Tom Stoneham's Reading List

"I chose Logic Primer by Colin Allen and Michael Hand for the reason that I taught from it for over a decade at the University of York. One of the interesting things about teaching logic at a university is that no logic teacher at a university is happy with anyone else’s textbook. This is why there are so many logic textbooks: everyone gets hyper-frustrated with the text they’re teaching and ends up writing their own. Now, I’m quite lazy, and I didn’t. I stuck to this book, though actually I changed it in lots of ways. When I teach with it, I reorder it, I delete sections, I add in new sections and new definitions of terms, so in practice the students are learning from my annotated version of the text. But this is why so many logic textbooks are written. The solution to that problem has arisen in our Web 2.0. I’ll mention it for reference, namely that there is now a logic textbook which is open-source and freely editable, called forallx . It’s online, and more and more logic teachers are saying ‘I’ll take that, and I can edit it in any way I like and use it.’ Anyone can freely access not only the original version of the text, but also any of its modifications. So there’s a Cambridge version of this textbook, a York version, a Calgary version, a SUNY version, a UBC version and probably many more I don’t know about. But the underlying formal language and system is the same in all of those. “Effectively, formal logic is a very general form of algebra” Let me go back to Logic Primer and why I like it so much. I like it because it doesn’t explain anything. Allen and Hand say, in the preface, that it’s intended to be used in conjunction with someone giving lectures who’ll do the explanations. They say they don’t really think you can learn logic from this book alone. I think that’s false—I’ve known students who failed to turn up to all my lectures who still managed to do well in the exam by teaching themselves from this book! This book presents a formal system of logic in its clearest, most structured form. I’ll just read from the preface, where they describe what they do: “The text consists of definitions, examples, comments and exercises.” As you go through the text, every paragraph is labeled as either a definition, an example, a comment or an exercise. Exactly. And if your mind is prepared to engage with that structure, then absolutely everything you need to learn logic is there. If something doesn’t work, if you keep getting an exercise wrong, you can go back to the definition and ask yourself, ‘Did I use the definition correctly?’ These definitions are incredibly carefully crafted. They’re not crafted to be easy to understand; they’re crafted to make sure that everything works perfectly if you follow the definitions strictly. Exactly. Most logic textbooks try to soften the blow of what a formal language is like, and how explicit and rulebound it is, by giving lots of examples, by trying to make it feel natural and comfortable. Many logic lecturers do the same: they’re worried that people are going to be put off, and so they try to say, ‘It’s OK, this isn’t too far out of your comfort zone’. Whereas this book, Logic Primer , doesn’t have any of that at all. It just says, ‘Here it is, bare bones, follow the rules, it’ll all work.’ All that is gone from this book. If you’re teaching from it, it’s great because you can put in as much or as little of that as you want. And if you’re wanting to teach yourself logic, you’ve got everything you need and nothing that you might not need in there. So that’s a really nice feature of it. The type of logic in this book—there are different types of formal logic, usually categorized by their proof system, i.e. how you manage to prove things in that logic—is called a natural deduction proof system. You might think that means it feels very natural when you use it. It doesn’t. The way you prove something in this system is you start with your premises and you end with your conclusion. All the bits in between can feel very unnatural, because it’s formal logic and you have to follow these very strict rules. Interestingly, the authors didn’t invent a new system—they used one that was in a previous textbook, E. J. Lemmon’s Beginning Logic , which was first published in 1965 and was the standard textbook in Oxford for a very long time. But it’s turgid. So, there are two books that you could use to learn exactly the same set of rules. (I’ll come back to this idea that there might be different rules and systems in my fifth choice.)"

"Yes, for someone who’s motivated and already has some aptitude, for example who enjoys mathematics. If you found algebra fun at school, you’re probably going to get on well with Logic Primer . My second choice is another textbook that you could use to learn logic yourself. In fact, I was given it by a maths teacher while I was at school, who thought I was getting bored in maths lessons. This is Wilfrid Hodges’ book, which is just called Logic . It’s a Penguin book and has been used by several universities as a textbook. This book sets logic more in the context of the humanities than mathematics. It’s written for someone who has an interest in the workings of language and the clever things you can do (and not do) with language. In that sense, yes, it’s still doing logic; it’s still going to be formal; it’s still going to have symbols; but it’s a much softer, gentler introduction, appealing to a different curiosity. It’s also a book that’s written in such a way that if you didn’t want to learn formal logic for the purpose of doing an exam in the subject—completing the exercises and the quizzes—but you wanted to get a really good sense of what it was like , you could read this book without having to learn all of the techniques. It has other virtues, as well. From the point of view of learning logic, I think it has the best discussion of relations. A sentence like ‘The ball is red’ has a subject (‘ball’) and what logicians call a predicate (‘is red’), which says the ball has a property. So the predicate ‘is red’ applies to one thing, or group of things like the apples in the bowl, but what it applies to is taken as a single subject. When I say ‘Mary is my daughter’, we have a relation there between two subjects. There’s my daughter and me. Then we’ve got a relation between the two, which in this example is a biological relation, a family relation. But there are lots of other relations: to the right of, larger than, smaller than. So, relations typically are parts of language that pick out not a feature of one thing or collection of things, as predicates do, but something structural holding between two or more things. Relations have their own logic. We can say, ‘If John is taller than Peter, and Peter is taller than Fred, then John is taller than Fred.’ That’s an inference in natural language and when we start using formal logic we also want to use such inferences. That would be the logic of relations. Hodges does this particularly well in his book, and of the textbooks I’ve looked at and used, I think Hodges’ account is the best. “As a form of mind-training it is very good because it forces you to pay attention to the details of exactly what is being said and exactly what is meant.” The other thing to say about this textbook in contrast to Logic Primer is it uses a different logical system. I said that Logic Primer is a natural deduction system; you start with your premises and you try to reach your conclusion, so you’re moving through steps to try to get to your conclusion. Hodges uses a different system, which is called a tree proof system. I won’t go into the details, but it’s very graphical, very visual. I talked about truth preservation and validity earlier. When trying to prove that some conclusion follows from certain premises—if you accept these premises, then you must accept this conclusion—that’s equivalent (nice logician’s term there) to saying that if you accept these premises and deny this conclusion, you’re committed to a contradiction. What a tree proof system does is it starts with the premises, denies the conclusion, and then tries to show that there’s no way of avoiding contradiction."

"The next book is Mark Sainsbury’s Paradoxes . I love this book. Whole university courses are taught around this book. It’s an absolute classic. Sainsbury starts with logical reasoning. I’ve talked about validity and defined it as a logical property. I’ve also talked about how when you learn some formal logic, you learn this very distinctive way of thinking or reasoning. What Sainsbury is saying is: let’s stay within that way of thinking, not ordinary or common sense reasoning, not what would be acceptable in a normal conversation, but a logician’s way of reasoning, where you’re sticking strictly to the truth, not deviating, not saying more or less. When doing this, it doesn’t matter if what you conclude is slightly absurd, as long as it’s true. Over the history of philosophy , philosophers have identified a group of puzzles or problems that are called paradoxes. Sainsbury introduces a logician’s definition of a paradox, which is: a paradox occurs when you start from some premises which seem obviously true, and you reach a conclusion which seems obviously false, by obviously good reasoning. This is a problem—it seems that you can use this special logical form of reasoning to go from apparent truths to apparent falsehoods. A very famous example is the liar paradox. Its simplest formulation is the statement, ‘This sentence is false.’ Now ask yourself, is that statement true or false? If it’s true, then what it says is the case. And what it says is that it’s false. So if it’s true, it’s false. So it can’t be true. What if it’s false? Well, if it’s false, then what it says is not the case. But what it says is that it’s false. If that’s not the case, it’s not false, so it must be true. So, if it’s false, it’s true. So it can’t be false. “Most universities in the world that teach philosophy teach logic as a compulsory course in the early stages.” We have a sentence here—a single sentence—which is a paradox. Because if it’s true it’s false, and if it’s false it’s true. We’re stuck. Every statement is either true or false, and it can’t be both. Yet here we have a statement that doesn’t seem to fit into that. That’s a very famous example of a paradox that’s been around for a very long time. It’s called the liar paradox because of a variation in which the Cretan Epimenides says ‘All Cretans are liars.’ If what he says is true, then he’s a liar, and so what he says is false… Sainsbury explores a selection of these paradoxes. Another (in)famous one is the paradox of the heap. You have a heap of sand and take away one grain of sand; it doesn’t stop being a heap of sand. A heap of sand less one grain is still a heap of sand. Take away another grain, it’s still a heap. Eventually, you’ll get down to one grain or no grains, and you definitely haven’t got a heap of sand. It seems like we’ve got an acceptable form of logical reasoning: if something is a heap of sand, then one grain fewer will still be a heap of sand. You just keep applying this and you get to a conclusion you can’t accept, which is that one grain of sand is a heap of sand. It is another example of where we appear to use logical reasoning to go from something we all accept to something we can’t accept. That’s the fun thing about the study of paradoxes. There’s no universal solution to all paradoxes, and there are many different types of paradox. In each case, we have to work out what the best solution is. It might be that the obvious truths we began with were mistaken. Something wasn’t as obviously true as we thought it was: perhaps 99 grains of sand is a heap but 98 grains is not. Or it might be that the logical reasoning we’ve used is faulty in some way and we have to revise it. Or it might be that the conclusion that we thought was unacceptable is something we just have to end up accepting and bite the bullet. With the liar paradox, the problem is if it’s true it’s false and if it’s false it’s true, and that looks like an unacceptable conclusion, because we can’t allow that it’s both true and false. Some logicians – called dialethists –conclude that there are some special statements which are both true and false, just a small set, and we can use tools like the liar paradox to identify them. They accept the apparently unacceptable conclusion. Others might say it’s neither true nor false. Others might try to challenge the reasoning. So there are different ways to respond to a paradox, but they quickly take us into very deep philosophical waters. Sainsbury takes the way of thinking you learn from doing and studying formal logic and shows that the traditional paradoxes are all cases of acceptable premises and acceptable reasoning leading to unacceptable conclusions. He then shows the different ways you might respond, and the philosophical interest of those different responses. That’s a way into logic where you can see that the application of logical thinking generates philosophical problems itself, and it tests our ability to think in this particular way about the truth. Take the paradox of the heap. In practical life, no one’s going to care about that. If you go on about it at the beach, someone’s just going to come and kick the sand in your face. But it generates a philosophical puzzle. That’s the interest of what Sainsbury’s doing. It’s a very different way into logic. You don’t need to know formal logic to grasp this book. He uses a bit of symbolization, but that’s fairly simple. If you’re okay with basic algebra, it won’t be unfamiliar. The way he writes is very easy to follow, but you need to be interested in this logical way of thinking to get the point of what he’s doing."

"Despite having a Latin title, it’s not written in Latin ; it’s written in German. Quite. In a way, this follows on from the Sainsbury book, because in it we see the limits of logical thinking. When struggling with the paradoxes we seem to have reached or even transgressed the limits of thinking. Wittgenstein’s book is about how we understand the thinkable and the unthinkable, which is a traditional philosophical problem. In this book, Wittgenstein approaches the problem from the point of view of formal logic. It’s worth reading Bertrand Russell’s preface to the book, where he summarizes how the book proceeds very simply: “The logical structure of propositions and the nature of logical inference are first dealt with. Thence, we pass successively to Theory of Knowledge, Principles of Physics, Ethics and finally the Mystical.” Support Five Books Five Books interviews are expensive to produce. If you're enjoying this interview, please support us by donating a small amount . This is a fascinating and puzzling book. It’s absolutely clear that Wittgenstein starts with an interest in formal logic and that distinctive way of thinking which is concerned with truth, accuracy and precision. He doesn’t take this as an end in itself, but thinks it is the route into solving the really big questions Russell mentions. He goes on to say, “[Wittgenstein] is concerned with the conditions for accurate Symbolism [Russell’s using ‘Symbolism’ here to mean symbolic representation of the world] i.e. for Symbolism in which a sentence ‘means’ something quite definite.” Wittgenstein is building his philosophy—trying to solve philosophical problems—by starting with the conception of what language can and should do that is embedded in formal logic. It’s not the natural language approach to talking about the world; it’s the formal logic approach to talking about the world. Wittgenstein uses this starting point to get to some very big conclusions. Wittgenstein’s approach reminds us of what I was saying earlier about the second way of thinking about formal logic, namely as a self-standing language. Wittgenstein is saying we all possess natural language, but when we want to focus on the precise and exact expression of truth and the relationship between truths, we need to move into these formal languages where everything is defined explicitly. He is claiming that when you do that, you can start solving the big philosophical problems. For me that’s the fascination of the book, but I should warn that there are very different interpretations of it around. I’d be very careful about that. The interpretation of the book is very controversial and has been increasingly so for the last 20 years. Most commentaries on the book are highly partisan, they’re driving an agenda, and therefore not particularly introductory. If you forced me to recommend one, it would be David Pears’ – it certainly helped me find my way through on first reading. Monk’s book is certainly helpful, but the TLP is more Euclidean than the aphoristic style of Wittgenstein’s later philosophy. The structure of it is seven numbered propositions. Under all of them except number seven—I’ll come to number seven in a second—we have sub-propositions. The first proposition is “The world is all that is the case”, and then under that we get proposition 1.1, “The world is a totality of facts, not of things.” So that’s an elucidation of 1. But then we get 1.1.1, so this is going into an elucidation of 1.1, and so on. A very useful way to read the book is one that wasn’t available to its original audience. We’re used to bullet points and collapsing bullet point structures and this consists in nested bullet points. One of the things I would recommend the reader is to go through and identify the seven master propositions, and then identify the propositions immediately below them, and so on. I’ll just mention proposition seven, which has no sub-propositions, and thus in a sense is the conclusion of the book. In the translation I tend to use, which is Pears and McGuinness, it is “What we cannot speak about we must pass over in silence.” This drives the historically dominant interpretation of Wittgenstein: that if you start with this logician’s conception of accuracy and precision of language, sticking to only what is true and only truth-preserving consequences, then there are some very, very sharp limits to what we can say. And that’s it. You’ve got to stop at that point. The controversy over the book’s interpretation is over what Wittgenstein thinks human beings may also be able to do as well as logic. There’s a suggestion by Wittgenstein that there may be other forms of human expression or intellectual activity which allow us to engage with the things we can’t engage with through logical languages. A famous early positivist criticism of the book was by Frank Ramsey, who pithily said, “What you can’t say, you can’t say, and you can’t whistle either.” That’s why Russell mentions ethics, because a lot of the immediate critics (and followers) of Wittgenstein thought he was pushing ethics into the non-factual and making it less important, subjective and a matter of taste. Whereas what we know of him is that this was not his intention at all. This dispute has driven the more recent interpretations which say Wittgenstein is showing the limits of truth-directed, fact-speaking – logical – discourse, not the limits of human expression and human engagement with reality."

"My fifth choice is Willard Van Orman Quine’s book Philosophy of Logic . I have introduced two books for learning formal logic, formal systems, and formal languages. I have discussed two books which apply the thinking that’s captured in formal languages, and not well-captured in natural languages, to philosophical problems. In contrast, Quine’s book is about when we construct a formal logic, when we create these formal languages, then we’re making philosophical decisions or choices about how we do it. The Philosophy of Logic is all about the philosophical arguments that underlie the decisions to do logic in one way or another. There are potentially an infinite number of different formal logics, and every textbook will be slightly different, so decisions have to be made. Quine is trying to pick out the most important types of decision made when creating a formal language, and looking at the philosophical considerations behind those. I’ll give an example from towards the end of the book. I talked earlier about the law of excluded middle, sometimes called tertium non datur . That’s the principle we came across when talking about the liar paradox: that if you’ve got a well-formed grammatical statement, which has the grammatical form that says something is true or false, then either it’s true or false. It’s not both and it’s not neither. Now, a classical logic—which is the sort of logic that’s in the books I’ve cited—will always stick to that. But when we’re thinking about the options in constructing a logic, we might wonder, ‘Is that right? Do we always want to do that?’ And the dialethists I mentioned are an example of philosophers who reject the principle of non-contradiction. Take the paradox of the heap. Take 14 grains of sand: is that a heap, or is that not a heap? In classical logic you have to decide. For any predicate either it applies or it doesn’t apply. There’s no choice and no alternative. With natural languages, that doesn’t always seem the case, and there may be other examples which are less paradoxical. Take cases where we’ve been mistaken about the existence of something. At one point in the history of astronomy, in order to explain some unusual features of the motions of Mercury, it was postulated that there was an unobserved planet which exerted a gravitational pull on Mercury. There was a hypothesis and the name ‘Vulcan’ was introduced for this planet. “We have a sentence here—a single sentence—which is a paradox. Because if it’s true it’s false, and if it’s false it’s true. We’re stuck.” Now consider the statement: Vulcan is a planet. Is that true or false? Well, it’s not true—because there is no planet Vulcan. But if we say it’s false, then surely we’d have to say that Vulcan is not a planet. Then what is it? An asteroid? Therefore we don’t want to say it’s not a planet either. So it looks like our statement has failed to say anything true or anything false. It’s failed to get into the truth-speaking game, despite being grammatically fine. If you decide that you want to be able to allow sentences like that in your formal logic, then you’re going to have to give up the law of excluded middle. You’re going to have to say, ‘Some statements can fail to be either true or false.’ Once you have done that, you will have to make other choices in your logic to keep it consistent. That is just one example and Quine is interested in the many different decisions logicians have to make. While some are basic choices about the syntax and vocabulary of formal logic, others raise complex philosophical issues. Quine is clear that these are decisions, and logicians can go alternative ways. He tries to persuade us that some options are preferable, and he talks about where our disagreement would lie if we made different choices. On fundamental questions, like the law of non-contradiction, he calls making different choices ‘changing the subject’. When you learn logic in a university context as a philosophy student, it’s the only exam you take where you can get a hundred per cent. Everything is either right or wrong. Consequently, it looks entirely objective and factual, but that’s only because the students taking that exam are learning one particular logic. Each logic is explicitly defined, so once you choose a logic, every exam answer is definitive. But that choice of logic is precisely where the interesting philosophy comes in. And personally I think you’re right, there are different logics. Going back to our starting point—the two different ways of thinking about formal logic—if you thought of formal logic as capturing the universal features of all languages, then you’d think there’s just one true logic, and that philosophers are arguing about which is the right logic, which are the correct choices to make. On that view, these are arguments about how to formalize natural languages to get at their hidden logical features. But when you get into the details of those philosophical disagreements, the view that there’s just one true logic seems wildly implausible. In contrast, if you think of a formal logic as a new language we’ve created for a particular purpose, then we have any alternative logics and some are good for some purposes, and others for different purposes. They are more like computer programming languages, as you said earlier. We might think that some logics, for example the dialethic logics I mentioned, in which some statements can be both true and false, would be very risky logics to use if you were a scientist or an engineer. Equally, fuzzy logic might be good for washing machine programmes but not for airplane safety systems. We may even conclude that some logics are ruled out for most humanly important purposes, but they’re still there, and you can study them and learn them. True, it’s not a case of anything goes in logic – if a logic allows arguments which are not truth-preserving (or that don’t preserve a truth-like property such as probability or provability) then it isn’t really a logic at all. What I am saying is that it’s a case of going back to understanding that formal logic is a tool for human purposes. When we do the philosophy of logic, we must move away from being mathematicians and back to being humanists. All these technical tools are fascinating, and enjoyable to study for their own sake, but the driving question should be: what can I use this one for and what can I use that one for? When will a formal language allow me to do something better or more easily than a natural language? Of course, I don’t want to denigrate the pure study of logic, which has both value in itself and for the student. However, we shouldn’t mistake the precision and clarity of formal logic for a deep insight into the laws of truth."