Paradoxes

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