Philosophy of Logic

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