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 âphilosophyâ, âlogicâ is a word with a lot of different currency and different uses, so the best way to nail this down is to say what weâre really talking about here is whatâs sometimes called âformal logicâ. There are two ways of understanding formal logic which are subtly and importantly different.

The first and most commonâthe one used in universities when teaching formal logicâis 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 itâs 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 itâs clearly very important. So letâs take those bits of the language where weâre concerned with truth and falsity and the relationships between truths, and see if we can make those properties explicit.

Itâs 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 whatâs true or whatâs false and have no other interests, then languages are perfectly inter-translatable. Thatâs pretty much definitional of this conception of formal logic.

Itâs interesting youâre talking about truth, because that makes it sound a bit like epistemologyâas in, how do we know that things are true? But logic is usually not thought of as a branch of epistemology.

Thatâs a very good point. Logic is not concerned with which sentences are true; itâs concerned with the patterns of truth. If we take the group of speech acts of asserting truths – âmaking a statementâ is often the favoured phrase – the question is what are the relationships between these different statements? Thatâs 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 didnât previously accept on the basis of those claims weâve started by agreeing on. Weâve taken a set of statements which are agreed to be true, and then weâve worked out which other statements we have to accept if weâve accepted those ones as true. That relationship between sets of statements is the primary interest. Itâs a very particular conception of argument that weâve appealed to here: the idea that weâre 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 âtruth preservationâ, for very good reason: by moving from some given truths to accepting more truths, youâre preserving truth. Youâre staying in the domain of truth. Itâs less about which statements are true than how to keep to the truth once youâve got some.

âI often say when Iâm teaching logic, âDonât use this at home or youâll end up unhappily single.ââ

But thereâs 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 whatâs called âthe law of excluded middleâ. Iâll 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 thatâs true by virtue of logic alone. You donât need to know anything about the moon to know that statementâs true: you have to understand the sentence âthe moon orbits the earthâ, but you donât need to know whether itâs true. The statement âeither the moon orbits the earth, or the moon does not orbit the earthâ is true by virtue of logic alone. So, as well as validityâthose relationships between premises and conclusionsâlogicians 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, weâre 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 termsâI introduced âvalidityâ as a technical term a few minutes agoâand 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: thereâs all this richness and complexity in natural language, letâs 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 âsymbolizationâ, sometimes called âformalizationâ) where we go from a bit of natural languageâit could be any languageâand 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, youâre using the fact that youâve 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 youâve 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 donât map particularly well onto these explicit definitions. If you get interested in logic, youâll 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 âorâ. People write books and papers about how you map the English word âorâ 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 weâre not really digging out the universal properties of all languages; perhaps what weâre trying to do is force an abstract structure onto our languages.

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Thereâs a very different way of thinking about formal logic, much more a mathematicianâs 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 canât do it very well in natural languagesâtheyâre 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 weâve created it, you understand it and we can teach it. Then what formal logic does is allow us to say, âHereâs another tool. Weâve got natural languagesâEnglish, 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.â So really, weâve just created a special language for a particular purpose.

Thatâs 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 logicâthis symbolization, formalization process which implies that formal logic is telling you a universal truth about all languages. Instead, we just say, âNo, itâs a new language we can all learn if we want to. And once youâve learned it, you can do new things with it.â

Yes, very much so. Itâs 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 âspeakerâ of both languages, you can move from one to the other for different purposes. The question of, âIs this a correct or accurate translation/symbolization/formalization?â isnât important. What is important is that we choose the correct linguistic tool for the job.

Thatâs very interesting, but whatâs the point? Why should anybody study logic?

Thatâs a good question. Often when philosophers are asked this, theyâll 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, youâll find theyâre not interested. It doesnât help them do their job. So itâs not clear that formal logic has a direct, practical application in that sense.

What is trueâand as we go through the books, I will come to this pointâis 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, âWell, letâs deal with this in formal terms.â

Sometimes, logic elucidates problems in other areas, but itâs not a universal panacea, and the idea that science would be so much better if we did it in formal logic isâcrazy (well, I think itâs crazy, at least). But it is the case that logic involves a very particular way of thinking, a very particular focus on truthâand the relationship between truth and staying within the domain of truthâthat raises interesting questions and weâll 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 canât do it unless youâre extremely precise about what you mean by the terms that youâre 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 Iâm teaching logic, âDonât use this at home or youâll end up unhappily single.â But there are particular contexts where it is very useful. We think of lawyers as having a particular skill in this area. A lawyerâs skill is with a particular purpose in mind and with a particular way of resolving disagreements, namely judicial systems. Whereas the logicianâs 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. Thatâs probably why most universities in the world that teach philosophy teach logic as a compulsory course in the early stages.

Letâs 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 elseâs textbook. This is why there are so many logic textbooks: everyone gets hyper-frustrated with the text theyâre teaching and ends up writing their own. Now, Iâm quite lazy, and I didnât. 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. Iâll mention it for reference, namely that there is now a logic textbook which is open-source and freely editable, called forallx. Itâs online, and more and more logic teachers are saying âIâll take that, and I can edit it in any way I like and use it.â Anyone can freely access not only the original version of the text, but also any of its modifications. So thereâs a Cambridge version of this textbook, a York version, a Calgary version, a SUNY version, a UBC version and probably many more I donât know about. But the underlying formal language and system is the same in all of those.

âEffectively, formal logic is a very general form of algebra.â

Let me go back to Logic Primer and why I like it so much. I like it because it doesnât explain anything. Allen and Hand say, in the preface, that itâs intended to be used in conjunction with someone giving lectures whoâll do the explanations. They say they donât really think you can learn logic from this book alone. I think thatâs falseâIâve 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. Iâll just read from the preface, where they describe what they do: âThe text consists of definitions, examples, comments and exercises.â As you go through the text, every paragraph is labeled as either a definition, an example, a comment or an exercise.

Itâs simple but fascinating, almost from a sociological or psychological point of view, to see somebody thinking that clearly or organizing things that clearly. Itâs almost like a surgeon getting ready to perform an operation: the scalpels are in this tray, the sutures are hereâitâs 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 doesnât work, if you keep getting an exercise wrong, you can go back to the definition and ask yourself, âDid I use the definition correctly?â

These definitions are incredibly carefully crafted. Theyâre not crafted to be easy to understand; theyâre crafted to make sure that everything works perfectly if you follow the definitions strictly.

In a sense, itâs showing as well as saying. Itâs 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: theyâre worried that people are going to be put off, and so they try to say, âItâs OK, this isnât too far out of your comfort zoneâ. Whereas this book, Logic Primer, doesnât have any of that at all. It just says, ‘Here it is, bare bones, follow the rules, itâll all work.’

Iâve never taught formal logic, but Iâve taught critical thinking. Thereâs this problem that whatever example you use, students get caught up in the details of the example and forget weâre talking about the particular move or paradox or whatever it is.

All that is gone from this book. If youâre teaching from it, itâs great because you can put in as much or as little of that as you want. And if youâre wanting to teach yourself logic, youâve got everything you need and nothing that you might not need in there. So thatâs a really nice feature of it.

The type of logic in this bookâthere are different types of formal logic, usually categorized by their proof system, i.e. how you manage to prove things in that logicâis called a natural deduction proof system. You might think that means it feels very natural when you use it. It doesnât. 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 itâs formal logic and you have to follow these very strict rules. Interestingly, the authors didnât invent a new systemâthey used one that was in a previous textbook, E. J. Lemmonâs Beginning Logic, which was first published in 1965 and was the standard textbook in Oxford for a very long time. But itâs turgid. So, there are two books that you could use to learn exactly the same set of rules. (Iâll 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 whoâs motivated and already has some aptitude, for example who enjoys mathematics. If you found algebra fun at school, youâre 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. Itâs 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. Itâs 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, itâs still doing logic; itâs still going to be formal; itâs still going to have symbols; but itâs a much softer, gentler introduction, appealing to a different curiosity.

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Itâs also a book thatâs written in such a way that if you didnât want to learn formal logic for the purpose of doing an exam in the subjectâcompleting the exercises and the quizzesâbut 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 âThe ball is redâ has a subject (âballâ) and what logicians call a predicate (âis redâ), which says the ball has a property. So the predicate âis redâ 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 âMary is my daughterâ, we have a relation there between two subjects. Thereâs my daughter and me. Then weâve 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, âIf John is taller than Peter, and Peter is taller than Fred, then John is taller than Fred.â Thatâs 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 Iâve looked at and used, I think Hodgesâ account is the best.

âAs 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.â

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 youâre moving through steps to try to get to your conclusion. Hodges uses a different system, which is called a tree proof system. I wonât go into the details, but itâs very graphical, very visual.

I talked about truth preservation and validity earlier. When trying to prove that some conclusion follows from certain premisesâif you accept these premises, then you must accept this conclusionâthatâs equivalent (nice logicianâs term there) to saying that if you accept these premises and deny this conclusion, youâre 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 thereâs no way of avoiding contradiction.

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

The next book is Mark Sainsburyâs Paradoxes. I love this book. Whole university courses are taught around this book. Itâs an absolute classic.

Sainsbury starts with logical reasoning. Iâve talked about validity and defined it as a logical property. Iâve 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: letâs stay within that way of thinking, not ordinary or common sense reasoning, not what would be acceptable in a normal conversation, but a logicianâs way of reasoning, where youâre sticking strictly to the truth, not deviating, not saying more or less. When doing this, it doesnât matter if what you conclude is slightly absurd, as long as itâs true.

Over the history of philosophy, philosophers have identified a group of puzzles or problems that are called paradoxes. Sainsbury introduces a logicianâs 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 problemâit 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, âThis sentence is false.â Now ask yourself, is that statement true or false? If itâs true, then what it says is the case. And what it says is that itâs false. So if itâs true, itâs false. So it canât be true.

What if itâs false? Well, if itâs false, then what it says is not the case. But what it says is that itâs false. If thatâs not the case, itâs not false, so it must be true. So, if itâs false, itâs true. So it canât be false.

âMost universities in the world that teach philosophy teach logic as a compulsory course in the early stages.â

We have a sentence hereâa single sentenceâwhich is a paradox. Because if itâs true itâs false, and if itâs false itâs true. Weâre stuck. Every statement is either true or false, and it canât be both. Yet here we have a statement that doesnât seem to fit into that. Thatâs a very famous example of a paradox thatâs been around for a very long time. Itâs called the liar paradox because of a variation in which the Cretan Epimenides says âAll Cretans are liars.â If what he says is true, then heâs 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 doesnât stop being a heap of sand. A heap of sand less one grain is still a heap of sand. Take away another grain, itâs still a heap. Eventually, youâll get down to one grain or no grains, and you definitely havenât got a heap of sand.

It seems like weâve 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 canât 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 canât accept.

Whatâs the reaction, then? Do you say âAh, well thereâs something wrong with my logic. Of course, the law of contradiction only holds in some circumstancesâ?

Thatâs the fun thing about the study of paradoxes. Thereâs 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 wasnât 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 weâve 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 itâs true itâs false and if itâs false itâs true, and that looks like an unacceptable conclusion, because we canât allow that itâs both true and false. Some logicians – called dialethists âconclude 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 itâs 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.

Thatâs quite a different way into logic.

Thatâs 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 oneâs going to care about that. If you go on about it at the beach, someoneâs just going to come and kick the sand in your face. But it generates a philosophical puzzle. Thatâs the interest of what Sainsburyâs doing. Itâs a very different way into logic. You donât need to know formal logic to grasp this book. He uses a bit of symbolization, but thatâs fairly simple. If youâre okay with basic algebra, it wonât 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 heâs 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 Wittgensteinâs first book, Tractatus Logico-Philosophicus.

Despite having a Latin title, itâs not written in Latin; itâs 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.

Wittgensteinâs 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. Itâs worth reading Bertrand Russellâs preface to the book, where he summarizes how the book proceeds very simply: âThe 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.â

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This is a fascinating and puzzling book. Itâs 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 doesnât 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, â[Wittgenstein] is concerned with the conditions for accurate Symbolism [Russellâs using âSymbolismâ here to mean symbolic representation of the world] i.e. for Symbolism in which a sentence âmeansâ something quite definite.â

Wittgenstein is building his philosophyâtrying to solve philosophical problemsâby starting with the conception of what language can and should do that is embedded in formal logic. Itâs not the natural language approach to talking about the world; itâs the formal logic approach to talking about the world. Wittgenstein uses this starting point to get to some very big conclusions.

Wittgensteinâs 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 thatâs the fascination of the book, but I should warn that there are very different interpretations of it around.

Is there a commentary that youâd recommend? Is there anything about the book to help somebody whoâs reading it on their own?

Iâd 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, theyâre driving an agenda, and therefore not particularly introductory. If you forced me to recommend one, it would be David Pearsâ â it certainly helped me find my way through on first reading.

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

Monkâs book is certainly helpful, but the TLP is more Euclidean than the aphoristic style of Wittgensteinâs later philosophy. The structure of it is seven numbered propositions. Under all of them except number sevenâIâll come to number seven in a secondâwe have sub-propositions.

The first proposition is âThe world is all that is the caseâ, and then under that we get proposition 1.1, âThe world is a totality of facts, not of things.â So thatâs 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 wasnât available to its original audience. Weâre 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.

Iâll 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 âWhat we cannot speak about we must pass over in silence.â This drives the historically dominant interpretation of Wittgenstein: that if you start with this logicianâs 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 thatâs it. Youâve got to stop at that point.

The controversy over the bookâs interpretation is over what Wittgenstein thinks human beings may also be able to do as well as logic. Thereâs 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 canât engage with through logical languages. A famous early positivist criticism of the book was by Frank Ramsey, who pithily said, âWhat you canât say, you canât say, and you canât whistle either.â

Which include ethics, presumably.

Thatâs 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 â logical â discourse, not the limits of human expression and human engagement with reality.

So thatâs 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. Letâs look at the last of the logic books you’ve chosen.

My fifth choice is Willard Van Orman Quineâs 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 thatâs captured in formal languages, and not well-captured in natural languages, to philosophical problems. In contrast, Quineâs book is about when we construct a formal logic, when we create these formal languages, then weâre making philosophical decisions or choices about how we do it. TheÂ 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 itâs clear what youâre saying?

Iâll give an example from towards the end of the book. I talked earlier about the law of excluded middle, sometimes called tertium non datur. Thatâs the principle we came across when talking about the liar paradox: that if youâve got a well-formed grammatical statement, which has the grammatical form that says something is true or false, then either itâs true or false. Itâs not both and itâs not neither. Now, a classical logicâwhich is the sort of logic thatâs in the books Iâve citedâwill always stick to that. But when weâre thinking about the options in constructing a logic, we might wonder, âIs that right? Do we always want to do that?â 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 doesnât apply. Thereâs no choice and no alternative. With natural languages, that doesnât always seem the case, and there may be other examples which are less paradoxical. Take cases where weâve 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 âVulcanâ was introduced for this planet.

âWe have a sentence hereâa single sentenceâwhich is a paradox. Because if itâs true itâs false, and if itâs false itâs true. Weâre stuck.â

Now consider the statement: Vulcan is a planet. Is that true or false? Well, itâs not trueâbecause there is no planet Vulcan. But if we say itâs false, then surely weâd have to say that Vulcan is not a planet. Then what is it? An asteroid? Therefore we donât want to say itâs not a planet either. So it looks like our statement has failed to say anything true or anything false. Itâs 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 youâre going to have to give up the law of excluded middle. Youâre going to have to say, âSome statements can fail to be either true or false.â 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 âchanging the subjectâ.

Itâs interesting. Throughout this discussion, itâs almost as if weâve been talking about logics plural. âLogicâ makes it sound as if thereâs one thing that gets taughtâIâm going to teach you logicâand thereâs only one way that logic can be because itâs this kind of crushing system that defeats everything else. But actually, whatâs emerged is a series of logics.

When you learn logic in a university context as a philosophy student, itâs 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 thatâs 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 youâre right, there are different logics.

Going back to our starting pointâthe two different ways of thinking about formal logicâif you thought of formal logic as capturing the universal features of all languages, then youâd think thereâs 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 thereâs just one true logic seems wildly implausible.

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In contrast, if you think of a formal logic as a new language weâve 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 theyâre still there, and you can study them and learn them.

Itâs not a case of anything goes, though.

True, itâs not a case of anything goes in logic â if a logic allows arguments which are not truth-preserving (or that donât preserve a truth-like property such as probability or provability) then it isnât really a logic at all. What I am saying is that itâs 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 donât want to denigrate the pure study of logic, which has both value in itself and for the student. However, we shouldnât mistake the precision and clarity of formal logic for a deep insight into the laws of truth.

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