise. (There are some exceptions, but we won’tcover them here).Besides the steps and their numbers, the example up above hadsome other useful symbols. (3) started off with this symbol: ∴ Thissymbol is read ‘hence’ or ‘therefore’, and it signals a conclusion.We know that (3) is the conclusion because of the ∴ symbol. Thismight seem obvious now — it’s the last step, so of course it’s theconclusion. But we’ll see more complicated arguments below, and ∴ will be very important. (If you can’t figure out how to make ∴ on your computer, just start the conclusion with the word ‘hence’instead.) There were also the numbers in parentheses after (3):(1,2). These numbers indicate logical support. They tell us thatstep (3) is supported by steps (1) and (2), taken together. Again,this is obvious in this simple example — so I won’t be using themregularly in this pamphlet — but it’ll be very useful when we getto more complicated arguments.For now, compare the earlier argument with this one:(4) All cats are mammals.(5) Whiskers is a cat.(6) ∴ Whiskers is a mammal.The two arguments look quite similar. We could even borrow fromalgebra and replace the nouns and names with letters:(7) All A s are B s.(8) x is a A .(9) ∴ x is a B .And we could go the other direction, replacing the generic letterswith nouns and names. (Actually, that’s how I thought up theexample about Whiskers.) When we have two arguments that aresimilar in this way, we say that they have the same logical form or pattern . Here’s another pair of examples in English, along withthe generic-letter version:(10) If Obama is Presidentthen Biden is VP.(11) Obama is President.(12) ∴ Biden is VP.(10  ) If  p then q .(11  ) p (12  ) ∴ q (10  ) If it’s raining then thesidewalks will be slippery.(11  ) It’s raining.(12  ) ∴ The sidewalks will beslippery.(The  is a ‘prime’, so for example (10  ) is ‘ten-prime’ and (11  )is ‘eleven double-prime’.) These arguments all have the same pat-tern, though of course it’s different from the pattern we saw withSocrates and Whiskers.The next thing to notice is that, in arguments with the samepattern, the premises are related to the conclusion in the sameway. The premises (10) and (11), taken together, support (12) inthe same way that (10  ) and (11  ), taken together, support (12  ).And that, in turn, means that we can ask and answer some interest-ing questions about an argument even if all we know is the logical  form  . By studying the generic version of the argument (10  − 12  ),and without even thinking about President Obama are whetherthe sidewalks really are slippery, we can learn something abouthow the premises, in each case, support the conclusion. And we3  can go on to apply that to other arguments with the same pat-tern, whether they’re about President Obama or the planet Marsor whatever.The move from English-language arguments (or, more generally,arguments in natural languages ) to their patterns is like themove from drawing shapes on a piece of paper to studying geom-etry. Indeed, the study of argument-patterns goes by two names: formal logic — because it studies logical form — and mathe-matical logic — because it’s studied mathematically. If you’rereading this pamphlet for a logic class, you’ll go on to learn someof that mathematics. (If you’re math-phobic, don’t worry: you canbe excellent at logic without knowing anything at all about algebraor calculus or any of that.) But, if you’re not, you can still use thetools we develop here to analyze arguments. In fact, that’s whyI wrote this: so students who haven’t learned any mathematicallogic at all can still use it.So far, we’ve only seen arguments with a single conclusion. Thesearguments are called simple arguments. We can build compound arguments by putting together simple arguments. Here’s an exam-ple:(13) Either Obama is President or McCain is President.(14) McCain isn’t President.(15) ∴ Obama is President. (13, 14)(16) If Obama is President then Biden is VP.(17) ∴ Biden is VP. (15, 16)In compound arguments, just like simple arguments, we arrangethe premises first and the conclusion last. But, unlike simple ar-guments, compound arguments will have steps in the middle, thatare neither independent premises nor the final conclusion. In thisexample, (15) isn’t an independent premise — it follows from (13)and (14) — but it’s also just a step on the way to (5). These in-termediate steps are called subconclusions . Like conclusions, wemark subconclusions with ∴ and numbers indicating logical sup-port. Notice that, after (17), we list only (15) and (16). Thisis because these only two steps, all by themselves, give support directly  to the conclusion. The other steps — (13) and (14) —support the conclusion indirectly  , by directly supporting (15). 1.2 Truth, validity, and soundness So far, we’ve developed some important concepts to understandthe ‘anatomy’ of an argument: its premises, conclusions, and pat-tern. Next, we want to evaluate arguments: do the premises, takentogether, give us a reason to believe the conclusion?The most familiar concept here is truth : we say that it’s truethat Obama is President, that it’s false that it’s raining, and so on.Defining truth is a huge philosophical problem; many philosophersthink of truth as something like ‘correspondence to reality’, butthere are plenty of philosophers who think this definition won’tdo. I’ll assume that truth is something like correspondence.Notice that, with this conception of truth, truth isn’t the sameas belief and opinion. Contrast these three sentences: • Juanita believes that humans are responsible for globalwarming. • Juan believes that humans aren’t responsible for globalwarming. • Humans are responsible for global warming.The first two sentences are about what people believe . These sen-4  tences are true if, and only if, these people actually believe thesethings. (If, and only if, the sentences reporting their beliefs cor-respond to what they believe.) The third sentence has nothing  todo with what anyone believes. It’s true if, and only if, humansactually are responsible for global warming. (If, and only if, thesentence blaming humans corresponds to the effects of our actions.)That means whether the third sentence is true has nothing to dowith whether people agree on it: Juanita’s belief doesn’t make thethird sentence true, and Juan’s belief doesn’t make the third sen-tence false. Their disagreement might make it hard for us to tell  whether or not the third sentence is true, but that’s a matter of our knowledge of the world, not the way the world is . Disagreement,by itself, doesn’t make anyone’s beliefs false. The last paragraph shouldn’t seem outrageous or surprising. Butnow I’m going to make a very controversial move. Contrast thesethree sentences: • Juanita believes that we shouldn’t eat meat. • Juan believes that we should eat meat. • We should eat meat.Notice, first, that the relationship between these three sentencesis the same as the relationship between the last three sentences:the first is about what Juanita believes, the second is about whatJuan believes, and the third doesn’t seem to have anything to dowith what anyone believes. The controversial move is to assumethat the third really  doesn’t have anything to do with what any-one believes. Like the claim that humans are responsible for globalwarming, ‘we should eat meat’ is true if, and only if, it correspondsto the way the world actually is.This probably strikes you as much more surprising and outra-geous than the global warming stuff. You’ve probably been taughtthat there’s a difference between fact and opinion , and you’veused this distinction to say things like ‘there’s a fact about whetherhumans are responsible for global warming’ and ‘whether we shouldeat meat is a matter of individual opinion’. But we often act asthough there are some facts about what everyone should  do: notcommit murder or rape; not be racist or sexist; pay taxes; protectsmall children. We believe that it’s true that you should pay yourtaxes, regardless of what anyone thinks; similarly, we believe thatsomeone who believes that murder is acceptable is wrong  , and hasa false belief.Philosophers call the facts corresponding to claims about whatwe should do (if there are any such facts) moral facts or moraltruths . And philosophers disagree about whether there are moralfacts. I’m going to assume that there are moral facts, but onlybecause they make thinking about the logic of what we should doeasier. If it’s simply true or false that we should eat meat, it’seasier to think about the logic of arguments for and against eatingmeat. Arguments cannot be true or false . The individual steps of an argument can be true or false, and usually the person giving anargument would like the conclusion to be true, but the argumentitself is neither true nor false. Only declarative sentences canbe true or false. Instead of talking about the truth of arguments,we talk about their validity . The two ideas come together in theidea of  soundness ; these are illustrated in the diagram below. truth of individualsentences soundness of wholeargument validity of wholeargumentAll of the arguments we’ve seen so far have been valid . An ar-5
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