Logic is an excellent form of mind-training because it involves a very particular way of thinking and focus on truth. But how does it work and what are its limitations? Tom Stoneham, a professor of philosophy at the University of York, picks some great books for anyone who wants to learn more about logic.

Before we get to the books, can I begin by asking the most obvious question, which is what is logic?

A bit like ‘philosophy’, ‘logic’ is a word with a lot of different currency and different uses, so the best way to nail this down is to say what we’re really talking about here is what’s sometimes called ‘formal logic’. There are two ways of understanding formal logic which are subtly and importantly different.

The first and most common—the one used in universities when teaching formal logic—is to think of it as a particular kind of study of the very general properties of languages; that is, natural languages, the languages which we all speak and write. One of the things that all languages do is allow us to speak truly or falsely. They also allow us to make connections between different truths we speak about. If we think one thing is true, then we may be committed to thinking something else is true. The most common conception of formal logic is that it’s saying all languages have this interest in truth. They may have lots of other things they do as well, but an interest in truth is common to all of them and it’s clearly very important. So let’s take those bits of the language where we’re concerned with truth and falsity and the relationships between truths, and see if we can make those properties explicit.

It’s a working assumption of this approach that when we make it explicit for one language, we could do the same thing for any other language. In other words, when speakers of different languages are engaged in talking about what’s true or what’s false and have no other interests, then languages are perfectly inter-translatable. That’s pretty much definitional of this conception of formal logic.

It’s interesting you’re talking about truth, because that makes it sound a bit like epistemology—as in, how do we know that things are true? But logic is usually not thought of as a branch of epistemology.

That’s a very good point. Logic is not concerned with which sentences are true; it’s concerned with the patterns of truth. If we take the group of speech acts of asserting truths – ‘making a statement’ is often the favoured phrase – the question is what are the relationships between these different statements? That’s what logicians study.

Why do logicians want to study that? The reason is often best explained in terms of arguments. For example, when I give an argument, I start with some claims upon which we both agree, and eventually we get to a point where you accept something that you didn’t previously accept on the basis of those claims we’ve started by agreeing on. We’ve taken a set of statements which are agreed to be true, and then we’ve worked out which other statements we have to accept if we’ve accepted those ones as true. That relationship between sets of statements is the primary interest. It’s a very particular conception of argument that we’ve appealed to here: the idea that we’re moving from some truths to some more truths.

One of the key concepts in formal logic is the concept of validity. An argument is valid, logicians say, when we have one set of statements which we call the premises and if they are true, then this other statement, the conclusion, must be true. Validity is a relationship between the first set of sentences and the conclusion. Sometimes validity is called ‘truth preservation’, for very good reason: by moving from some given truths to accepting more truths, you’re preserving truth. You’re staying in the domain of truth. It’s less about which statements are true than how to keep to the truth once you’ve got some.

“I often say when I’m teaching logic, ‘Don’t use this at home or you’ll end up unhappily single.’”

But there’s always an exception! Once we start doing logic, we discover that there are some statements which have to be true whatever. These are sometimes called the logical truths. Take an instance of what’s called ‘the law of excluded middle’. I’ll try to take a fairly uncontroversial one: either the moon orbits the earth, or the moon does not orbit the earth. Now, it looks like that’s true by virtue of logic alone. You don’t need to know anything about the moon to know that statement’s true: you have to understand the sentence ‘the moon orbits the earth’, but you don’t need to know whether it’s true. The statement ‘either the moon orbits the earth, or the moon does not orbit the earth’ is true by virtue of logic alone. So, as well as validity—those relationships between premises and conclusions—logicians are also interested in the logical truths, and how they get to be true.

To go back to where I was on this conception of formal logic, we’re saying there are sentences of every language (like that one about the moon orbiting the earth) which are logical truths and that there are arguments in every language which are valid, or truth-preserving. These properties of logical truth, of validity, occur in every language that can be used to speak truths or falsehoods. What formal logic does is it tries to capture those properties into a series of explicit definitions. The way we do this is by introducing new terms—I introduced ‘validity’ as a technical term a few minutes ago—and new symbols. Unlike most natural languages, these terms and symbols have very explicit definitions which everyone starts by agreeing to keep to. In natural languages we let meaning develop and emerge and then dictionaries try to capture some of that and we discover how rich and complex it is, and so on. What formal logic tries to do is say: there’s all this richness and complexity in natural language, let’s introduce some special terms and symbols, where we all agree on these explicit terms and explicit definitions and rules for using them. This begins the process (sometimes called ‘symbolization’, sometimes called ‘formalization’) where we go from a bit of natural language—it could be any language—and we convert it into these new symbols and terms and explicit definitions. And because they have explicit definitions, you can then manipulate them and find out new things about what has been said.

It becomes more like mathematics or algebra, at that point.

Yes. At that point, you’re using the fact that you’ve got an explicit set of definitions to draw upon the techniques of mathematics and algebra. Effectively, formal logic is a very general form of algebra.

I certainly understand that sense of logic that you’ve described. What was the other sense, the second way of approaching logic that you mentioned?

One of the problems with that first sense of logic is that natural languages don’t map particularly well onto these explicit definitions. If you get interested in logic, you’ll find that there are libraries full of philosophers arguing about how to map the terms of natural language onto the terms and symbols of formal logic. Take a very simple word like ‘or’. People write books and papers about how you map the English word ‘or’ onto the logical symbol for disjunction; it turns out to be quite controversial, and there are heated disagreements.

Something we face when we teach logic is precisely that problem: we have to fudge a little this process of symbolization or formalization to hide the controversies. That can make you suspicious that we’re not really digging out the universal properties of all languages; perhaps what we’re trying to do is force an abstract structure onto our languages.

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There’s a very different way of thinking about formal logic, much more a mathematician’s way of thinking about it, which is that we create a new language; we say that existing natural languages are wonderful for many things, but they have imperfections. If our obsession is just truth, the relationship between truths, valid arguments and logical truths, we can’t do it very well in natural languages—they’re not made for that sort of project.

So on this conception, what logicians do is create artificial languages with lots of explicit definitions and rules. We make all the meanings and the grammatical rules absolutely explicit. We start by defining the exact use of every symbol, making clear that any use outside this exact way is nonsense, in this artificial language. So this language is not going to be nearly as expressive as natural language, but because we’ve created it, you understand it and we can teach it. Then what formal logic does is allow us to say, ‘Here’s another tool. We’ve got natural languages—English, French German, Spanish, Chinese or Arabic. And we can use them for some purposes. But for other purposes, we must move into this formal language.’ So really, we’ve just created a special language for a particular purpose.

That’s a different way of thinking about formal logic which gets away from those difficult questions about how to translate effectively from natural language into formal logic—this symbolization, formalization process which implies that formal logic is telling you a universal truth about all languages. Instead, we just say, ‘No, it’s a new language we can all learn if we want to. And once you’ve learned it, you can do new things with it.’

Yes, very much so. It’s like computer language, like mathematical language, like particular branches of mathematics. You just have to learn this language and then you can do interesting things with it. As a competent ‘speaker’ of both languages, you can move from one to the other for different purposes. The question of, ‘Is this a correct or accurate translation/symbolization/formalization?’ isn’t important. What is important is that we choose the correct linguistic tool for the job.

That’s very interesting, but what’s the point? Why should anybody study logic?

That’s a good question. Often when philosophers are asked this, they’ll either say it helps you reason better, or it helps you do science better, or something like that. But the truth is, if you try to teach logic to a microbiologist, you’ll find they’re not interested. It doesn’t help them do their job. So it’s not clear that formal logic has a direct, practical application in that sense.

What is true—and as we go through the books, I will come to this point—is that when you learn formal logic, you learn to engage in a particular way of thinking. And that particular way of thinking can then allow you to engage in certain philosophical questions. It can also help sometimes with particular problems about a disagreement in another area. You can say, ‘Well, let’s deal with this in formal terms.’

Sometimes, logic elucidates problems in other areas, but it’s not a universal panacea, and the idea that science would be so much better if we did it in formal logic is—crazy (well, I think it’s crazy, at least). But it is the case that logic involves a very particular way of thinking, a very particular focus on truth—and the relationship between truth and staying within the domain of truth—that raises interesting questions and we’ll talk about some of those later.

Great. From my point of view, it also forces a kind of precision on you as a thinker, because you can’t do it unless you’re extremely precise about what you mean by the terms that you’re using.

Yes. 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. That can be very useful. It can also be immensely irritating for your partner.

Or anyone!

I often say when I’m teaching logic, ‘Don’t use this at home or you’ll end up unhappily single.’ But there are particular contexts where it is very useful. We think of lawyers as having a particular skill in this area. A lawyer’s skill is with a particular purpose in mind and with a particular way of resolving disagreements, namely judicial systems. Whereas the logician’s skill and attention lies with a different purpose, which is truth-preservation rather than agreement, and with a different method of resolving disagreements. So it does train the mind well. That’s probably why most universities in the world that teach philosophy teach logic as a compulsory course in the early stages.

Let’s move on to the logic books you’ve chosen. The first one is called Logic Primer.

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.

It’s simple but fascinating, almost from a sociological or psychological point of view, to see somebody thinking that clearly or organizing things that clearly. It’s almost like a surgeon getting ready to perform an operation: the scalpels are in this tray, the sutures are here—it’s all clearly organized.

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.

In a sense, it’s showing as well as saying. It’s actually demonstrating the virtues of precision as well as talking about it.

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.’

I’ve never taught formal logic, but I’ve taught critical thinking. There’s this problem that whatever example you use, students get caught up in the details of the example and forget we’re talking about the particular move or paradox or whatever it is.

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.)

What’s your second choice in your list of logic books? The first one sounds like something that could really work for the motivated auto-didact.

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.

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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.

What are 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.

Brilliant. That could actually lead quite neatly into the next book.

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.

What’s the reaction, then? Do you say ‘Ah, well there’s something wrong with my logic. Of course, the law of contradiction only holds in some circumstances’?

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 quite a different way into logic.

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.

Your next choice is a notoriously difficult book to understand in its entirety, but possibly relatively simple to understand the key message, which presumably is about the limits of thought, or the meaning of thought. This is Wittgenstein’s first book, Tractatus Logico-Philosophicus.

Despite having a Latin title, it’s not written in Latin; it’s written in German.

Parts of it might as well have been . . .

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.”

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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.

Is there a commentary that you’d recommend? Is there anything about the book to help somebody who’s reading it on their own?

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.

Maybe the context given by Ray Monk’s biography would be useful, and also culturally explain why he wrote it in the style that he did, which is aphoristic.

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.”

Which include ethics, presumably.

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.

So that’s obviously a classic book with a lot of depth in it, and everybody would get something from it, but to take in the whole book would take years of work. Let’s look at the last of the logic books you’ve chosen.

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.

Could you give an example of that, so that it’s clear what you’re saying?

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’.

It’s interesting. Throughout this discussion, it’s almost as if we’ve been talking about logics plural. “Logic” makes it sound as if there’s one thing that gets taught—I’m going to teach you logic—and there’s only one way that logic can be because it’s this kind of crushing system that defeats everything else. But actually, what’s emerged is a series of logics.

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.

It’s not a case of anything goes, though.

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.

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