Logic 1

Struggling with logic?

This page is aimed at students of Logic 1h at the University of Edinburgh who are finding the course difficult - though it may be of use to anyone struggling with introductory logic. Of course, there are bits of the course that are quite tricky, and if you have some specific problem you should email me or ask me about it during, before or after a tutorial. This page, however, is inteneded for students who are struggling with most or all of the course. If that's you, then - chances are - one, some or all of the following will apply to you:

• You're confused by logic because it looks completely different from any other subject you've studdied.
• You're concerned that logic is like maths, and didn't find maths easy at school.
• You can't see any connection between logic and the real world.
• You can't see any connection between logic and your other philosophy courses.

Below, I deal with these four problems in turn. If you're in one of my tutorial groups and, after reading through this page, you don't feel more confident that you can pass Logic 1, or if you don't think I've identified the root of your problems with logic then tell me. I can help you, but not if I don't know that you're having problems. If you're not in one of my tutorial groups, then I'm still interested in any feedback you have about this page, but for specific help and advice you should talk to your own tutor.

If your problems with logic do fall into one of the four categories then I hope what I say below will help alleviate them. But you can also draw reassurance from another source, namely that you are not alone. Many students have found logic bewildering at first, but have gone on to pass the course.

What's your problem?

The most important thing is not to panic. If you've studied the same subjects for a number of years at school then each new topic tends to build on what has gone previously and it's easy to forget just how confusing it can be when you start something completely new. But try to cast your mind back to the last time you did learn something new - like how to swim, ride a bike, speak French or complete a Sudoku grid. At first, you probably felt there was so much you didn't understand and couldn't do that you could hardly even imagine yourself being able to succeed. But you did, and now you probably find it almost as hard to imagine not having the skill. Logic can be just the same.

Perhaps the biggest mistake you can make at this stage is to convince yourself that you can't do logic. Being confused at first does not mean that you will still be confused at the end of the course, and it certainly does not mean that you're not clever enough to pass it. Anyone intelligent enough to get to university is intelligent enough to pass Logic 1. Logic can appear very different from things you've done before, but that's no reason to be scared of it. Presumably, part of why you came to university in the first place was to learn new things. Once it clicks, Logic 1 is simple. For much of the course, we give you some reasonably simple rules to follow, you follow the rules, and you can't go wrong. But if you allow yourself to believe that it's too hard for you to understand, then you'll overcomplicate matters and look for some tricky, subtle and obscure point ... that you won't find because it doesn't exist!

Logic 1 uses letters and symbols. Because of that, students with more background in other subjects that use letters and symbols, such as maths and science, tend to feel more comfortable with it more quickly. But the difference is just that - feeling more comfortable - rather than having actual knowledge that helps with the subject. Logic 1 does not assume mathematical knowledge, and whilst logic is fundamental to maths, it's fundamental to almost all other academic disciplines as well. That's because logic is about reasoning - reasoning about anything. True, it is mostly in maths and philosophy (and perhaps computer science) that we formalize logic, but that doesn't mean it doesn't have an important role to play in other subjects.

Fortunately for students not already comfortable with using letters and symbols, the symbolization we use in logic is very simple. In propositional logic, with which we start, letters such as 'P' and 'Q' are used to stand for statements about the way the world could be (or else propositions, but you don't need to worry about the difference between statements and propositions until you do some philosophy of logic at the end of the course). So 'P' could stand for the statement that the world is round, that it is snowing, that Paris is the capial of France, that the moon is made of cheese or any other statement at all. We use 'P' rather than any of these other specific statements because we just don't care which specific statement is meant, or even if any is, and so writing "P" is quicker, more convenient and clutters up the page less. One interesting thing we find out when we start doing propositional logic is that the actual content of the statements often has little to do with whether arguments that involve them are valid. Asking what statements the letters 'P' and 'Q' stand for in logic is a bit like interrupting a joke that starts "Three men walked into a bar ..." by asking what the men's names were - it doesn't matter, it's irrelevant to the joke, they're just men. Similarly, it doesn't matter what statements 'P' and 'Q' stand for, it's irrelevant to the argument, they're just statements - any statements.

Of course, we don't just use letters to stand for statements, we also use symbols to stand for connectives: "¬" for "not", "∧" for "and", "∨" for "or", "→" for "if ... then ..." and, finally, "↔" for "if and only if". We could just use the English words, of course, but there are two main reasons we don't. First, as with the letters, it's quicker, more convenient and clutters up the page less to use the symbols. Second, whilst the logical connectives mean roughly the same as their English-language counterparts, and can replace them in many sentences and arguments, the logical connectives are actually simplified versions of English-language connectives and we use the symbols to distinguish them from the latter. Consider "and", for example. The sentence "I saw a pink elephant and closed my eyes" implies something quite different from "I closed my eyes and saw a pink elephant". But this is a complexity we just don't need - assessing arguments is difficult enough already - so when we start doing logic we introduce simple versions of the connectives and just specify the circumstances under which sentences involving them are true or false (that's what the general truth tables for the connectives do). For example, we specify that both "I saw a pink elephant and closed my eyes" and "I closed my eyes and saw a pink elephant" are true when "I saw a pink elephant" is true and "I closed my eyes" is true, and otherwise they're false. Of course, this can lead to problems when using the logical connectives to formalize an English-language argument, and logicians do worry about this. But you're just starting out, so you don't have to (at least until you get to the philosophy of logic section of the course).

Finally, we put the letters and the symbols together to make what we call "formulas", which sounds a bit mathematical. But we could call the logical formulas "sentences" instead, and, indeed, some logic books do. We don't, as that can cause confusion because one formula can stand for many different sentences. For example, "P ∧ Q" can stand for "I closed my eyes and saw a pink elephant", "It's Saturday and October", "Calcium and mercury are both metals", "The moon is made of cheese and I am a green banana", any other sentence that makes two claims separated by an "and", or, indeed, all such sentences. But that's all the formulas do - they stand for sentences.

People use logic all the time in everyday life. For example, "It had to be you or me that used the last of the milk. It certainly wasn't me so it must have been you", "If you'd locked the door then it wouldn't have been open when I came home, but it was, so you must have forgotten" or "She's only ever nice to me when she wants a favour, so this invitation means she's after something". Often, it's pretty clear when arguments people give are good and when they're not, so why do we bother with formalizing them and developing long-winded ways of testing them, as we do in logic? Well, first, it is often clear when arguments are okay, but often our intuitions about logic let us down. One famous demonstration of this is the Wason Card Problem, but examples of poor reasoning abound in the media (see Julian Baggini's online column Bad Moves for some fun examples). Second, we notice that many of the arguments we recognize as good have the same sort of form, and that what is important is not so much what the argument is about as the form that argument takes. So if we can learn to recognize good argument forms, and bad argument forms, we can say immediately whether an argument is good or bad without having to worry about the details of the subject matter. Third, distinguishing between the assumptions, or premises, of an argument and the reasoning from them to a conclusion helps us assess arguments in a more systematic fashion. If the argument is good, then we know that what we have to do, if we disagree with the conclusion, is attack the assumptions. If it isn't good, then we can accept the assumptions and still disagree with the conclusion. Fourth, all this raises the question of what exactly makes an argument good or bad. Why are the examples given at the beginning of this paragraph okay, but "If it rains then the pavements get wet, and the pavements are wet. Therefore, it must have rained." not okay? The answer seems to be that, in the last example, it's possible that things are just as the first sentence says and yet the second sentence be false: the pavement could be wet for some other reason - because it had been washed, for example. In the first set of examples, on the other hand, it's impossible that the assumptions are correct and the conclusion reached false. And this is just the notion of a valid argument that we use in logic.

Sometimes students get confused about the connection between logic and the real world because they think logic is trying to do something that it's not. There's not much, if anything, that logic alone can tell us about the way the world actually is*. Logic alone can't tell us that, for example, Edinburgh is either in Scotland or England, or that Edinburgh is not in England, but, when put together with those two facts, it can tell us that Edinburgh is in Scotland. Another way of thinking about it is that logic helps determine what is possible and impossible. That's what a truth table, for example, does: it shows the possible combinations of truth values of a particular set of sentences. In the absence of any factual information at all, we can still know, for example, that "It's either Monday or not Monday" is true, that "It's both raining and not raining" isn't true and that "It's neither October nor December" and "It's October" cannot be true at the same time. We can also know that if "If dalmations are dogs, then they're mammals" and "Dalmations are dogs" are both true, then it's impossible that dalmations are not mammals. If we can put this knowledge together with some factual knowledge - that the two quoted statments in the previous sentence are true - then we can know that dalmations must be mammals. Obviously, this isn't a very impressive bit of knowledge, but exactly the same principle holds for more complex arguments about more important subjects.

Finally, the above may still not have convinced you that logic is useful. Or else, you may be able to see that understanding what makes reasoning good or bad and what must follow from given assumptions is useful, but not be able to see how what you've learned in lectures and tutorials can help you accomplish that for any but the most basic and dull arguments. In either case, it's fine to worry about such things, but not to the exclusion of actually learning logic. It may not immediately obvious how logic is useful, but then you shouldn't expect it to be. You're only just starting, after all, and you need to learn a fair bit about what logic is before you can see how it can be useful. Complaining that you can't see how logic is useful when all you've learned are some very basic bits of it is a bit like learning a basic stocking stitch in knitting and then complaining that it's not useful for making a jumper because, well, how can you shape armholes? So don't expect too much of either yourself or the very basic logic you're learning. Certainly you can try to relate it to the real world, but do that on your own time. For now, your priority should be to learn what the lecturer is teaching you without worrying too much about why. Much of what you're learning is basic and mechanical, and it's perfectly possible to learn how to do it without knowing why you're doing it. Once you get to to the end of the course, you can think again about whether and how logic is useful, and you'll actually have some information to base your judgement on.

* If you do more philosophy courses then you will probably come across the debate about what, if anything, logic alone can tell us about the world.

When you start doing philosophy, you generally start by doing Introduction to Philosophy 1 and Logic 1. At first glance, the subjects appear very different. In Introduction to Philosophy 1, you read and discuss texts written in English and will be assessed on the basis of essays similarly written in English. In Logic 1, we have all those letters and symbols, and in the examination you have to construct truth tables and semantic tableaux. Furthermore, unless you opt to do further courses in logic later in your degree, most of the philosophy you will do will look more like Introduction to Philosophy 1 than like Logic 1. Why, then, do we insist on your doing Logic 1?

First, the superficial dissimilarity between the courses hides a deeper connection. Both are about arguments. True, in Introduction to Philosophy 1 you will be looking at specific arguments rather than arguments in general as in Logic 1, but basic concepts that we explore in Logic 1, such those of validity, consistency, necessary and sufficient conditions and counter-examples, are fundamental to all philosophy. Almost every text you read in philosophy will be setting out an argument - making certain assumptions and drawing conclusions from them - or criticizing the argument some other philosopher has given (which is just to argue that some other philosopher's argument is flawed). Whenever you read a philosophical text, you should be asking yourself: "What are the assumptions, here?", "What is the argument?", "Is it valid?". Because you don't yet have the knowledge you'll gain in Logic 1, we don't expect you to do much more - at first - in Intorduction to Philosophy than use your intuitions about what moves in the argument you're studying are good or bad, but, as mentioned above, untutored intuitions about inferences can lead us astray, and the more complex the arguments are, the less reliable they are. So, at the same time as starting you studying some typical philosophical arguments in Introduction to Philosophy, we introduce you to fundamental concepts associated with arguments in general and the formal techniques for assessing them in Logic, so that, in time, you will develop the skills to assess the arguments you encounter in philosophy more systematically and rigourously. That is not to say that you will be able to formalize the arguments in the philosophical texts that you encounter - they are usually too complex for that, though you probably will come across basic argument forms such as modus ponens, modus tollens, disjunctive syllogism and reductio ad absurdum. However, even without being able to formalize the arguments, the concepts and rigour you learn in Logic 1 will make you better at spotting arguments and holes in arguments, and help you to construct better arguments yourself.

Of course, logic is important in more subjects than just philosophy. If a historian were to claim "World War II would not have started had the Nazis not come to power, but the resentment in Germany caused by excessive reparations imposed after World War I may have been enough to trigger WWII even under a less extremist regime", then that would be as problematic as a philosopher making two conflicting claims. So why is do we study logic in philosophy and not other subjects? The main reason is that the focus on arguments in philosophy is far more central - it is what is distinctive about philosophy. A student of history will spend a lot of time learning historical facts, a physics student learning physical facts, a biology student learning biological facts and so on. By contrast, there are few, if any, analogous philosophical facts that a student of philosophy can learn (most of the facts you learn will be facts about what various philosophers have argued!). Instead, much of what we do in philosophy is take facts - or alleged facts - from other academic disciplines, or just commonly held assumptions, and ask what follows from them, and whether they lead us into conflict with other alleged facts or commonly held assumptions. A couple of examples may help:

• In philosophy, we might ask whether the different societies' possession of different moral codes means that there can be no absolute right and wrong. Obviously, if it weren't plausible that different societies do have different moral codes then we probably wouldn't bother asking the question, but whether they do or not is not a philosophical question - that's something for historians, anthropologists and so on to confirm or deny. The peculiarly philosophical part of the problem is whether it follows from societies' having different moral codes that there is no absolute right or wrong.
• Another philosophical problem is whether, if all phyisical events have a sufficient physical cause, what goes on in our minds (such as making choices) can make any difference to the physical world - whether there can be such a thing as 'mental causation'. Again, that all physical events have a sufficient physical cause is a plausible enough assumption, but whether it is actually true is more properly the concern of scientists - probably physicists. For philosophers, the key concern is whether mental causation is consistent with all physical events having a sufficient physical cause.

The questions addressed in philosophy are incredibly diverse. What ties them together - makes them all philosophy - is the way these problems are addressed: by means of identifying and assessing the assumptions and arguments involved. The ability to critically assess arguments, and develop good arguments of your own, is perhaps the most important skill you will develop over the course of your philosophy degree. The study of logic, which is the study of arguments in general - reasoning in general - is an essential step in developing this skill.

In summary...

Hopefully the foregoing has been useful to you. I'll conclude with a quick summary of the most important tips for cracking Logic ! mentioned above.

• Don't convince yourself you can't do Logic. You can, as long as you remain calm and determined to succeed.
• Don't overcomplicate things. What we are asking you to do in Logic 1 is, for the most part, mechanical and straighforward. It can take a little while to see what we want of you, but it will click if you don't start thinking that, because you don't see it immediately, we must be asking you to do something tricky, subtle and complicated.
• Don't let the superficial similarities with mathematics mislead you into thinking that, if you found maths difficult at school, then you will find Logic 1 similarly difficult. Logic is about reasoning - it does not assume any mathematical knowledge.
• Don't spend your time worrying about why you're doing logic to the exclusion of learning how to do logic. It may not be immediately obvious how logic connects with either everyday concerns or the rest of philosophy, but until you have learned what logic you shouldn't expect to see why it's useful. Just focus on learning what is required of you, then you can think again about its application.

Finally, I'll repeat something I said at the top of this page. If you are struggling with the course (and are one of my students), then tell me. I can help you, but not if I don't know that you're having problems.

Acknowldgements

This page is based on numerous conversations with both tutors and students of logic, but particular thanks are due to David Harris and Lynne Turnbull.