is a senior lecturer in philosophy at the University of York in the UK, teaching modules in logic, metaphysics, the philosophy of language and the philosophy of science. His book Properties and Propositions (2021) investigates the metaphysics of higher-order logic.
is a senior lecturer in philosophy at the University of York in the UK, teaching modules in logic, metaphysics, the philosophy of language and the philosophy of science. His book Properties and Propositions (2021) investigates the metaphysics of higher-order logic.
Argument 1: Every time anyone has jumped in the air, they have fallen back to the ground. So, the next time someone jumps in the air, they will fall back to the ground.
Argument 2: The Moon is made of cream cheese. If the Moon is made of cream cheese, then all dogs are history professors. So, all dogs are history professors.
I’m guessing there’s a pretty good chance that you think Argument 1 is better than Argument 2. If someone asked you how you know that the next person to jump in the air is bound to fall back to the ground, you would probably give something like Argument 1.Argument 2, on the other hand, just looks silly. The Moon isn’t made of cream cheese, and even if it were, that would have no bearing on the academic careers of dogs.
However, there is an important way in which Argument 2 is actually the better one. Understanding how it is better sheds light on the very idea of logic.
What is a valid argument?
An argument is essentially a collection of sentences. One of the sentences is the ‘conclusion’, and all the rest are the ‘premises’. The conclusion is meant to follow from the premises. But what exactly does that mean?
Here’s one possibility: a conclusion follows from some premises if they make the conclusion likely or probable. That’s how Argument 1 works. The premise (‘Every time anyone has jumped in the air, they have fallen back to the ground’) makes the conclusion (that the next time someone jumps, they will fall back down) overwhelmingly likely.
But it doesn’t make the conclusion certain. The truth of the premise does not guarantee the truth of the conclusion. As unlikely as it might sound, it is possible that we got the laws of physics all wrong, and that, just sometimes, when a person jumps in the air, they will simply float away. Or imagine that it turned out that we live in a hyper-realistic computer simulation and that, one day, some cheeky hacker turned the gravity off. I’m not saying that you should take these farfetched possibilities seriously. The point is that the premise of Argument 1 makes the conclusion only very likely, not certain.
Argument 2, on the other hand, is different. The premises of that argument are obviously ridiculous. But if they were true, then the conclusion would have to be true too. The truth of the premises would guarantee the truth of the conclusion.
Logicians call arguments like this ‘valid’. When an argument is valid, that means it is absolutely impossible for the conclusion to be false if the premises are true. Validity is the fundamental logical notion. (Most logicians would define logic as the study of validity.) Argument 2 is valid, and Argument 1 is not. That is the important sense in which Argument 2 is better.
It matters whether an argument is valid or not. Imagine that a friend of yours claims to have an argument against the existence of God. And let’s also say that you happen to believe in God. How should you reply to your friend? Well, that depends on whether their argument is valid or not. If it isn’t, then you might well just shrug it off. Perhaps you will be willing to concede that their argument makes it seem less likely that God exists. But if you still have lots of good reasons for believing in God, then you may still think that, on balance, you should continue believing.
However, everything changes if your friend’s argument is valid. In that case, you can’t just shrug it off. If your friend’s premises are all true, then their conclusion – that God doesn’t exist – must be true as well. So, if you want to maintain your belief, you need to explain which of their premises is false.
Validity doesn’t just matter for big philosophical questions, like whether God exists. We all make arguments all the time: whenever you offer a reason for believing something, or for doing something, you are making an argument. For example, if you find yourself explaining why you disagree with some government policy, or even why you think a friend’s comment at lunch was rude, you are making an argument. Sometimes, those arguments will be valid in the sense we are interested in here. When they are, they will be all the more impactful: if the premises are true, then the conclusion must be true too.
Now that we have the notion of validity on the table, and have some sense of why it matters, we can start thinking about how to get better at checking the logic of an argument.
Key points
A valid argument is an especially strong type of argument. When an argument is valid, it means that, if its premises were true, then its conclusion would have to be true, too.
Consider an argument even if you reject a premise. An argument might be valid and informative even when you think one of its premises is questionable.
Eliminate ambiguities. To determine whether an argument is valid, you need clarity about what its words and premises actually mean.
Use your imagination. An argument is invalid if there is a possible version of the world where the premises are true, yet the conclusion is false. To test an argument, see if you can think of such a world.
Avoid known fallacies. Learn about informal fallacies (like the straw-man fallacy) as well as more formal kinds to help you spot deceptive flaws in an argument.
Consider an argument even if you reject a premise
First, let’s appreciate something that is clearly illustrated by Argument 2, about dogs and a cream-cheese moon: an argument can be valid even if it has what seem like false premises.
Of course, it’s best if an argument is valid and has true premises. Logicians call arguments like that ‘sound’. Sound arguments are even better than valid ones: if an argument is valid, then its conclusion must be true if its premises are true; but if an argument is sound, then its conclusion must be true, full stop. So, it might be tempting to automatically dismiss an argument if you think it starts with any false premises. But that would be a mistake, for two reasons.
The first reason is that it would reflect a fundamental misunderstanding of logic itself. Logic does not care about what happens to be true. It cares about the structure of truth, about the laws of truth. Logic concerns itself with validity, the way that the truth of some premises would be enough to guarantee the truth of some conclusion.
The second reason is more practical. You might think that a premise of some argument is false, but you might be talking to someone who disagrees. Let’s say that I believe lying is always wrong, no matter what. You might well disagree: you think it’s bad to lie in general, but you insist that there are some special circumstances where lying is acceptable, such as when it’s the only way to keep someone else’s secret. Even though you disagree with me, you can still treat me with intellectual respect and take my belief seriously. And that involves working out what validly follows from it. By working that out, you would get a better sense of how the world looks from my point of view. For example, if lying is always wrong, then it validly follows that it is wrong to lie even if telling the truth comes at a personal cost. So you can conclude that, if I have the courage of my convictions, I should be willing to pay some price in my pursuit of honesty. (Or, if it turns out that I’m not, then I should at least be willing to take some criticism for my behaviour!)
Eliminate ambiguities
To figure out whether an argument is indeed valid (or not), you need to be absolutely clear on what it would take for the premises to be true – and whether the truth of the premises would guarantee the truth of the conclusion.
That’s easier said than done. A big part of the problem is that ordinary language is often ambiguous. Most obviously, we sometimes use words that have more than one meaning. Consider this example:
Argument 3: Joe had a picnic at the bank. Sharon deposited her money at the bank. So, Joe had a picnic where Sharon deposited her money.
Whether this argument is valid or not depends on whether we are using the word ‘bank’ in the same way throughout the argument. If we are, then it is. But that would be pretty strange. It is more natural to think that Joe had a picnic at a river bank, and Sharon deposited her money at a financial bank. If that is what we meant, then the argument is not valid.
That is one kind of ambiguity. Here is another:
Argument 4: Joe had a picnic, and Joe went to the shops, or Joe visited a museum. So, Joe had a picnic.
Whether this argument is valid depends on how we clear up an ambiguity in the structure of the premise. We might be saying that Joe had a picnic, plus he either went to the shops or visited a museum. In that case, the argument would be valid. But if we’re instead saying that either Joe had a picnic and went to the shops or he just visited a museum, the argument would not be valid.
If you want to check whether an argument is valid, you need to do everything you can to eliminate any ambiguities from the premises and conclusions. That is partly a matter of making sure that you are clear about how you are using individual words: what do you or the person you’re arguing with mean when you use words like ‘wrong’, ‘good’, ‘healthy’ or other key terms? Eliminating ambiguity is also a matter of making sure that it’s clear how sentences are structured, and thus what the premises and conclusion are actually saying. Being absolutely clear about these things can sometimes make an argument seem more than a little pedantic. But pedantry is often the price of validity.
Use your imagination
Even when you have done all you can to chase out ambiguity from the premises and conclusion, it can still be hard to tell whether an argument is valid. Somehow, you need to figure out whether the truth of the premises would guarantee that the conclusion is true. How do you do that?
A big part of it involves using your imagination. You need to do your best to imagine a possible world – a way the world could be – where the premises are true but the conclusion is false. The possible worlds you come up with do not need to be especially realistic. An argument is valid only if there is no possible world, no matter how outlandish, where the premises are true and the conclusion is false. (Think back to Argument 1: I introduced some pretty outlandish possible worlds to show that that argument was not valid.)
This exercise might strike you as silly, or a waste of time. Suppose you and I are discussing something important. Maybe we are still arguing about the wrongness of lying. I tell you that lying is always wrong (premise), and people should avoid doing wrong things (premise) – so lying should never be permitted (conclusion). Would it be a waste of time to assess that argument by trying to imagine a possible world where the premises are true and yet the conclusion is false?
I want to make clear that this would not be a waste of time. In this example, you and I might both come to recognise that my argument is invalid, because (you point out) there’s an imaginable situation where someone must choose between two wrong things – such as lying to keep someone’s secret, or instead causing someone great harm by breaking their trust. In light of this possibility, it seems that, even if my premises are true, the conclusion might still be false. Lying might sometimes be permissible.
I could then try to revise my conclusion to make the argument valid. Maybe I settle on something like: lying should never be permitted, unless it is the least wrong thing to do. Perhaps my new argument is valid, perhaps it isn’t. To figure that out, we’d need to get back to work and see if we can come up with any possible worlds that would make this new conclusion false while still making the premises true. But whether this argument is ultimately valid or not, we can clearly make progress in our discussion together by exploring these possible worlds.
It is important to remember why valid arguments are special: if an argument is certified valid, then it just isn’t an option to accept the premises and reject the conclusion. But a valid argument can provide this kind of absolute guarantee only because there is no possible world where the premises are true and the conclusion is false. So, if we want our arguments to be valid, we need to get good at imagining as wide a variety of possible worlds as we can.
Of course, having said all of this, it’s worth remembering that, even if an argument is not valid, the premises may still make the conclusion very likely. That is how things went for Argument 1. And, for many purposes, that is good enough. But it is not good enough for logic. In logic, we set our sights higher: we demand validity, and nothing less.
Avoid known fallacies
When you are trying to figure out whether an argument is valid, it can help to be on the lookout for fallacies. A fallacy is any style of argument that isn’t valid but wins people over anyway. Some of the best-known fallacies are informal. For example, the ad hominemfallacy involves suggesting that a conclusion is false because it is believed by people you don’t like. Another common one, the straw-man fallacy, involves criticising a distorted version of someone’s view, a version that no one really holds. The straw-man fallacy is especially common in political debates: we’ve all seen politicians misrepresent their opponents’ views in absurd and exaggerated terms.
But not all fallacies are informal; there are formal fallacies too. These are invalid arguments that look valid (at least to most people). My favourite example* comes from the beginning of Aristotle’s Nicomachean Ethics, where he made something like this argument:
Argument 5: Everyone aims to do some good. So, there is some good – the ultimate good – that we’re all aiming for.
Plenty of philosophers have been convinced by this argument. But it is really invalid. To see why, just compare Argument 5 with this one:
Argument 6: Everyone was born on some day. So, there is some day – the ultimate birthday – that everyone was born on.
This argument must be invalid, because the conclusion does not follow from its (true) premise. It’s true that everyone was born on some day, ie, everyone has a birthday. But different people often have different birthdays. This style of invalid argument has a name: it is a quantifier-shift fallacy. Quantifiers are words that we use to make generalisations, like ‘everything’, ‘everyone’, ‘something’, ‘someone’, etc. It turns out that the order of quantifiers really matters (every-some versus some-every), and you can’t just swap them around and expect your argument to be valid. But that is exactly what Argument 5 does. Even if it turned out that we all aim to do some good, different people might be aiming for completely different kinds of good.
There are a lot of logical fallacies, and many philosophers and logicians have compiled lists of them that you can refer to. One of the best known, and earliest, is On Sophistical Refutations, which was written by Aristotle himself.
* Some Aristotle scholars deny that he ever really meant to give this argument. But it is still a good example.
Final notes
As useful as lists of fallacies can be, putting together such lists is not something that philosophers or logicians tend to do very much anymore. Instead, we have developed general tools for evaluating the validity of any argument that we care to consider. These tools are called formal logic. So, my final bit of advice for anyone who would like to dive deeper is to learn some formal logic.
There are lots of logic textbooks out there, and they can be expensive. However, that has started to change. There has been a massive increase in the number of freely available logic resources on the internet. I am a particularly big fan of a logic textbook called forallx (2010), originally written by P D Magnus, a philosopher at the University at Albany, State University of New York. In an extreme act of generosity, he released his textbook for free under a Creative Commons Licence. This means that everyone can download it without charge and even modify it however they like. I use my own revised version at the University of York, and that is also free for anyone to download.
If you do take the plunge into formal logic, you will not only learn more about testing the validity of arguments. The way that you think about how individual sentences are put together, and about how multiple sentences hang together in an argument, will change for good. This new way of thinking can be difficult, and it is sometimes a little slow, but it is precise, powerful and beyond reproach.
