Arguments: A User's Guide

A "no tears" intro to informal logic

Every intro-level course in analytic philosophy features a primer on arguments.

This is my primer.

It’s written for (i) people new to philosophy and (ii) wizened philosophers feeling nostalgic. Enjoy!

Philosophers love arguments—attempts to persuade you of some conclusion by laying out supporting reasons. Arguments come in all kinds of forms, from quick and breezy to long and technical. Some are mind-blowing. Some are…meh.

Here are some examples of arguments you’d find in the wild:

• You shouldn’t go in there! That’s where the Yeti lives.

• The flat earth theory is easy to disprove. During a lunar eclipse, the earth casts a shadow over the moon. That shadow is roughly circular the entire time. But that would only happen if the earth were roughly spherical.

• I subscribed to Big iff True this morning, and then I won the lottery. Clearly, subscribing to Big iff True is a shrewd way to get rich.

These are all arguments, not in the sense of being heated disputes, but in the sense that they give reasons for a conclusion.

How can you tell if an argument is any good? What do you do if you’re confronted with an argument that’s fishy, incomplete, or ambiguous? How do you make sure your own arguments are clear and compelling?

Standard Form

Arguments have two main parts: premises and a conclusion. A premise is a reason given in support of the conclusion; the conclusion is the thing that the argument is trying to establish.

To make it clear which part is which, most philosophy classes ask you to write out arguments in standard form. This just means that you write a list of the premises, followed by a clearly marked conclusion. For example, here’s how we would put some of the above arguments in standard form:

1. The Yeti lives in there.

   —————

   Therefore: you shouldn’t go in there.

Credit: Jason Snell

1. The earth casts a constant round shadow during a lunar eclipse.

2. If the earth is flat, the earth does not cast a constant round shadow during a lunar eclipse.

   —————

   Therefore: The earth is not flat.

Now, it doesn’t really matter how exactly you mark off the conclusion. You might preface it with “so, “therefore,” or “∴” for short. You might add in a line between the premises and conclusion. Whatever you like. The important thing about standard form is that you can tell, just by looking at your argument, which things are premises and which are conclusions.

In standard form, conclusions go at the end. But beware. In the wild, conclusions might arrive at the start of an argument, they could pop up in the middle. One virtue of standard form is that it forces us to un-scramble our arguments.

If your argument is a proof, you can cap it off with a “Halmos”—∎, □, or ▯—named after the great Paul Halmos, author of classics like Naive Set Theory

Validity, Soundness, Persuasiveness

So what makes an argument good?

An ideal argument has true premises that support the conclusion in a clear and persuasive way. To make this more precise, we need some new concepts.

A deductively valid argument—or just “valid” for short—is one whose premises guarantee the truth of the conclusion. The inference from the premises is “truth-preserving.” What this means is that there’s no possible way for the premises to be true and the conclusion to be false.

For example, here’s a nice valid argument:

1. All cats meow.

2. Snowball is a cat.

   —————

   Therefore: Snowball meows.

And here’s a valid argument actually used by the ancient Greeks:

1. During a lunar eclipse, the earth casts a constant round shadow.

2. If the earth is flat, then during a lunar eclipse, the earth does not cast a constant round shadow.

   —————

   Therefore: The earth is not flat.

Now here’s an invalid argument.

1. The Yeti lives in there.

   —————

   Therefore: You shouldn’t go in there.

This isn’t a terrible argument. Presumably, if the Yeti really does live there, it’s a dangerous place. But the premise doesn’t actually guarantee the truth of the conclusion—so the argument isn’t valid. Same with:

1. I subscribed to Big iff True this morning.

2. I won the lottery this afternoon.

   —————

   Therefore: All subscribers to Big iff True win the lottery.

And same for:

1. Most cats meow.

2. Snowball is a cat.

   —————

   Therefore: Snowball meows.

Why are these invalid?

Because in each case, there is some possible way for the premises to be true while the conclusion is false. It’s possible that the Yeti is a gracious host, so maybe you should go hang out in his cave. It’s possible that Snowball is one of a minority of cats that don’t meow. And it’s possible—indeed, it’s very likely—that even if I win the lottery, other subscribers do not.

It’s worth noting that the premises of an “invalid” argument can still lend support to a conclusion. If most cats meow, then given that Snowball is a cat, Snowball probably meows.

But “probably” isn’t enough to make the argument valid, in our sense (“deductive validity”). Validity is a very strict standard. That’s what makes it so helpful. If you can give a (deductively) valid argument in standard form, then it’s guaranteed, not just probable, that the conclusion follows from the premises.¹

Now here are some quiz questions. Give them the old college try, then read the answers below.

Quiz #1

Q1. Is the following argument valid? Why or why not?

1. All pigs fly

2. Porky is a pig.

   —————

   Therefore: Porky flies.

Q2. Is the following argument valid? Why or why not?

1. Some Australians live in Melbourne.

   —————

   Therefore: Some Australians live in Melbourne.

Q3. Can you add a premise to the following argument to make it valid?

1. The Yeti lives in the cave.

   —————

   Therefore: You shouldn’t go in the cave.

Answers

Q1.

The answer is yes. The argument is valid because if the premises are true, then the conclusion must be, too. If (1) all pigs fly, and (2) Porky is a pig, Porky cannot but fly.

Of course, the argument is a bit fishy because it has a clearly false premise—namely, premise 1. But a false premise doesn’t make an argument invalid, only unsound.

A sound argument is a valid argument whose premises are all true. If an argument is sound, then the conclusion has to be true. (You have true premises and a truth-preserving inference to the conclusion.) That is why philosophers are always trying to convince you that their arguments are sound. And that is why, if you want to dispute an argument’s conclusion, you have to show that the argument is not really sound. Either it has a false premise, or the conclusion doesn’t follow from the premises—or worse, both!

Q2.

This argument has the conclusion as a premise:

1. Some Australians live in Melbourne.

   —————

   Therefore: Some Australians live in Melbourne.

But it’s still valid! Remember: the test for being valid is whether the conclusion can be false in a possible scenario where all the premises are true. But clearly, in any scenario where some Australians live in Melbourne, some Australians will be living in Melbourne!

Moreover, the argument is perfectly sound (which, again, means it’s valid with true premises). Clearly some Australians live in Melbourne!

The problem with this argument is that it isn’t persuasive. A persuasive argument—as I’ll define it—is one that could rationally convince someone of the conclusion. This means someone might believe the argument’s premises without yet believing the conclusion; the argument persuades by drawing the link. But if the “link” is painfully obvious, then there is no need for the argument, and the argument has no persuasive power. The above argument is an extreme version of this. Anyone who believes the premise automatically believes the conclusion!

There are other, less extreme ways in which an argument can be unpersuasive (even if it’s valid or even sound). For example, it might have very controversial premises, or it might have premises that are more controversial than the conclusion. Consider:

1. Everything on Big iff True is true.

   —————

   Therefore: The latest Big iff True post is true.

It’s going to be very hard to convince anyone of that big first premise. Let’s hope that there are other, more persuasive ways of defending my blogging.

Q3.

Here we have an invalid argument:

1. The Yeti lives in the cave.

   —————

   Therefore: You shouldn’t go in the cave.

There are plenty of ways—in fact, infinitely many—to make an argument valid by adding premises. The most straightforward way is to add an “if” statement that links the prior premises to the conclusion, as in:

2. The Yeti lives in the cave.

3. You shouldn’t go into any place with the Yeti in it.

   —————

   Therefore: You shouldn’t go in the cave.

This argument has some hope of being persuasive. By contrast, the argument would be totally unpersuasive if we just added the conclusion:

1. The Yeti lives in the cave.

2. You shouldn’t go in the cave.

   —————

   Therefore: You shouldn’t go in the cave.

Same goes for:

1. The Yeti lives in the cave.

2. Nobody should ever go anywhere.

   —————

   You shouldn’t go in the cave.

And this has the added problem of an implausible second premise.

Premises: Weak and Strong

No wonder it’s hard to come up with good arguments. If the premises are too weak, the argument won’t be valid. But if they’re too strong, the argument won’t be persuasive (in the sense of possibly persuading someone who agrees with the premises while being unsure of the conclusion).

In logic, we say one sentence P is “stronger” than a sentence Q if P implies Q though Q doesn’t imply P. We say P and Q are “equivalent” in strength if they imply each other, and “incomparable” if neither is stronger nor are they equivalent.

Some examples:

• “I had at least three coffees” is stronger than “I had at least one coffee.”

• “She has a dog” is weaker than “She has a gigantic dog.”

• “She has a dog and a cat” is equivalent to “She has a cat and a dog.”

• “She has a dog” is incomparable with “She has a cat.”

In a valid argument, the conclusion is never stronger than the premises. But you want the premises to be as weak as you can get them. That way, people will be more likely to agree with your premises, and you’ll be able to persuade more people of your conclusion!

Validity in virtue of form

Putting arguments into standard form is a craft, but also something of an art, requiring you to use your judgment to state your premises and conclusion clearly, distinctly, and succinctly.

Let me give you a tip. In general, you want your arguments to be not just valid, but obviously valid.

Consider this argument:

1. Some of Oscar’s products are hot dogs.

   —————

   Therefore: Some of Oscar’s products are sandwiches.

Is this valid?

It’s not obvious! To know if the conclusion follows, we have to settle a contentious question: do hot dogs necessarily count as sandwiches? (In favor of yes: they feature meat with bread on either side. In favor of no: the bread is connected.)

More seriously, consider this very famous argument:

1. Every human fetus has a right to life.

   —————

   Therefore: It is wrong to kill a human fetus.

Whether this argument is valid crucially depends on whether it is, as a matter of moral fact, necessarily wrong to kill something with a right to life. Some people will see the argument as valid; others won’t.

It would be better to make the argument’s validity undeniable, as in:

1. Some of Oscar’s products are hot dogs.

2. All hot dogs are sandwiches.

   —————

   Therefore: Some of Oscar’s products are sandwiches.

And:

1. Every human fetus has a right to life.

2. If a being has a right to life, then it is wrong to kill that being.

   —————

   Therefore: It is wrong to kill a human fetus.

These arguments are obviously valid. What makes them obviously valid is that they are valid not just in virtue of some controversial definitions (like the hot-dog inclusive definition of sandwiches) or principles (like the wrongness of killing anything with a right to life). The arguments are valid in virtue of form.

The form of the hot dog argument might be represented like so:

1. Some Fs are G.

2. All Gs are H.

   —————

   Therefore: Some Fs are H.

Think of this “form” as an argument skeleton. We get a genuine argument when we put on the meat, which we can do by plugging in whatever we want for F, G, and H. (As long as we are consistent; we can’t plug in “Oscar’s products” for the first F and then plug in “Oscar’s phobias” for the second.)

Skeletons have logical words: “and,” “or,” “if,” “some,” “all,” and so on. They don’t have content words: “hot dogs,” “sandwiches,” et cetera.

Which words count as “logical constants” depends on what logic you’re using, and it’s controversial which logic (if any) is the One True Logic. For more on this issue, I recommend Gillian Russell on logical pluralism.

When an argument is valid in virtue of form, that means that any argument with that form is valid. Another way to put it: an argument must be valid if it has the same skeleton as this one. Once you do a bit of philosophy (or logic), you will get quite good at recognizing a validity-inducing skeleton.

For practice, here are some more arguments that are valid in virtue of form, along with the names up top and skeletons on the side.

Conditional Elimination (AKA, Modus Ponens)

1. It’s raining. (P)

2. If it’s raining, it’s pouring. (If P, then Q.)

   —————

   Therefore: It’s pouring. (Q)

Disjunctive Syllogism

1. It’s raining or it’s snowing. (P or Q.)

2. It’s not raining. (Not-P.)

   —————

   Therefore: It’s snowing. (Q.)

Conjunctive Syllogism

1. It’s raining and it’s hot. (P and Q.)

   —————

   Therefore: It’s raining. (P.)

“Conditional” means we’re talking about “ifs.” “Disjunction” means “or.” “Conjunction” means “and.” For anyone who needs a few more syllables to stretch your papers out: you’re welcome.

Quiz #2

Q1.

Is the following argument valid in virtue of form?

1. Some cats are mammals.

2. All mammals are dogs.

   —————

   Therefore: Some mammals are dogs.

Q2.

What about this one? Is it valid in virtue of form?

1. All cats are mammals.

2. Some mammals are dogs.

   —————

   Therefore: Some cats are dogs.

Q3.

Is it possible to use a single example of a single possible scenario to show that an argument is valid in virtue of its form? Can you use one such example to show that an argument is not valid in virtue of its form?

Answers

Q1.

The argument is valid in virtue of form. If some Fs are Gs, and all Gs are Hs, then some Fs are Hs.

Q2.

The argument is not valid (and therefore can’t be valid in virtue of its form). Even if all Fs are Gs, and some Gs are Hs, that doesn’t show that all Fs are Hs. You can see this just by thinking about the actual case of cats and dogs: all cats are mammals, and some mammals are dogs, but no cats are dogs.

Q3.

In general, you can’t show that an argument is valid (or valid in virtue of form) just by producing one example. Validity requires that all possible scenarios where the premises are true are ones where the conclusion is also true. Equivalently, validity means that there’s no scenario where the premises are all true and the conclusion is false.

But if you do have a possible case where the premises are true and the conclusion false, then that will suffice to show that the argument is invalid. And obviously an invalid argument can’t be valid in virtue of form.

Fallacies and Flaws

Now you know about standard form, logical form, validity, soundness, persuasiveness, and strength. Not bad!

Unfortunately, a bunch of arguments—even some valuable ones—aren’t really sound. They might not even be valid. Or worse: there might be problems that make it impossible to tell, straightaway, whether we’re dealing with a valid and sound argument.

When an argument contains some kind of error of reasoning, we say that it is fallacious, and the most memorable errors are called fallacies. There are also other flaws that an argument might have, such as being unpersuasive, ambiguous, or just plain difficult to understand. Here we’ll cover some all-too-common flaws and fallacies.

Ad Hominem: The Argument and the Fallacy

“Ad hominem” means “addressed to the person.”

Now, there’s nothing wrong with addressing yourself to a specific audience when giving arguments. In fact, it’s often a good idea. Suppose you are trying to convince someone who believes in natural healing that they ought to take vaccines. You might say:

1. We should take medicines that effectively interact with our own natural immune systems.

2. Vaccines work by effectively interacting with our own natural immune systems (in particular, they stimulate the production of antibodies and T-cells).

   —————

   Therefore: We should take vaccines.

This can be a powerful argument even if you don’t believe the first premise. Maybe you don’t think it matters if a medical treatment is “natural,” so long as it’s effective. But if your audience thinks it matters, it could be smart to give them an argument based in naturalness. The argument will seem sound to them, given their background beliefs, even if it doesn’t seem sound to you.

This is called an ad hominem argument: an argument whose premises are taken from the beliefs of the person to whom the argument is being addressed, rather than from the beliefs of the person giving the argument. It’s a valuable tool in any longstanding debate. In ethics, for example, utilitarians (who think you should maximize happiness) and deontologists (who think rights also fundamentally matter) often have different starting points. But deontologists could still give ad hominem arguments to utilitarians (“You should keep promises because it makes people happy”), and vice versa, so that the two sides can convince each other of specific moral claims without first having to bicker about the foundations of morality.

There is another maneuver that goes by the name of “ad hominem,” however, which is also worth noting. That is the so-called ad hominem fallacy: the fallacy of inferring that something is false just because it came from a certain source you distrust. Behold:

1. The mainstream media said that it’s raining.

   —————

   So: It’s not raining.

Terrible argument!

Now, as it turns out, I don’t think many people actually make arguments like this. Most actual, real-world examples of the so-called “ad hominem fallacy” are less blatant, and they seem to suffer from other problems.

Consider:

• Sure, the protestors say we should change our abortion laws, but I don’t buy it. You know they’re all just paid actors.

• You think it’s morally permissible to sell and purchase coffee for money? Yeah, right. That’s what capitalists say—and you can’t trust capitalists.

These arguments seem to include reasons why some source can’t be trusted. And we could easily put these arguments in standard form and add a premise to make them valid, as in:

1. Capitalists say that it’s OK to sell coffee.

2. If capitalists say something, it’s false.

   —————

   Therefore: It’s false that it’s OK to sell coffee.

The problem with this argument isn’t the logic of it. It’s that the second premise is too strong. Nobody’s always saying falsehoods!

We could weaken the premise, which of course requires us to weaken the conclusion to maintain validity:

1. Capitalistssay that it’s OK to sell coffee.

2. If capitalists say something, it’s probably false.

   —————

   Therefore: It’s probably false that it’s OK to sell coffee.

Now, this argument has some potential. But it’s still not very persuasive, since it will only persuade people who (i) already distrust capitalists while (ii) not yet buying the anti-capitalist conclusion. But presumably, if you already distrust capitalists, you’ve already got suspicions about the capitalist view that it’s fine to sell coffee in exchange for money.

The fundamental problem is that we’re just assuming that you can’t trust the “hominem”—in this case, the capitalist. This sort of argument only really works when your audience already agrees with your choice of sources. If they don’t, you’ll need to try something else, something that illuminates the issue at stake. Instead of engaging in an “ad hominem evasion,” say something about the issue itself. Don’t just evade the issue by focusing entirely on who’s talking about it.

Of course, sometimes it’s OK to dismiss bad sources (“don’t feed the trolls”), just as it can be smart to trust good sources. But if everyone is doing ad hominem evasions all the time for no good reason, no one’s arguments will be very illuminating, and we won’t learn anything except what comes from our own preferred gurus. This is the problem of “echo chambers”—a pretty timely one. An echo chamber is the last place where you’d want to think about philosophy!

Tip #1: subarguments

Suppose you’ve got a premise that you like, but which your audience doesn’t yet believe. What can you do?

One tried-and-true solution is to support your premise by making it into the conclusion of another argument, called a “sub-argument.” In mathematics, the conclusion of a subargument is called a “lemma,” as distinct from a “theorem,” which is always the conclusion of a main argument (whose premises have to be axioms and which is also valid in virtue of its logical form).

The Fallacy of Equivocation

Consider the following argument:

1. Daniel teaches philosophy in North Carolina.

2. Daniel was the star of Harry Potter.

   —————

   Therefore: The star of Harry Potter teaches philosophy in North Carolina.

Is this valid? Is it sound?

Well, that depends. Who do we mean when we say “Daniel”?

Suppose we mean me—the guy writing this blog. Then premise 1 is true (I do teach philosophy in NC), but premise 2 is false (I’m no movie star). To lay it out explicitly:

1. Daniel Muñoz teaches philosophy in North Carolina.

2. Daniel Muñoz was the star of Harry Potter.

   —————

   Therefore: The star of Harry Potter teaches philosophy in North Carolina.

Alas, unsound. On the other hand, if “Daniel” means Daniel Radcliffe, then premise 1 is false, and premise 2 is true, as in:

1. Daniel Radcliffe teaches philosophy in North Carolina.

2. Daniel Radcliffe was the star of Harry Potter.

   —————

   Therefore: The star of Harry Potter teaches philosophy in North Carolina.

Here is the crucial dilemma for this kind of argument. Unless we change the meaning of “Daniel” midstream, the premises can’t all be true. But if we change meanings, the argument can’t be valid! Just think about it. This is not a valid argument:

1. Daniel Muñoz teaches philosophy in North Carolina.

2. Daniel Radcliffe was the star of Harry Potter.

   —————

   Therefore: The star of Harry Potter teaches philosophy in North Carolina.

So the original argument could only look sound given a kind of illusion, produced by a hidden swap of meanings.

This kind of swapping is called the Fallacy of Equivocation: you switch meanings in the middle of an argument while acting as if you were just using one constant meaning the whole time. (To “equivocate” is to use one word to mean different things.)

To avoid this fallacy, it is extremely important that you use clear language, and that you resolve any relevant ambiguities in your argument. When you spot an argument that equivocates between two meanings, do what I did, and spell out the different versions of the argument: one where you use just the first meaning throughout, another where you use the second throughout, and a third where you explicitly switch meanings. Usually, what you’ll find is that none of the arguments are sound. Switching meanings tends to undercut validity, and forcing consistency can make one of the premises false.

Tip #2: be consistent!

When putting arguments in standard form, try to use simple and consistent language as much as possible, but always use different words when it helps reduce ambiguity.

For example:

1. Killing adult humans is wrong.

2. If murdering grownups is impermissible, then infanticide is not acceptable.

   —————

   Therefore: Taking the life of a human child is immoral.

This argument might be valid, but it’s hard to tell, because we don’t know if all these near-synonyms really mean the same thing. Is killing the same as murdering/taking a life? Is being wrong the same as being unacceptable/immoral/impermissible? The argument is better if it’s put more consistently:

1. Killing adult human beings is wrong.

2. If killing adult human beings is wrong, then killing human children is wrong.

   —————

   Therefore: Killing human children is wrong.

When there isn’t any danger of ambiguity, use the same simple words each time as much as possible. When the word you’re tempting to use is ambiguous, be careful not to use it different ways during the same argument, and consider using a distinct word for each meaning, so that it’s always clear which meaning you have in mind.

(Micro-quiz: is premise 1—about killing—ambiguous? If so, between what and what?)

Conclusion

There are plenty of other fallacies and flaws to talk about—indeed, there is an entire field that studies the logic of arguments—but I hope you’ve gotten a feel for how to work with arguments in a more rigorous way.

Here are the key tips and concepts you’ll want to remember:

• Standard form: list the premises and demarcate the conclusion.

• Validity and soundness: valid arguments have premises whose truth would ensure the conclusion, and sound arguments are valid ones with true premises.

• Validity in virtue of forms: some arguments have a logical “skeleton” that ensures their validity. These sorts of arguments are super useful, because they’re not just valid but obviously valid.

• Persuasiveness: an argument can’t be persuasive unless someone might believe the premises without yet believing the conclusion. (An argument persuades by illuminating the link between a set of plausible premises and a conclusion that follows from them.)

• Ad hominems: it’s fine to give “ad hominem arguments” that address your opponent, using their beliefs as your premises, but you won’t be persuasive if you overuse “ad hominem evasions,” dismissing sources rather than engaging with the substance of the issue at hand.

• Good wording: use simple, clear, and consistent language in your arguments, and don’t equivocate on key terms. If you think an argument is ambiguous, try “disambiguating” by creating multiple versions of the argument where the meaning is clear. You may find that some versions are valid, while others have true premises, though none are both valid and blessed with true premises.

Good luck, and happy arguing!

1. In one of the greatest pranks on formal logic ever, the word “deduction” is most associated with Sherlock Holmes, whose inferences are almost always nondeductive. Holmes is making shrewd guesses about what is merely hinted at by his evidence—e.g. guessing someone’s line of work from a quick look at their fingernails. Obviously this sort of leap is not going to be deductively valid.