Systematic Philosophy

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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 (<…>).

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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

What Is An Argument?

In order to gain knowledge of most truths, arguments are critical. As a result, it is important for us to know everything we can about arguments. We need to know what they are, how to construct them, how to extract them from texts and conversation and how to assess them. Let’s being by looking at what arguments are.

What an argument is

What is an argument? Arguments have three components. First, every argument involves one or more propositions. Second, every argument has exactly one final conclusion. That final conclusion is one of the propositions. Third, every argument represents one or more of the propositions as entailing the final conclusion. It may or may not represent the other propositions as entailing or being entailed by one another.

That’s it. That’s all there is to an argument.

Two examples

Let’s look at some examples. First, we said that every argument involves one or more propositions. Here is a list of propositions:

<Socrates is a person>
<all people are mortal>
<Socrates is mortal>
<all mortal things have parts>
<Socrates has parts>
<all things that have parts are made of particles>
<Socrates is made of particles>

This list of propositions is not an argument. It is just a list. To make it into an argument, we still need to add two components. One of those components is a final conclusion. We said above that every argument has a single final conclusion and that that final conclusion had to be one of the propositions in the argument. So let’s select one of the propositions as a final conclusion:

Final Conclusion: <Socrates is made of particles>

We still do not have an argument yet. One component remains. In particular, as we said above, every argument must represent its final conclusion as being entailed by one or more of the propositions in the argument. The argument may or may not also represent other relations of entailment. Let’s add the following:

    • <Socrates is a person> and <all people are mortal> together are meant to entail <Socrates is mortal>
    • <Socrates is mortal> and <all mortal things have parts> together are meant to entail <Socrates has parts>
    • <Socrates has parts> and <all things that have parts are made of particles> together are meant to entail <Socrates is made of particles>

If we put these components together, we have an argument. There are various different ways to present arguments. This argument, for instance, presented in numbered format, looks like this:

  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]

We have now given one example of an argument. We chose this example first in order to give an example of an argument that makes sense. But it is not necessary for an argument to make sense in order for it to be an argument. The following is an argument as well, presented in text format:

Rabbits are good. And squares are shapes. These two propositions entail our final conclusion, which is that pickles are bad.

Obviously, this is a terrible argument. Nevertheless, it is still an argument. It has one or more propositions. It has a single final conclusion, which is one of the propositions. It represents one or more of the propositions as entailing the final conclusion. This is all that is needed. It fits the description we gave above, so it is an argument.

Looks okay to me.

What an argument is not

Different people use the word “argument” in different ways. Some people use it to mean “fight”. Some people use it to mean “conversation in which people express opposing viewpoints”. Some people use it as we have described above.

All of these different uses are fine. People can use words however they want to. Since we have a particular purpose here, however, and because we want to speak with precision, we hereby stipulate that when we use the word “argument”, we mean only what we have indicated above.

As far as we’re concerned, this is not an argument.

This means that as far as we’re concerned, an argument is not a fight. It is not a discussion. It is not even a multi-person activity. It is not an activity at all. It is a collection of propositions with a final conclusion and some purported relations of entailment. The final conclusion must be one of the propositions. One of more of the propositions must be represented as entailing the final conclusion. That’s all.

Next up: Steps, Premises, Conclusions, Etc.

Written by Geoff Anders

May 11, 2011 at 10:14 pm

Posted in Uncategorized