Before I get to the books, may I start by asking the most obvious question, which is what is logic?
A bit like ‘philosophy’, ‘logic’ is a word with many different currencies and different uses, so the best way to concretize this is to say what we’re really talking about here is what is sometimes called ‘ formal logic. there are two ways of understanding formal logic that are subtly and importantly different.
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The first and most common, the one used in universities when teaching formal logic, is to think of it as a particular type of study of the very general properties of languages; that is, the natural languages, the languages that we all speak and write. one of the things all languages do is allow us to speak true or false. they also allow us to make connections between the different truths we speak of. if we think one thing is true, then we can commit to thinking something else is true. the most common conception of formal logic is that it says that all languages have this interest in truth. they may have many other things to do as well, but the interest in the truth is common to all of them and is clearly very important. so let’s take those bits of language where we care about truth and falsity and the relationships between truths, and see if we can make those properties explicit.
It is a working assumption of this approach that when we make it explicit for one language, we could do the same for any other language. In other words, when speakers of different languages are engaged in talking about what is true or false and have no other interests, then the languages are perfectly intertranslatable. that is quite defining of this conception of formal logic.
It’s interesting that you’re talking about truth, because that makes it sound a bit like epistemology, as in, how do we know things are true? but logic is not usually thought of as a branch of epistemology.
That’s a very good point. logic is not concerned with which sentences are true; it has to do with patterns of truth. If we take the truth-asserting group of speech acts – ‘making a statement’ is often the favored phrase – the question is what are the relationships between these different statements? that’s what logicians study.
why would 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 statements that we both agree on, and eventually we get to a point where you accept something that you didn’t previously accept based on those statements that we started by agreeing on. we have taken a set of statements that are accepted as true, and then we have worked out what other statements we have to accept if we have accepted them as true. that relationship between sets of statements is of primary interest. it is a very particular conception of the argument that we have appealed to here: the idea that we are 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 a set of statements which we call 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. validity is sometimes called “truth preservation”, for a very good reason: by going from some given truths to accepting more truths, you are truth preserving. you stay in the realm of truth. it’s less about which statements are true than how to maintain the truth once you have a few.
“I often say when teaching logic, ‘don’t use this at home or you’ll end up unhappily single.'”
but there is always an exception! once we start doing logic, we discover that there are some statements that have to be true in any case. these are sometimes called the logical truths. Let’s take an example of what is called “the law of the excluded middle”. I’ll try to take a fairly uncontroversial one: either the moon goes around the earth, or the moon doesn’t go around the earth. now, that seems to be true only by virtue of logic. You don’t need to know anything about the moon to know that that statement is true: you have to understand the sentence “the moon orbits the earth,” but you don’t need to know if it’s true. the statement ‘either the moon revolves around the earth, or the moon does not revolve around the earth’ is true only by virtue of logic. so in addition to validity, those relationships between premises and conclusions, logicians are also interested in logical truths and how they come to be true.
To get back to where I was in this conception of formal logic, we are saying that there are sentences in all languages (such as the one about the moon going around the earth) that are logical truths and that there are arguments in all languages that they are valid, or preservers of truth. These properties of logical truth, of validity, occur in any language that can be used to say truths or falsehoods. what formal logic does is try to capture those properties in 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 that everyone starts by agreeing to abide by. in natural languages we let the meaning develop and emerge and then dictionaries try to capture some of it and we find out how rich and complex it is, and so on. what formal logic is trying 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 starts the process (sometimes called ‘symbolization’, sometimes called ‘formalization’) where we go from some natural language, it could be any language, and we turn it into these new symbols and explicit terms and definitions. and because they have explicit definitions, you can manipulate them and discover new things about what has been said.
It becomes more like math or algebra, at that point.
yes. at that point, you are using the fact that you have an explicit set of definitions to take advantage of the techniques of math and algebra. indeed, formal logic is a very general form of algebra.
I certainly understand that sense of logic you’ve described. what was the other sense, the second way of approaching the logic that you mentioned?
One of the problems with that first sense of logic is that natural languages don’t fit these explicit definitions particularly well. if you are interested in logic, you will find that there are libraries full of philosophers discussing how to map the terms of natural language to the terms and symbols of formal logic. take a very simple word like ‘or’. people write books and articles on how to map the English word ‘or’ to the logical disjunction symbol; 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 rig this process of symbolization or formalization a bit to hide the controversies. that may make you suspect that we are not really investigating the universal properties of all languages; perhaps what we are trying to do is force an abstract structure into our languages.
There is a very different way of thinking about formal logic, much more the way of thinking of a mathematician, 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 only the truth, the relationship between truths, valid arguments and logical truths, we can’t do it very well in natural languages, they are not made for that kind of project.
then, according to this conception, what logicians do is create artificial languages with many definitions and explicit rules. we make all meanings and grammatical rules absolutely explicit. we begin by defining the exact use of each symbol, making it clear that any use outside of this exact form is nonsense, in this artificial language. so this language is not going to be as expressive as natural language, but because we have created it, you understand it and we can teach it. so what formal logic does is allow us to say, ‘here’s another tool’. we have natural languages: English, French, German, Spanish, Chinese or Arabic. and we can use them for some purposes. but for other purposes we must pass to this formal language.” so actually we just created a special language for a particular purpose.
that’s a different way of thinking about formal logic that moves away from those hard questions about how to effectively translate from natural language to formal logic: this process of symbolization and formalization 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.”
It’s a bit like computer programming.
yes, a lot. 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 switch between them for different purposes. the question of “is this a correct or accurate translation/symbolization/formalization?” It’s not important. the important thing is that we choose the right language tool for the job.
That’s very interesting, but what’s the point? why should anyone study logic?
That’s a good question. often when philosophers are asked this, they will say that it helps you reason better, or that it helps you do science better, or something of the sort. but the truth is that if you try to teach a microbiologist logic, he will discover that he is not interested. It doesn’t help them do their job. so it is 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 get 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 sometimes help with particular problems about a disagreement in another area. you can say, ‘well, let’s deal with this formally’.
Logic sometimes clears up 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 less) . 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 realm of truth—that raises interesting questions, and we’ll talk about some of them more. ahead.
great. from my point of view, it also imposes a kind of precision on you as a thinker, because you can’t do that unless you’re extremely precise about what you mean by the terms you’re using.
yes. as a form of mental training it is very good because it forces you to pay attention to the details of exactly what is said and exactly what is meant. which can be very useful. it can also be immensely irritating to your partner.
or anyone!
I often say when teaching logic, “don’t use this at home or you’ll end up unhappily single.” but there are particular contexts in which it is very useful. we think that lawyers have a particular skill in this area. A lawyer’s skill has a particular purpose in mind and a particular way of resolving disagreements, namely the court systems. while the logician’s skill and attention are met with a different purpose, which is the preservation of truth rather than agreement, and with a different method of settling disagreements. so train your mind well. That’s probably why most universities in the world that teach philosophy teach logic as a required course in the early stages.
Let’s move on to the logic books you’ve chosen. the first is called the logic primer.
I chose Colin Allen and Michael Hand’s Basic Logic Manual because I taught it for over a decade at York University. One of the interesting things about teaching logic in a university is that no logic professor in a university is happy with anyone else’s textbook. that’s why there are so many logic textbooks: everyone gets so frustrated with the text they’re teaching and ends up writing their own. Now, I’m pretty lazy, and I didn’t. I stuck to this book, although I actually changed it in many ways. when I teach with it, I rearrange it, remove sections, add 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 this problem has emerged in our web 2.0. I’ll mention it for reference, namely that there is now a logic textbook that is open source and freely editable, called forallx. it’s online, and more and more logic teachers say ‘I’ll take it, and I can edit it any way I want and use it’. Anyone can freely access not only the original version of the text, but also any of its modifications. So there is a Cambridge version of this textbook, a York version, a Calgary version, a Suny version, a UBC version, and probably many more that I am unaware of. but the underlying formal system and language is the same in all of them.
“indeed, formal logic is a very general form of algebra.”
let me go back to the logic manual 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 is meant to be used in conjunction with someone who is lecturing and explaining. They say they don’t really think you can learn logic from this book alone. I think that’s false: I’ve met students who didn’t attend all my lectures and still managed to do well on the exam learning from this book on their own.
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This book presents a formal system of logic in its clearest and most structured form. I’ll just read the preface, where they describe what they do: “The text consists of definitions, examples, comments, and exercises.” As you progress through the text, each paragraph is labeled as a definition, an example, a comment, or an exercise.
It is simple but fascinating, almost from a sociological or psychological point of view, to see someone thinking so clearly or organizing things so clearly. it’s almost like a surgeon preparing to perform an operation: the scalpels are on this tray, the sutures are here, everything is neatly organized.
exactly. and if your mind is ready 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 crafted with incredible care. they are not designed to be easy to understand; they are designed to ensure that everything works perfectly if you strictly follow the definitions.
In a sense, it’s both showing and telling. he’s actually demonstrating the virtues of precision as well as talking about it.
exactly. most logic textbooks try to soften the impact of how formal a language is, and how explicit and rule-bound it is, by giving lots of examples, trying to make it feel natural and comfortable. a lot of logic professors do the same thing: they worry that people will get discouraged, so they try to say, “okay, this isn’t too far out of your comfort zone.” while this book, a basic manual of logic, has none of that. it just says, “here it is, basic, follow the rules, everything will work”.
I have never taught formal logic, but I have taught critical thinking. there is the problem that whichever example you use, students get caught up in the details of the example and forget that we are talking about a particular move or paradox or whatever.
everything is gone from this book. if you’re teaching from it, that’s great because you can put in as much or as little as you want. and if you want to learn logic by yourself, you have everything you need and nothing you don’t need there. so that’s a very nice feature.
The type of logic in this book (there are different types of formal logic, generally categorized by their proof system, i.e. how you go about proving things in that logic) is called the proof system of natural deduction. you might think that means it feels very natural when you wear it. It doesn’t. The way you prove something in this system is that you start with your premises and 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.g. J. Lemmon’s Early Logic, which was first published in 1965 and was the standard textbook at Oxford for a long time. but it is turgid. so there are two books you could use to learn the exact same set of rules. (I’ll come back to this idea that there can be different rules and systems in my fifth option).
what is your second option in your list of logic books? the former sounds like something that could really work for the motivated autodidact.
yes, for someone who is motivated and already has some aptitude, for example, who enjoys math. If you thought algebra was fun in school, you’ll probably do well with the logic manual.
My second option is another textbook that you could use to learn logic yourself. in fact, it was given to me by a math teacher while I was in school, who thought I was getting bored in math lessons. This is Wilfrid Hodges’ book, which is simply called Logic. is a penguin book and has been used by several universities as a textbook.
This book places logic more in the context of the humanities than mathematics. it is written for someone who has an interest in how language works and the clever things you can do (and not do) with language. in that sense, yes, he continues to make logic; it’s still going to be formal; it’s still going to have symbols; but it’s a much softer and gentler introduction, appealing to a different curiosity.
It’s also a book that is written in such a way that if you didn’t want to learn formal logic in order to take a test on the subject (complete the exercises and tests), but wanted to get a very good sense of what it was like, you could read this book without having to learn all the techniques. it also has other virtues. From a logic learning point of view, I think it has the best discussion of relationships.
what are relationships?
A sentence like “the ball is red” has a subject (“ball”) and what logicians call a predicate (“is red”), which says that 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 ‘maria is my daughter’, we have a relationship between two subjects. we are my daughter and me. so we have a relationship between the two, which in this example is a biological relationship, a family relationship. but there are many other relationships: to the right of, greater than, less than. therefore, relations are typically parts of language that select not a feature of a thing or collection of things, as predicates do, but something structural that holds between two or more things.
relationships 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 relationships. Hodges does this particularly well in his book, and of the textbooks I’ve seen and used, I think Hodges’ account is the best.
“As a form of mental training it is very good because it forces you to pay attention to the details of what exactly is said and what is exactly meant.”
The other thing to say about this textbook in contrast to the logic manual is that it uses a different logic system. I said that the logic primer is a system of natural deduction; you start with your premises and try to reach your conclusion, so you work your way through the steps to try to reach your conclusion. hodges uses a different system, which is called a tree proof system. I won’t go into details, but it’s very graphic, very visual.
I talked earlier about preserving truth and validity. in trying to prove that some conclusion follows from certain premises, if you accept these premises, then you must accept this conclusion, that is equivalent (excellent logical term) to saying that if you accept these premises and deny this conclusion, you are committed to a contradiction. what a tree proof system does is start with the premises, negate the conclusion, and then try to show that there is no way around the contradiction.
bright. that could actually lead to the next book quite nicely.
The next book is Mark Sainsbury’s Paradoxes. I love this book. Entire college courses are taught around this book. is an absolute classic.
sainsbury begins with logical reasoning. I have 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 us stay in that way of thinking, not in ordinary or common sense reasoning, not in what would be acceptable in normal conversation, but in the way of reasoning of a logician, where you stick strictly to the truth, without deviating, without saying more or less. when you do this, it doesn’t matter if what you conclude is a bit absurd, as long as it’s true.
Throughout the history of philosophy, philosophers have identified a group of puzzles or problems that are called paradoxes. Sainsbury introduces the logical definition of a paradox, which is: A paradox occurs when you start from some premises that seem obviously true and arrive at a conclusion that 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 paradox of the liar. its simplest formulation is the statement, “this sentence is false”. now ask yourself, is that statement true or false? if it is true, then what it says is the case. and what he says is that it is false. So if it’s true, it’s false. so it can’t be true.
what if it’s fake? well, if it is false, then what it says is not the case. but what he says is that it is false. if that is not the case, it is not false, then 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 required course in the early stages.”
We have a sentence here, a single sentence, which is a paradox. because if it’s true it’s a lie, and if it’s a lie it’s true. we are stuck each statement is either true or false, and cannot be both. however, here we have a statement that doesn’t seem to fit that. that is a very famous example of a paradox that has been around for a long time. it is called the liar paradox because of a variation in which the Cretan Epimenides says “all Cretans are liars”.
sainsbury explores a selection of these paradoxes. another (in)famous one is the heap paradox. you have a lot of sand and you remove a grain of sand; it’s just a pile of sand. a heap of sand minus a grain is still a heap of sand. remove another pimple, it’s still a bunch. eventually, you’ll get a grain or no grain, and you definitely won’t have a lot of sand.
It seems that we have an acceptable form of logical reasoning: if something is a pile of sand, one less grain will still be a pile of sand. you keep applying this and you come to a conclusion that you cannot accept, which is that a grain of sand is a heap of sand. is another example of where we seem to use logical reasoning to go from something we all accept to something we can’t accept.
what’s the reaction then? 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 of studying paradoxes. There is no universal solution to all paradoxes, and there are many different kinds of paradoxes. in each case, we have to find out what is the best solution. it could be that the obvious truths we started with were wrong. something was not as obviously true as we thought: maybe 99 grains of sand is a lot, but 98 grains is not. or it may be that the logical reasoning we have used is flawed in some way and we need to revise it. or it may be that the conclusion we thought was unacceptable is something we have to end up accepting and bite the bullet.
With the liar paradox, the problem is that if it’s true it’s false and if it’s false it’s true, and that seems like an unacceptable conclusion, because we can’t allow it to be both true and false. some logicians, called dialecticians, conclude that there are some special statements that are both true and false, just a small set, and we can use tools like the liar’s paradox to identify them. accept the seemingly unacceptable conclusion.
others might say that it is neither true nor false. others might try to challenge the reasoning. so there are different ways to respond to a paradox, but they quickly lead us into very deep philosophical waters.
sainsbury takes the way of thinking that is learned by doing and studying formal logic and shows that traditional paradoxes are all cases of acceptable premises and acceptable reasoning leading to unacceptable conclusions. then shows the different ways you might respond and the philosophical interest of those different responses.
that’s quite a different way to go into logic.
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That’s a way of getting into logic where you can see that the application of logical thinking raises philosophical issues of its own, and tests our ability to think in this particular way about truth.
take the heap paradox. in practical life, no one is going to care about that. if you continue that on the beach, someone will come and kick the sand in your face. but it generates a philosophical puzzle. that is the interest of what sainsbury is doing. it is a very different way of entering logic. you don’t need to know formal logic to understand this book. uses a bit of symbolization, but that’s pretty simple. if you’re good with basic algebra, it won’t be unfamiliar. the way he writes is very easy to follow, but you have to be interested in this logical way of thinking to understand what he is doing.
your next pick is a notoriously difficult book to understand in its entirety, but possibly relatively simple to understand the key message, which is presumably 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 is not written in Latin; it is written in German.
parts of it could also have been . . .
quite. In a way, this follows Sainsbury’s book, because in it we see the limits of logical thought. struggling with paradoxes we seem to have reached or even transgressed the limits of thought.
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 is worth reading the preface to Bertrand Russell’s book, where he summarizes how the book proceeds very simply: “The logical structure of propositions and the nature of logical inference are dealt with first. from there we pass successively to the theory of knowledge, principles of physics, ethics and finally mysticism.”
This is a fascinating and perplexing book. It is absolutely clear that Wittgenstein begins with an interest in formal logic and that distinctive way of thinking that he is concerned with truth, correctness, and precision. he doesn’t take this as an end in itself, but he thinks it’s the way to solve the really important questions that russell mentions. He goes on to say, “[Wittgenstein] is concerned with the conditions for a precise symbolism [Russell uses ‘symbolism’ here to refer to the symbolic representation of the world], that is, for symbolism in which a sentence ‘means’ something quite definite “.
wittgenstein is building his philosophy, trying to solve philosophical problems, beginning with the conception of what language can and should do that is embedded in formal logic. it is not the natural language approach to talking about the world; it is the formal logic approach to talking about the world. Wittgenstein uses this starting point to reach some very important conclusions.
Wittgenstein’s approach reminds us of what he said earlier about the second way of thinking about formal logic, that is, as an independent language. Wittgenstein is saying that we all have a natural language, but when we want to focus on the precise and exact expression of truth and the relationship between truths, we must move to these formal languages where everything is explicitly defined. he claims that when you do that, you can start to solve the big philosophical problems.
For me, that is the fascination of the book, but I must warn that there are very different interpretations.
Are there any comments you would recommend? Is there anything in the book that might help someone who is reading it on their own?
I would be very careful with that. the interpretation of the book is highly controversial and has become increasingly so over the last 20 years. most of the comments on the book are very partisan, are pushing an agenda, and therefore not particularly introductory. if you forced me to recommend one, it would be david pears; It certainly helped me get my bearings on the first reading.
perhaps the context given by ray monk’s biography would be helpful, and would also explain culturally why he wrote it in the style he did, which is aphoristic.
monk’s book is certainly helpful, but tlp is more euclidean than the aphoristic style of wittgenstein’s later philosophy. its structure is seven numbered propositions. under all of them except number seven, I’ll get to number seven in a second, we have subpropositions.
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 things”. so that’s an elucidation of 1. but then we get 1.1.1, so this goes to an elucidation of 1.1, and so on. a very useful way to read the book is one that was not available to its original audience. we are used to collapsing bullets and bullet structures and this consists of nested bullets. one of the things I would recommend to the reader is to go through and identify the seven master propositions, and then identify the propositions immediately below them, and so on.
I will only mention proposition seven, which has no sub-propositions and is therefore in a sense the conclusion of the book. in the translation i tend to use, which is pears and mcguinness, it is “what we cannot speak of, we must pass over in silence”. this drives wittgenstein’s historically dominant interpretation: that if you start with language’s logical conception of accuracy and precision, holding fast only to what is true and only to truth-preserving consequences, then there are some very, very definite limits to what what can we say and that is. you have to stop at that point.
the controversy over the interpretation of the book is about what wittgenstein thinks human beings can also do besides logic. There is a suggestion from Wittgenstein that there may be other forms of human expression or intellectual activity that allow us to engage with things that we cannot engage with through logical languages. A famous early positivist critique of the book was by Frank Ramsey, who tersely 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 many of Wittgenstein’s immediate critics (and followers) thought he was pushing ethics into the non-factual and making it less important, subjective, and a matter of taste. whereas what we know about him is that that was not his intention at all. This dispute has fueled more recent interpretations that Wittgenstein is showing the limits of logical, fact-based, truth-driven discourse, not the limits of human expression and human engagement with reality.
It’s obviously a classic book with a lot of depth, and everyone would get something from it, but the whole book would take years of work to digest. Let’s take a look at the last of the logic books you’ve chosen.
my fifth choice is the book philosophy of logic by willard van orman quine. I have presented two books to learn formal logic, formal systems and formal languages. I have discussed two books that apply thinking that is captured in formal languages, and is not captured well in natural languages, to philosophical problems. In contrast, Quine’s book is about when we build formal logic, when we create these formal languages, then we are making philosophical decisions or choices about how we do it. the philosophy of logic has to do with the philosophical arguments that underlie decisions to do logic one way or another.
There are potentially an infinite number of different formal logics, and each textbook will be slightly different, so decisions must be made. who is trying to pick out the most important kinds of decisions that are made in creating a formal language and discusses the philosophical considerations behind them.
Could you give an example of that, so it’s clear what you’re saying?
I’ll give you an example of towards the end of the book. I talked earlier about the law of the excluded middle, sometimes called tertium non datur. that’s the principle we’re running into when we talk about the liar’s paradox: that if you have a well-formed grammatical statement, that has the grammatical form that says something is true or false, then it’s true or false. it is not both and it is neither. now a classical logic, which is the kind of logic that’s in the books I’ve quoted, will always stick to that. but when we’re thinking about the options to build a logic, we can ask ourselves, ‘is that correct? do we always want to do that?’ and the dialectists I mentioned are an example of philosophers who reject the principle of non-contradiction.
take the heap paradox. take 14 grains of sand: is that a lot, or is it not a lot? in classical logic you have to decide. for any predicate it applies or it does not apply. there is no choice or alternative. with natural languages this does not always seem to be the case, and there may be other, less paradoxical examples. take cases where we have been mistaken about the existence of something. At one point in the history of astronomy, to explain some unusual features of Mercury’s motions, it was postulated that there was an unobserved planet exerting 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 a lie, and if it’s a lie 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 vulcan planet. but if we say that it is false, surely we would have to say that vulcano is not a planet. then what is that? an asteroid? therefore, we also do not want to say that it is not a planet. so it seems that our statement has not said anything true or false. he failed to get into the game of telling the truth, despite being grammatically correct. if you decide that you want to be able to allow sentences like that in your formal logic, then you will have to give up the law of the excluded middle. you will have to say, “some statements may not be true or false”. once you’ve done that, you’ll have to make other decisions in your logic to maintain consistency.
that’s just an example and who is interested in the different decisions logicians have to make. while some are basic choices about the syntax and vocabulary of formal logic, others raise complex philosophical questions. who is clear that it is about decisions, and logicians can go through alternative paths. it tries to persuade us that some options are preferable and talks about where we would disagree if we made different decisions. on fundamental issues, such as the law of non-contradiction, he calls for different choices to “change the subject”.
It’s interesting. Throughout this discussion, it is almost as if we have been talking about logics in the plural. “logic” makes it sound like there’s one thing that’s taught, I’m going to teach you logic, and there’s only one way logic can be because it’s this kind of crushing system that trumps everything else. but in reality, what has 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 100 percent. everything is right or wrong. consequently, it seems completely objective and factual, but that is only because the students who take that exam are learning a particular logic. each logic is explicitly defined, so once you choose a logic, each answer on the exam is final. 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 that formal logic captures the universal features of all languages, then you would think that there is only one true logic, and that philosophers are arguing about what is the correct logic, what are the correct choices to make. From that point of view, these are arguments about how to formalize natural languages to get to their hidden logical features. but when you get into the details of those philosophical disagreements, the view that there is only one true logic seems wildly implausible.
On the contrary, if you think of a formal logic as a new language that we have created for a particular purpose, then we have alternative logics and some are good for some purposes and others for different purposes. they are more like computer programming languages, as you said before. we might think that some logics, for example the dialectical 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. likewise, fuzzy logic might be good for washing machine programs, but not for airplane security systems. we can even conclude that some logics are discarded for most humanly important purposes, but they are still there, and you can study and learn them.
however, it is not a case of anything goes.
That’s right, it’s not a case of anything-goes logic: if a logic allows arguments that don’t preserve truth (or that don’t preserve a truth-like property, like probability or provability), then it doesn’t is really a logic at all. What I am saying is that it is a question of re-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 of these technical tools are fascinating and enjoyable to study on their own, but the main 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 do not want to denigrate the pure study of logic, which has value in itself and for the student. however, we must not confuse the precision and clarity of formal logic with insight into the laws of truth.
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