Systematic Philosophy

to know what can be known

The Purpose Of An Argument

Arguments can be used to entertain. They can be used to baffle. They can be used to provoke outrage or dismay. If arguments are spatial objects like most people believe, they can be used to fill books or prop up leaning tables. But while arguments can be used for many purposes, there is one purpose that is especially important for people who wish to gain knowledge of many truths. We will call this purpose the “primary” purpose of an argument.

The primary purpose

The primary purpose of an argument is to give people knowledge of that argument’s final conclusion.

An example

For example, consider the following argument, which is presented in numbered format:

  1. Socrates is a person.
  2. All people are mortal.
  3. Therefore, Socrates is mortal. [1,2]
  4. All mortal things have parts.
  5. Therefore, Socrates has parts. [3,4]
  6. All things that have parts are made of particles.
  7. Therefore, Socrates is made of particles. [5,6]

Here, the final conclusion is <Socrates is made of particles>. This means that the primary purpose of this argument is to give people knowledge of the fact that Socrates is made of particles.

Other purposes

We said that the primary purpose of an argument is to give people knowledge of that argument’s final conclusion. Not everyone will agree. Some people maintain that the primary purpose of an argument is to give people the justified belief that the argument’s final conclusion is true or to coerce people into believing the argument’s final conclusion. This is fine. We are not saying that arguments have objective purposes and that the objectively most important purpose of every argument is to give people knowledge of that argument’s final conclusion. We are simply saying that this is the purpose we will focus on most.

Purpose and quality

Understanding the purpose of arguments is useful for understanding how we assess arguments. After all, when we assess arguments, we assess them in terms of how close they come to fulfilling their primary purpose.

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Written by Geoff Anders

May 24, 2011 at 11:05 pm

Posted in Uncategorized

What Is Transparent Validity?

When assessing arguments, philosophers often talk about validity. Validity is important. But more important than validity is a particular species of validity: transparent validity. Let’s talk about that.

What transparent validity is

Transparent validity is very similar to validity. To say that an argument is “valid” is to say that in every case where that argument represents a conclusion as being entailed by one or more steps, those steps actually do entail that conclusion. To say that an argument is “transparently valid” is to say that in every case where that argument represents a conclusion as being entailed by one or more steps, those steps immediately entail that conclusion.

From the definitions of “validity” and “transparent validity”, it follows that every argument that is transparently valid is also valid. The reverse is not true, however. Some arguments that are valid are not transparently valid. Let’s call any argument that is valid but not transparently valid “opaquely valid”.

How to test for transparent validity

To test for transparent validity, you need to be able to test for immediate entailment. This means that you need to know how to tell whether some collection of propositions immediately entails some specific proposition. If you know how to test for immediate entailment, testing for transparent validity is a straightforward process.

The process is as follows. Take the argument in question. Look at its first conclusion. Examine the steps the argument says are supposed to entail that conclusion. Merely by considering those steps and the concept of immediate entailment, can you see that those steps immediately entail the conclusion in question? If not, the argument is not transparently valid. On the other hand, if you can see that the steps immediately entail the conclusion, then go on to the next conclusion. Repeat the process. If you make it through the argument and find immediate entailments in every case, the argument is transparently valid.

Some examples

Consider the following argument:

  1. Every thing loves my baby.
  2. My baby does not love any thing but me.
  3. Therefore, I am my baby. [1,2]

Surprisingly, this argument is valid! Steps 1 and 2 really do entail step 3. But while it is valid, it is not transparently valid. Consider only steps 1 and 2 and the concept of immediate entailment. Do not derive any further propositions. On this basis alone, can we see that steps 1 and 2 together immediately entail step 3? The answer is no. To get from steps 1 and 2 to the final conclusion, we have to derive at least one further proposition.

Now consider this:

  1. Every thing loves my baby.
  2. Therefore, my baby loves my baby. [1]
  3. My baby does not love any thing but me.
  4. Therefore, I am my baby. [2,3]

What about this argument? Consider only steps 1 and 2 and the concept of immediate entailment. Do not derive any further propositions. Merely on that basis, can we see that step 1 immediately entails step 2? The answer is yes. So step 1 immediately entails step 2. Now consider only steps 2, 3 and 4 and the concept of immediate entailment. Do not derive any further propositions. Merely on that basis can we see that steps 2 and 3 together immediately entail step 4? Again the answer is yes. So steps 2 and 3 immediately entail step 4. These are all of the entailments the argument represents. It follows that in every case where the argument represents steps as entailing a conclusion, those steps immediately entail the relevant conclusion. It follows that the argument is transparently valid.

Your mind is blown.

The second argument above is transparently valid. The first is not. For other, more striking examples, consider almost any argument whose premises are the axioms of arithmetic, which does not draw any intermediate conclusions and whose final conclusion is some interesting fact about numbers that follows from the axioms. Such an argument will be valid but not transparently valid. Now fill in all of the missing intermediate conclusions. The argument will become transparently valid.

Haggling about the details

Some people will look at the first argument above and declare that they can see that the final conclusion follows without deriving any intermediate conclusions. What do we say about that? First, it matters whether such people are right. If they are wrong, then the example is just fine as it stands. If they are right, then it matters whether the same is true for everyone. If some people can see that the final conclusion follows without deriving any intermediate conclusions but others can’t, then we should say that the first argument above is transparently valid to some people and not transparently valid to others. If it turns out that everyone can see, merely by considering the first two steps and the concept of immediate entailment, that the first two steps entail the third, then the example is simply a bad example.

Whichever way any particular example comes out, though, this entire discussion should make it clear what it is for an argument to be transparently valid. That was our original purpose here.

Generating transparent validity

There is a simply expressible recipe for making transparently valid arguments opaque or making opaquely valid arguments transparent. To make a transparently valid argument into an opaquely valid one, just delete one or more of the argument’s intermediate conclusions. This will usually work. To render an opaquely valid argument transparent, simply add the right intermediate conclusions back in.

Transparent validity and argument quality

Transparent validity is one of the two qualities that must be checked when assessing an argument. Furthermore, it is always possible to check an argument for transparent validity. Thus transparent validity is extremely important. Gaining the ability to check for transparent validity is absolutely essential.

In terms of necessary and sufficient conditions, transparent validity is necessary for an argument to be good, but is not sufficient on its own to make an argument good. In this regard, transparent validity is just like validity.

Next up: […]

Written by Geoff Anders

May 23, 2011 at 4:38 pm

Posted in Uncategorized

What Is Soundness?

There are a few technical terms philosophers use to talk about the quality of an argument. One of these terms is “soundness”.

What soundness is

To say that an argument is “sound” is to say that that argument is valid and that all of its premises are true. That’s all. Is the argument valid? Are all of its premises true? If so, it is sound. If not, it is unsound.

Some examples

Let’s look at a few examples. First, consider this argument:

  1. All cats are animals.
  2. All animals are things.
  3. Therefore, all cats are things. [1,2]

First we check for validity. Is the argument valid? To be valid, all of the entailments the argument represents must actually hold. The argument represents the proposition <all cats are animals> and the proposition <all animals are things> as jointly entailing the conclusion <all cats are things>. This is the only entailment the argument represents, and those propositions actually do entail that conclusion. It follows that the argument is valid. Now we check for the truth of the premises. Are all cats animals? Yes. Are all animals things? Yes. So all of the premises are true. Thus the argument is valid and has only true premises. It follows that the argument is sound.

Now consider this argument:

  1. All cats are animals.
  2. All dogs are animals.
  3. Therefore, all cats are dogs. [1,2]

Again the first step is to check for validity. Is the argument valid? This argument represents the proposition <all cats are animals> and the proposition <all dogs are animals> as jointly entailing the conclusion <all cats are dogs>. But these propositions do not actually entail that conclusion. It follows that this argument is not valid. Because it is not valid, the argument is automatically unsound.

Lastly, consider this:

  1. All cows can fly.
  2. All things that can fly are triangular.
  3. Therefore, all cows are triangular. [1,2]

To check for soundness, we first check for validity. The argument is in fact valid, so we next check whether the premises are true. Can all cows fly? No. Are all things that can fly triangular? No. None of the premises of the argument are true. For it to be sound, all of the premises need to be true. It follows that while the argument is valid, it is unsound.

Why people care about soundness

If an argument is valid, then if its premises are true, its final conclusion must also be true. If an argument is sound, then it is valid and has only true premises. It follows that if an argument is sound, its final conclusion has to be true. In fact, if an argument is sound, then all of its steps have to be true.

If it is sound, all of its steps are true.

The fact that soundness guarantees truth has led many philosophers to conclude that a sound argument is the same thing as a good argument. This is not true. In fact, the relation between soundness and the quality of an argument is somewhat complex. It is true that in order to be flawless, an argument must be sound. But as we will discuss, it is also the case that some sound arguments are bad and some good arguments are unsound.

Next up: […]

Written by Geoff Anders

May 22, 2011 at 11:41 am

Posted in Uncategorized

What Is A Statement?

When an argument is presented in numbered format, every proposition needs to be expressed in the form of a statement. What is a statement?

What a statement is

A “statement” is a unit in physical or mental language that we would judge from our pre-theoretical perspective to be capable of truth or falsity. Is it an instance of language? Would we pre-theoretically say it can be true or false? If so, then it is a statement.

Examples

The following are examples of statements:

  • “The sky is blue.”
  • “All people are happy.”
  • “Nothing exists.”
  • “Pegasus exists.”
  • “You should not lie.”
  • “I want you to win.”
  • “Two plus two equals four.”
  • “Caesar is a prime number.”
  • “It is not the case that the sky is blue.”
  • “The sky is blue and two plus two equals four.”
  • “I want you to win or I want you to tie.”
  • “If Pegasus exists, then it is not the case that nothing exists.”
  • “This statement is false.”

Each of the examples here is an instance of language. Before reflecting a lot, we would judge each of these examples to be capable of truth or falsity. Thus each is a statement.

A caption might be redundant.

What a statement is not

We have explained what statements are. It may be helpful to note some things that are not statements. First, questions are not statements. Questions can be good or bad, helpful or unhelpful, clear or confused. But they are not the sort of thing we pre-theoretically think can be true or false. Suppose someone asks “When was the Internet invented?” It does not make sense, pre-theoretically, to respond “True!” or “False!”

Second, commands are not statements. Commands can be right or wrong. They can be appropriate or inappropriate. But they are not the sort of thing we pre-theoretically think can be true or false. Suppose someone says “Fetch some water.” From our pre-theoretical perspective, it does not make sense to answer “True!” or “False!”

Third, a large number of physical objects are not statements. Rabbits, boxes and trees, for instance, are not statements. They are not units of language and we do not pre-theoretically believe they can be true or false.

Not a statement.

Why we say “pre-theoretically”

We have been using expressions like “pre-theoretically” and “from our pre-theoretical perspective”. Why? The answer is that once people start theorizing, people often come to hold very different views about many of the examples we gave above. Some come to believe that statements with proper names that do not refer to anything, statements like “Pegasus exists”, cannot be true or false. Some conclude that statements that attribute the wrong sorts of properties to things, statements like “Caesar is a prime number”, cannot be true or false. Some argue that the statement in the Liar’s Paradox (“this statement is false”) cannot be true or false, lest there be a contradiction. Some maintain for various reasons that no statements are capable of truth or falsity.

But whatever we discover while investigating, before we begin investigating there are bits of language we believe to be capable of truth or falsity. We will call those things “statements”, whether or not we eventually decide that all of them are capable of truth or falsity.

The quotation marks convention

When we mention specific statements, we will use quotation marks (“…”). This will allow us to distinguish between statements, which we mark with quotation marks, and propositions, which we mark with angle brackets (<…>).

Next up: […]

Written by Geoff Anders

May 22, 2011 at 3:49 am

Posted in Uncategorized

What Is Validity?

In everyday speech, the word “valid” is often used to mean “good”, “reasonable” or “correct”. Throw all that away. In philosophy, “valid” is a technical term with a very specific meaning. And it doesn’t mean “good”, “reasonable” or “correct”.

What validity is

In philosophy, “validity” is a property of arguments. Every argument represents one or more conclusions as each being entailed by various steps in the argument. Now this representation can be correct or incorrect. An argument may say that step C is entailed by steps A and B, but steps A and B may or may not actually entail step C. To say that an argument is “valid” is to say that in every case where the argument represents a conclusion as being entailed by one or more steps, that conclusion actually is entailed by those steps.

More simply, to say that an argument is “valid” is just to say that all of the entailment relations the argument represents actually hold.

How to test for validity

To test for validity, you need to be able to test for entailment. That is to say, you need to know how to tell whether some collection of propositions entails some specific proposition. Once you know how to test for entailment, testing for validity is a straightforward process.

That process is as follows. Take the argument in question. Look at its first conclusion. Look at the steps the argument says are supposed to entail that conclusion. Do those steps actually entail the conclusion? If not, the argument is invalid. If so, go on to the next conclusion. Repeat this process. If you make it to the end and find that every represented entailment actually holds, the argument is valid.

Several examples

Let’s look at a few examples. Consider the following argument:

  1. All cats are animals.
  2. All animals are things.
  3. Therefore, all cats are things. [1,2]

This argument has just one conclusion: step 3. It represents step 3 as being entailed by steps 1 and 2. Do steps 1 and 2 actually entail step 3? If so, the argument is valid. If not, the argument is invalid.  Of course, steps 1 and 2 do entail step 3. So the argument is valid.

Next, consider this argument:

  1. All cats are animals.
  2. All dogs are animals.
  3. Therefore, all cats are dogs. [1,2]

This argument is like the first in several ways. It has just one conclusion: step 3. It represents that conclusion as being entailed by steps 1 and 2. But unlike the first argument, in this case steps 1 and 2 do not actually entail step 3. It follows that this argument is invalid.

Now consider this:

  1. All cows can fly.
  2. All things that can fly are triangular.
  3. Therefore, all cows are triangular. [1,2]

As before, this argument has only one conclusion: step 3. It represents this conclusion as being entailed by steps 1 and 2. Do steps 1 and 2 entail step 3? The answer is yes, so the argument is valid. This does not mean that the argument is good. On the contrary, the argument is terrible. How can a terrible argument be valid? Remember, as we are using the term “valid”, validity and goodness are not the same. What matters for validity is only whether the represented entailments actually hold. In this case, the only represented entailment does hold, so the argument is valid.

 

Let’s consider a slightly more complex example:

  1. All cats are robots.
  2. No dogs are robots.
  3. Therefore, no dogs are cats. [1,2]
  4. If no dogs are cats, then all cats are fish.
  5. Therefore, all dogs are fish. [3,4]

This argument has two conclusions: step 3 and step 5. To assess it for validity, we start with the first conclusion, step 3. The argument represents step 3 as being entailed by steps 1 and 2. Do steps 1 and 2 actually entail step 3? The answer is yes. So far, so good. Now we move on to the next conclusion, step 5. The argument represents step 5 as being entailed by steps 3 and 4. Do these steps actually entail step 5? The answer is no. So the argument is invalid.

Finally, consider this example:

  1. Therefore, Santa Claus exists. [1]

Shockingly, according to our definition of “argument”, this is an argument. Is it a good argument? Clearly not. Is it valid? The argument represents only one entailment: step 1 entailing itself. Does that entailment hold? All propositions entail themselves, so the answer is yes. Thus the argument, despite being bad, is valid.

Validity and arguments

As we saw above, validity alone is not enough to make an argument good. Arguments can be both valid and bad. Still, there is some relation between validity and the quality of an argument. As we will see, every good argument is valid. This is because every good argument is transparently valid, and any argument that is “transparently valid” is also valid.

Next up: What is soundness?

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Written by Geoff Anders

May 20, 2011 at 11:36 pm

Posted in Uncategorized

Text Format And Numbered Format

The very same argument can be presented in different formats. It can be presented in regular prose, in formal logic or in other ways. Each of these ways has its advantages. But there is one format that stands above the rest, one format that is the clearest and most useful. We call it “numbered format”.

Text format

Arguments are usually presented as big blocks of text. We call this “text format”. Here is an argument presented in text format:

Socrates is composed of particles. How do we know? Socrates is a person. This means that, like all people, he is mortal. Now all mortal things have parts. This and the fact that Socrates is mortal together entail that Socrates has parts. Which parts? His left hand and his right hand, for instance. Or his thoughts and his emotions. Now every thing that has parts is composed of particles. It follows that Socrates is composed of particles.

Numbered format

Take an argument in text format. Strip away absolutely everything but the argument. Express each step in the argument in the form of a statement. Place each step on its own line. Number each of the lines. Put a “Therefore, …” in front of each conclusion. Place bracketed numbers after each conclusion to indicate which steps are meant to entail that conclusion. Wherever possible, put steps some place above the conclusions they are meant to entail. Put the final conclusion at the bottom. Voilà! The argument is now in numbered format.

Performing these operations on the argument above, we get:

  1. Socrates is a person.
  2. All people are mortal.
  3. Therefore, Socrates is mortal. [1,2]
  4. All mortal things have parts.
  5. Therefore, Socrates has parts. [3,4]
  6. All things that have parts are made of particles.
  7. Therefore, Socrates is made of particles. [5,6]

Why we use numbered format

How many steps are there in an argument? Which steps are meant to entail which? Which proposition is the final conclusion? Numbered format makes these things clear instantly. The argument above has seven steps. The first and second are meant to the entail the third. The third and fourth are meant to entail the fifth. The fifth and sixth are meant to entail the seventh. The seventh is the final conclusion. These features constitute the structure of the argument and they are conveyed with crystal clarity.

Now look at the text argument above. Eleven sentences. How many propositions? Nine… maybe. We might say more or less, depending on how we interpret some of the sentences. How many propositions are actually part of the argument? Are all of the relevant propositions explicitly stated or are some tacitly implied? Which propositions entail which? Compared to numbered format, an argument in text format looks like a big jumble of words.

After a while, this is what text format feels like.

Why do we use numbered format? It cuts away everything inessential. It exhibits the structure of an argument clearly and concisely. It makes it easy to check arguments for transparent validity. And we can easily see what numbered arguments are saying as well. This is not true for arguments stated purely in formal logic.

Numbered format is the best we have. That is why we use it.

A checklist for numbered format

  • Has every step in the argument been expressed in the form of a statement?
  • Has every step been put on its own line?
  • Has every step been given its own number?
  • Does every line with a conclusion begin “Therefore, …”?
  • Does every line with a conclusion end with bracketed numbers that indicate the steps that are meant to entail that conclusion?
  • As far as is possible, does every line occur above any lines it is supposed to entail?
  • Does the final conclusion occur at the bottom?
  • Has absolutely everything else been stripped away?

Next up: […]

Previous: Steps, Premises, Conclusions, Etc.

Written by Geoff Anders

May 19, 2011 at 10:50 am

Posted in Uncategorized

Steps, Premises, Conclusions, Etc.

Once you know what an argument is, it is useful to learn the names of some of the different parts of an argument. Learning these names makes it easier to talk about arguments and the parts of arguments.

Four terms

We will now introduce four terms: “step”, “premise”, “conclusion” and “intermediate conclusion”. The definitions we will give assume that you already understand what a proposition is, what an entailment relation is and what a final conclusion is.

First, “step”. Every argument includes one or more propositions. A “step” in an argument is simply one of the propositions in that argument.

Second, “premise”. In addition to propositions, every argument also includes one or more relations of entailment. That is to say, every argument represents one or more of its propositions as entailing one or more of its propositions. A “premise” in an argument is a proposition in that argument that the argument does not represent as being entailed by any of the steps in that argument.

Third, “conclusion”. As we just said, every argument represents one or more of its propositions as entailing one or more of its propositions. A “conclusion” of an argument is a proposition in that argument that the argument represents as being entailed by one or more of its steps.

Fourth, “intermediate conclusion”. An “intermediate conclusion” of an argument is simply any conclusion of that argument other than its final conclusion.

It follows from the definition of “argument” and from the definitions above that every argument has at least one step. With the exception of some infinite arguments and some circular arguments, every argument has at least one premise. Every argument must have at least one conclusion. Every step in an argument is either a premise or a conclusion. One of the conclusions of an argument must be the final conclusion. All of the other conclusions, if there are any, are intermediate conclusions.

An example

Let’s consider an example:

This example is in numbered format. Every proposition here is a step in the argument. The premises, conclusions, intermediate conclusions and final conclusion are clearly marked.

Next up:  Text Format And Numbered Format

Previous: What Is An Argument?

Written by Geoff Anders

May 18, 2011 at 9:56 pm

Posted in Uncategorized