Philosophy 112

“Crime is common. Logic is rare. Therefore it is upon the logic rather than upon the crime that you should dwell.” — Sherlock Holmes

What is Logic?

Logic is the study of correct and incorrect reasoning. Logicians want to understand what makes good reasoning good and what makes bad reasoning bad. Understanding this helps us to avoid making mistakes in our own reasoning, and it allows us to evaluate the reasoning of others. It makes us better thinkers.

Aristotle (384-322 BCE)
Aristotle (384-322 BCE)

The first logician was Aristotle (384-322 BCE). But the logic we use and study these days—sometimes called “Modern Logic” or “Contemporary Logic”—was developed in the 19th and early 20th centuries. Important early works include George Boole’s The Laws of Thought (1854), and Gottlob Frege’s Begriffsschrift (“Concept Script”) (1879) and Grundgesetze der Arithmetik ("Basic Laws of Arithmetic) (1892).

George Boole (1815-1864)
George Boole (1815-1864)

Logic lies at the foundation of mathematics, where it allows us to provide a clear and rigorous account of mathematical proof. It also plays a central role in philosophy, where we use it to help reason as clearly and rigorously as possible about hard questions about ourselves, about knowledge, reality, truth, and beauty, and about right and wrong, good and bad. It also lies at the foundation of computer science: a computer is a logic machine. And a mind is, at least in part, a logic machine too, so logic lies at the foundation of cognitive science and philosophy of mind. It also lies at the foundation of linguistics, providing the tools we use for thinking about linguistic structure (syntax) and linguistic meaning (semantics).

Gottlob Frege (1848-1925)
Gottlob Frege (1848-1925)

Logic is one of the traditional sub-disciplines of Philosophy and one of the seven traditional “liberal arts”, alongside arithmetic, geometry, astronomy, music, grammar, and rhetoric. In a medieval university, students would begin by studying grammar, logic, and rhetoric, before going on to study the four other liberal arts. Advanced students might then study philosophy and theology. (At the time, “philosophy” included disciplines we would now categorize as “sciences”, like physics, biology, and psychology.)

So there are many good reasons to study logic. It will make you more medieval. It will give you insight into linguistics, the foundations of mathematics, and computer science. It will make you a better philosopher. And it will make you a better thinker.

Also, it can be a lot of fun.

Representing Reasoning

Logic aims to develop a theory of good and bad reasoning. But before we can evaluate a given piece of reasoning, we need a way to represent it.

The Structure of Reasoning

When we reason, we reason from some given information to some new information. We might represent this using a visual diagram:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise given information conclusion new information premise->conclusion:n

We will call a piece of reasoning an argument. We will call the information you reason from the premises, and the information you reason to the conclusion:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise premises conclusion conclusion premise->conclusion:n

So, for example, given the information that Andrew is vegetarian, you might reason to the conclusion that Andrew does not eat chicken:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Andrew is vegetarian conclusion Andrew does not each chicken premise->conclusion:n

But this diagram leaves a lot of important information out. When you reasoned from the premise to the conclusion, you relied your background knowledge about vegetarians and chickens. This gave you additional information, and so additional premises. Here is a more explicit representation of your reasoning:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Andrew is vegetarian Vegetarians don't eat animals Chickens are animals conclusion Andrew does not each chicken premise->conclusion:n

Notice that the conclusion does not follow from any one of the premises taken by itself. It only follows from all the premises taken together. So we cannot represent this as three separate arguments for the same conclusion.

We can also reason from one or more premises to one or more conclusions:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Alice, Bob, and Christina are teammates    Alice is the tallest player on her team conclusion Alice is taller than Bob Alice is taller than Christina premise->conclusion:n

Notice that here each conclusion follows separately from the premises. So we can represent this as two separate arguments, from the same premises, but to different conclusions:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Alice, Bob, and Christina are teammates    Alice is the tallest player on her team conclusion Alice is taller than Bob premise->conclusion:n

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise Alice, Bob, and Christina are teammates    Alice is the tallest player on her team conclusion Alice is taller than Christina premise->conclusion:n

When we reason from some premises to more than one conclusion, it is always possible to represent the reasoning in this way, with separate arguments. So, for the sake of simplicity, we will do this, and assume that an argument always has exactly one conclusion.

These diagrams provide a nice visual way of representing arguments. We began with them to help emphasize the basic structure of an argument: an argument is a piece of reasoning, from some given information, to a conclusion.

Premise-Conclusion Form

In practice, it is much easier to represent an argument as a simple list: first the premises, followed by the conclusion. So, for example, we can represent the argument above as,

  1. Alice, Bob, and Christina are teammates.
  2. Alice is the tallest player on her team.
  3. ∴ Alice is taller than Christina.

Here the premises are (1) and (2), and the conclusion is (3). We know that (3) is the conclusion because it is the last sentence in the list. We also know that it is the conclusion because it is marked by a special symbol, ‘∴’ which stands for “therefore”. We will always mark conclusions using ‘∴’.

When we represent arguments like this—as a list of premises, followed by the conclusion—we say that we have put the argument in premise-conclusion form. Putting arguments in premise-conclusion form is a common philosophical exercise. It forces you to clearly separate out the premises and the conclusion.

Test Your Understanding

Research shows that it is easy to read something and feel like you understood it, even when you didn’t. Most of us feel like we understand so long as we know what most of the words in a sentence mean, even if we don’t really grasp the ideas being communicated.

So, throughout these supplements, you will be provided with opportunities to stop and test your understanding. Please don’t skip this. If you get a question wrong, go back and re-read the preceding paragraphs with an eye to figuring out what you missed.

Arguments are everywhere. You can find them in blogs, magazine articles, textbooks, and newspaper editorials. They often pop up in conversations. But they are not usually found in premise-conclusion form, and important premises are often left unstated. Sometimes even the conclusion is left unstated, as in this example, from Confucius:

If there be righteousness in the heart,
there will be beauty in the character.
If there be beauty in the character,
there will be harmony in the home.
If there be harmony in the home,
there will be order in the nation.
If there be order in the nation,
there will be peace in the world.1

Can you put this argument in premise-conclusion form? What is the conclusion? When you think you know the answer, click on the gray box below.

Presumably the conclusion is,

  • If there be righteousness in the heart, there will be peace in the world.

So the argument, in premise-conclusion form, would be,

  1. If there be righteousness in the heart, there will be beauty in the character.
  2. If there be beauty in the character, there will be harmony in the home.
  3. If there be harmony in the home, there will be order in the nation.
  4. If there be order in the nation, there will be peace in the world.
  5. ∴ If there be righteousness in the heart, there will be peace in the world.

Here is an argument:

Some pigs have wings. Everything with wings can fly. So, some pigs can fly.

What are the premises of this argument? What is its conclusion? Can you put it in premise-conclusion form? When you think you know the answer, click on the gray box below:

The premises are ‘Some pigs have wings’ and ‘Everything with wings can fly’. The conclusion is ‘Some pigs can fly’. In English, the words ‘so’ and ‘hence’ and ‘therefore’ are often used to indicate a conclusion. In this case, the fact that the last sentence begins with ‘So’ tells us that it is the conclusion.

  1. Some pigs have wings.
  2. Everything with wings can fly.
  3. ∴ Pigs can fly.

Here is another argument:

We need to raise the capital gains tax. We need to do this because a low capital gains tax provides a disproportionate benefit to the wealthiest citizens. It serves only to increase the gap between the super-wealthy and the rest of us, and we need to decrease that gap.

What are the premises of this argument? What is the conclusion?

The conclusion is, ‘We need to raise the capital gains tax.’ Notice that when people give arguments, they don’t always state the conclusion last, and they don’t always mark it with a word like ‘so’ or ‘therefore’.

The first premise is ‘A low capital gains tax provides a disproportionate benefit to the wealthiest citizens.’ Notice the way the word because is used to indicate that this is a premise—a bit of information that helps support the conclusion. Another word that often indicates a premise is ‘since’.

Here is the argument in premise-conclusion form:

  1. A low capital gains tax provides a disproportionate benefit to the wealthiest citizens.
  2. A low capital gains tax serves only to increase the gap between the super-wealthy and the rest of us.
  3. We need to decrease the gap between the super-wealthy and the rest of us.
  4. ∴ We need to raise the capital gains tax.

Evaluating Arguments

“A deduction is speech in which, certain things having been supposed, something different from those supposed results of necessity because of their being so.” — Aristotle

Good and Bad Arguments

Not all reasoning is good reasoning, and so not all arguments are good arguments. What makes a good argument good and a bad argument bad?

Recall the basic structure of a piece of reasoning:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise premises conclusion conclusion premise->conclusion:n

There are two ways things can go wrong here.

First, we can plug bad information into to the premise box:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise false premises! conclusion conclusion premise->conclusion:n

You are probably familiar with the saying “Garbage in, garbage out”. When you reason from false premises, it doesn’t matter how well you reason. You’ll have no reason to suppose that your conclusion is true.

But suppose the information you plug into the premise box is true. Still things can go wrong: you might reason badly from those premises to your conclusion:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise premises conclusion conclusion premise->conclusion:n

 bad reasoning!

When you reason badly, it doesn’t matter that you began with true premises. You’ll have no reason to suppose that your conclusion is true.

And, of course, it is possible to reason badly from bad information!

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise false premises! conclusion conclusion premise->conclusion:n

 bad reasoning!

So what we want in a good argument is good reasoning from good information:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise true premises! conclusion conclusion premise->conclusion:n

 good reasoning!

When we reason correctly from true premises, we can be confident that our conclusion is true.

Now for some examples. Consider the argument,

  1. No mammals lay eggs.
  2. All platypus are mammals.
  3. ∴ All platypus don’t lay eggs.

Here the reasoning is good. The conclusion follows from the premises: if they were both true, it would have to be true too. But the first premise is false. So this is a case of reasoning well from bad information. As you can see, it led us to a false conclusion.

Or consider,

  1. All ducks are mammals.
  2. All mammals are birds.
  3. ∴ All ducks are birds.

Again, the reasoning is good. The conclusion follows from the premises: if they were both true, it would have to be true too. But again, we are reasoning from false premises. So this is a case of reasoning well from bad information. As you can see, we happened to be led to a true conclusion, but only by accident.

Here is an argument with true premises:

  1. Some ducks are birds.
  2. No birds are mammals.
  3. ∴ No ducks are mammals.

Here the premises are true, and the conclusion is true. But does the conclusion follow from the premises, or is this a case of bad reasoning?

The conclusion does not follow. Suppose that some ducks were birds and some ducks were mammals. Then the premises would be true, but the conclusion false. So this is an example of bad reasoning from good information. The conclusion happens to be true, but only by accident.

One last example. Consider the argument,

  1. Obama is President.
  2. If Obama is President, then a Democrat is President.
  3. ∴ A Democrat is President.

The premises are both true. And the conclusion follows from the premises: there is no way that they could be true but it false. So the conclusion is true, and it doesn’t just happen to be true. It’s truth was necessitated by the truth of the premises. So, finally, we have an example of good reasoning from good information!

Soundness and Validity

As we have seen, arguments are kind of like friends. A friend can be good in one way but bad in another—a good listener but bad at keeping secrets, for example. And an argument can be good in one way but bad in another—good reasoning but bad premises, or good premises but bad reasoning.

It is helpful to introduce some technical terms. We will call an argument that has good reasoning a valid argument. And we will call an argument that has good reasoning from good premises a sound argument.

So this argument is sound:

  1. Obama is President.
  2. If Obama is President, then a Democrat is President.
  3. ∴ A Democrat is President.

And these arguments are valid but not sound:

  1. No mammals lay eggs.
  2. All platypus are mammals.
  3. ∴ All platypus don’t lay eggs.
  1. All ducks are mammals.
  2. All mammals are birds.
  3. ∴ All ducks are birds.

While this argument is neither valid nor sound:

  1. Some ducks are birds.
  2. No birds are mammals.
  3. ∴ No ducks are mammals.

It is clear enough what it is for an argument has good or bad premises: a good premise is a true premise, and a bad premise is a false premise. But what does it mean to say that an argument has good reasoning?

Reasoning is good when the conclusion “follows”. But even that is just a metaphor. When we put an argument in premise-conclusion form, the conclusion always comes after the premises, so in a sense it always “follows” the premises. But that isn’t what we mean when we say that the conclusion follows from the premises. So what do we mean?

Roughly, what we mean is that, if the premises were true, then the conclusion would have to be true too. That is, we mean that there is no possible situation in which all the premises are true, but the conclusion false. So let this be our official definition of “validity”:

An argument is valid just in case it is impossible for all of its premises to be true but its conclusion false.

Here is an example of an invalid argument:

  1. If Obama is President, then a Democrat is President.
  2. A Democrat is President.
  3. ∴ Obama is President.

As I write this, (1), (2) and (3) are all true. But the reasoning is bad; the conclusion does not follow from the premises; the argument is not valid. This is because there is a possible situation in which the premises are all true, but the conclusion false. Can you describe such a situation?

Any situation in which some Democrat other than Obama is President will do the trick.

In general, you cannot tell whether or not an argument is valid just by considering whether or not its premises and conclusion are actually true or false. Validity is about what is or isn’t possible, not just about what happens to be the case. So you have to consider all the possibilities; you have to use your imagination.

Now that we have an official definition of validity, it is easy to provide an official definition of soundness:

An argument is sound just in case it is valid and all of its premises are true.

Remember, a sound argument is an argument with good reasoning—that is, a valid argument—from good information—that is, true premises.

In philosophy—and, indeed, in daily life—we want our arguments to be sound. As responsible reasoners, we want to be reasoning validly from true premises, since that ensures that we have reached a true conclusion. We want to put good information into our reasoning machine, and we want our machine to process that information correctly, so we get good information out of the machine.

But there is no general logical theory that allows us to assess whether or not a given premise is true or false. If you want to know whether or not all mammals lay eggs, you need to ask a biologist, not a logician. And there is no way to determine, using logic alone, who happens to be President, whether Andrew is vegetarian, and so on.

What logic can provide is a theory of good reasoning—a theory of validity, a theory of what follows from what.

Augustus De Morgan (1806-1871)
Augustus De Morgan (1806-1871)

Whether the premises be true or false, is not a question of logic, but of morals, philosophy, history or any other knowledge to which their subject-matter belongs. The question of logic is: does the conclusion certainly follow if the premises be true?

Let’s work through some more examples.

No Cars Allowed
No Cars Allowed
  1. No cars are allowed in the park.
  2. All police cruisers are cars.
  3. Therefore, all police cruisers are not allowed in the park.2

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

It is valid. It is hard to say whether or not it is sound. Premise (2) is true, but without further information about the park and its rules, we don’t know whether or not premise (1) is true. But if we can trust the sign, then the premises are both true, and the argument is valid, so it is sound.

  1. Some pigs have wings.
  2. Everything with wings can fly.
  3. ∴ Some pigs can fly.

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

It is valid, but it is not sound, because both of the premises are false (can you explain why is (2) false?)

  1. A good university must have a good library.
  2. ISU has a good library.
  3. ∴ ISU is a good university.

Is this argument valid? If it is not valid, can you describe a situation in which the premises are all true, but the conclusion false? If is valid, is it also sound?

It is not valid. Consider a possible situation in which ISU has a good library, and a good university must have a good library and a good philosophy department, but ISU does not have a good philosophy department. In that situation, both premises are true, but the conclusion is false.

(Since it is not valid, it can’t be sound, so we don’t need to worry about whether or not the premises are actually true.)

Can a valid argument have a false conclusion? If so, what would an example be?

Yes, a valid argument can have a false conclusion. We’ve already seen an example:

  1. Some pigs have wings.
  2. Everything with wings can fly.
  3. ∴ Some pigs can fly.

To check if the example you came up with was correct, ask yourself two questions:

  1. Is the conclusion false?
  2. If all the premises were true, would the conclusion have to be true?

If the answer to both of these question is yes, then your example was correct.

Can a valid argument with all true premises have a false conclusion? If so, what would an example be?

No! If the argument is valid, then there is no possible situation in which all the premises are true, but the conclusion false.

Can a valid argument have false premises but a true conclusion? If so, what would an example be?

Yes! Here is one example:

  1. Fish are mammals.
  2. Whales are fish.
  3. ∴ Whales are mammals.

Suppose you know that an argument is sound. What do you know about its conclusion? Stop and think about this—the answer is implicit in the definitions of validity and soundness just given.

If an argument is sound, all of its premises are true and it is valid. That is (plugging in the definition of validity), all of its premises are true and it is impossible for all of its premises to be true but its conclusion false. So if an argument is sound, we know that its conclusion is true.

It is easy to come up with a valid argument for the existence of God, but hard to come up with an uncontroversially sound argument for the existence of God. Try it! Try to formulate a valid argument whose conclusion is ‘God exists’. Are the premises uncontroversially true?

That was an open-ended question, so I hope you didn’t click on this box hoping for an answer! Here is one example of a valid argument for the existence of God:

  1. The moon is made of green cheese.
  2. If the moon is made of green cheese, then God exists.
  3. ∴ God exists.

This argument is valid—-if the premises were true, the conclusion would have to be true too. But the first premise is clearly false, so the argument is not sound.

Here is another example of a valid argument for the existence of God:

  1. The Bible says God exists.
  2. Everything the Bible says is true.
  3. ∴ God exists.

Again, this is valid—if the premises were true, the conclusion would have to be true too. But is it sound? Are the premises true? Some Christians believe that it is—they believe that both premises are true. But many Christians and most non-Christians will reject premise (2), claiming that it is false. Who’s right and who’s wrong? Ask the philosophers and theologians: that’s not a question that logic can answer.

Famously, Paul says, in Titus 1:12-13,

One of themselves, even a prophet of their own, said, the Cretans are always liars, evil beasts, slow bellies. This witness is true.

But what the Cretan said cannot be true! This is an example of the Liar Paradox (though it is not clear whether Paul meant it to be so). When Paul says “This witness is true,” that is, that what the Cretan said was true, he says something false.

So perhaps logic can show that not everything said by Paul in the Bible is true. Does that show that (2) is false?

Again, it is easy to come up with a valid argument for the non-existence of God, but hard to come up with an uncontroversially sound argument for that same conclusion. Try it!

Here is an example of a valid argument that is obviously unsound:

  1. The moon is made of green cheese.
  2. If the moon is made of green cheese, then God doesn’t exist.
  3. ∴ God doesn’t exist.

Here is an example of a valid argument whose soundness is unclear:

  1. If God existed, there would be no evil in the world.
  2. There is evil in the world.
  3. ∴ God does not exist.

Some philosophers argue that (1) and (2) are both true, and so the argument is sound. Others reject the truth of (1) or (2), claiming that the argument is valid but not sound. Who’s right and who’s wrong? Again, not a question that can be answered by logic alone.

Review and Loose Ends

Weve introduced five key terms: argument, premise, conclusion, valid, sound. The first three are defined in terms of each other:

Arguments, Premises, Conclusions
An argument is piece of reasoning that can be represented as a list of sentences, called the premises, followed by a sentence, called the conclusion.
Validity
An argument is valid just in case it is impossible that all the premises be true but the conclusion false.
Soundness
An argument is sound just in case it is valid and all of its premises are true.

Finally, we should briefly wrap up two loose ends.

First Loose End: Sentences?

First, the sentences that play the role of premises or conclusions must be sentences that express information. So they have to be declarative sentences—the kinds of sentences that are true or false. They cannot be questions or commands, since questions and commands are not used to express information.

So this is not an argument:

  1. It is important to eat fruits and vegetables.
  2. ∴ Eat your fruits and vegetables!

Why not?

This is not an argument because the conclusion is an imperative sentence—a sentence used to give a command—rather than a declarative sentence—a sentence used to say something true or false.

It is possible to develop a logic that includes imperatives. Computer programming languages often involve imperatives. Consider:

  1. If the user clicks on the icon, then open the app!
  2. The user has clicked on the icon.
  3. ∴ Open the app!

We might think of this as an argument from one imperative—the command expressed by (1)—to another imperative—the command expressed by (3). But the logic of imperatives (like the logic of questions) is an advanced topic, to be considered after you have mastered the material of this course.

Second Loose End: Good Reasoning that is not Valid

Second, we said above that good reasoning is valid reasoning, and we gave an official definition of validity. But not all good reasoning is valid in our sense.

Consider an eighteenth century European biologist, who has made a wide and careful study of mammals, and never encountered a mammal that lays eggs. Suppose she infers that no mammals lay eggs:

<!DOCTYPE svg PUBLIC “-//W3C//DTD SVG 1.1//EN” “http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd”> G premise all mammals that I have observed gave live birth conclusion no mammals lay eggs premise->conclusion:n

This seems like a reasonable inference for her to have made: it was based on a large set of good data, carefully gathered and considered. But it is not a valid inference: it is quite possible that the premises all be true but the conclusion false, as actually happened in this case.

Or consider a scientist who is interested in explaining the Cretaceous–Paleogene extinction event: the mass extinction that wiped out all the dinosaurs (except the birds, of course). The evidence strongly suggests that the extinction was caused, at least in part, by the impact of a large asteroid. So the scientist infers that the extinction was probably caused by a large asteroid.

Again, this seems like a reasonable inference. But it is not a valid inference in our sense. It is possible that, despite all the evidence, the extinction was caused by something else.

These examples show that not all good reasoning is valid reasoning. Some good reasoning is probabilistic or inductive: the given information supports a certain conclusion, but not absolutely, as there is still the chance that the conclusion is false despite the evidence.

The study of this kind of reasoning is the domain of inductive logic and probability theory. We are not going to be studying inductive logic or probability theory. We are going to be studying deductive logic. Deductive logic attempts to provide a theory of validity—iron-clad reasoning where the premises leave no possibility that the conclusion be false.

  1. Borrowed from Pospesel, p. 76.↩

  2. I borrow this example, and the associated photograph, from Pospesel and Marans, Arguments: Deductive Logic Exercises.↩