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Categorical Syllogism

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A categorical syllogism is an argument which has two categorical propositions for premises and one categorical proposition as the conclusion. (Each proposition has two different terms.) A categorical syllogism has exactly three terms, each occurring in two propositions. The term which occurs in both premises is called the middle term. The term which occurs as the predicate term in the conclusion is called the major term. The premise which has the major term is called the major premise. To put the argument into standard form, one must place the major premise first. The subject of the conclusion is called the minor term and the premise with the minor term is called the minor premise. In standard form the minor premise is placed second. In other words, if the predicate term of the conclusion is in the second premise, the argument is not in standard form and it must be rewritten with the premises switched to put it in standard form.

Mood and Figure

Since categorical propositions come in 4 kinds, each premise and conclusion of a categorical syllogism may be of 4 kinds. By putting syllogisms in standard form we can identify the kinds of proposition each premise and conclusion is, merely by stating the vowel names in order. This is called stating the mood of the argument. For example if the mood of a categorical syllogism is given as AEO, we can know that the major premise is an A statement, because the major premise comes first in standard form. Likewise we can see that the minor premise would be an E statement, and the conclusion an O statement. By knowing the figure of the syllogism, in addition to its mood, one can know the complete logical form of any categorical syllogism. The logical form, not the content, completely determines whether the argument is valid or invalid and so is very important. The figure of the syllogism refers to the configuration of the middle terms in the premises. If the middle term is the subject term of the major premise and the predicate of the minor premise, then the figure is 1st. If the middle term is predicate of both premises, then the figure is 2nd. If the middle term is subject of both, then the figure is 3rd. If the middle term is predicate of the major and subject of the minor, the figure is 4th. In the following table M is always the middle term; P is always the major term, and S is always the minor term. The words which provide the copula, quality and quantity have been deliberately left out of this table as they are irrelevant to figure.
1st2nd3rd4th

Major Premise

M PP MM PP M

Minor Premise

S MS MM SM S

Conclusion

S PS PS PS P

There are 256 syllogistic forms given that each proposition may be one of 4 forms (A, E, I, or O) and the figure may be one of 4 too (that is 4 X 4 X 4 X 4). At most only 24 of these are valid.

Validity

The words 'valid' and 'invalid' are often used in a loose way. Someone might say, "She makes a valid point." and mean perhaps that what she said was true or even merely that what she said needs to be considered. It would be less confusing if the word 'valid' were reserved for use in its more technical sense. We shall use the word in this technical sense: an argument is valid if and only if having a false conclusion is impossible when the premises are true. In other words, in a valid argument the truth of the premises necessitates the truth of the conclusion, indeed guarantees the truth of the conclusion. In this technical sense validity is a property of arguments, not of "points", sentences or statements. The relationship between validity and truth and falsity is very specific and not entirely what you might intuit or expect. The following table indicates which combinations of truth and falsity are possible for the premises and conclusions of valid and invalid arguments. When we speak of true premises we mean that all the premises are true; when we speak of false premises we mean that at least one premise is false.

PremisesConclusionInvalid ArgumentsValid ArgumentsFalseFalseYesYesFalseTrueYesYesTrueTrueYesYesTrueFalseYesNo

Only the combination of true premises with a false conclusion in a valid argument is impossible. The validity of an argument is determined by its logical form rather than by its content. If an argument having a certain form is valid then all arguments having the same form are equally valid no matter how different the content may be. Likewise if an argument having a certain form is invalid then all other arguments with the same form will be invalid. A sound argument is a valid argument with all true premises.

The Rules of Validity

An argument must meet all of the following conditions to be valid. Failing to meet one or more conditions shows an argument to be invalid.

The middle term must be distributed at least once.
If a term is distributed in the conclusion, then it must be distributed in its premise.
If one of the premises is negative, then the conclusion must be negative, and if the conclusion is negative, then one of the premises must be negative.
There must not be two negative premises.

Categorical Syllogisms

Categorical syllogisms are a special class of deductive arguments containing two categorical propositions as premises and a third as the conclusion.
Categorical propositions are propositions which express a simple relation of being or becoming between a subject and a predicate. They do not contain any logical connectives such as “if, then” They contain four elements: a subject, a predicate, a copula, and an expression of quantity.

The subject is a the portion of the proposition that refers to a set of persons, places, or things.
The predicate is some quality which the subject is supposed to have.
The copula is a term connecting the subject and the predicate of the proposition.
The expression of quantity tells us whether the proposition is particular to some set or is universal to all things like the subject.

Standard form refers to a rendering of the propositions such that it appears with a standard quantifier followed by the subject, followed by the predicate. Rendering statements into standard form allows us to make deductions using the techniques of formal logic. By making the arguments explicit in standard form, we can avoid the ambiguity of natural language.

Standard Labels Affirmative Negative Universal A: All S are P E: No S is P ParticularI:Some S are PO:Some S are not P

The Square of Opposition

The square of opposition was devised by classical logicians to model the relationships between the four types of standard categorical propositions (A, E, I, O). Pay special attention to Summary Box 8.1 on page 206 of Kelley. We can use the square of opposition to determine the truth values of certain propositions given the truth values of related propositions. The relations are described below.
- Contraries are pairs of propositions in which both cannot be true, but both can be false. A-statements and E-statements are contraries.
- Contradictories are pairs of propositions in which both cannot be true and both cannot be false. A-statements and O-statements are contradictory. E-statements and I-statements are contradictory.
- Subalternate propositions are particular propositions which must be true if the universal propositions of the same form are also true. This is a one-way relationship from universal to particular statements. I-statements are subalternate to A-statements. O-statements are subalternate to E-statements.
- Subcontraries are pairs of statements in which both cannot be false, but both can be true.

Taken from David Kelley's The Art of Reasoning, Third Edition, page 206.

Existential Import

Existential import refers to the notion that some statements by their very nature refer to the existence of certain people, places, and things as existing. All I and O statements refer to their subjects as existing and having certain qualities; however, this is not the case for A and E statements. This is because a claim such as, “All gremlins are destructive.” is a standard form universal statement, but there is nothing in the statement that necessitates the existence of gremlins. The reason for the shift from the classical square of opposition to the modern square is that the modern square leaves no room for ambiguity. There are no questions of content to hinder the operation of the modern square, and this is a goal of formal logic.

Consider the following examples:

No pennies are attracted by magnets. Pennies are made of metal. Hence some metallic objects are not attracted by magnets.
In this example, both of the premises are universal statements the first is an E-statement, the second and A-statement. The conclusion (an I-statement) is only guaranteed to be true, given these premises, if we add the knowledge that pennies exist.
All of the honest politicians were at the fundraiser. Everyone who attended the fundraiser made a donation. Therefore, some of those who donated at the fundraiser were honest politicians.

In this example, we have two A-statements as premises and an I statement as the conclusion. Here again, the argument is only deductively valid if we add the knowledge that honest politicians exist. The same procedure is required if we are talking about unicorns, chimera, dragons, etc.

The main point to remember about existential import is that it is not always the case that the truth of a universal statement implies the truth of a particular statement with the same subject and predicate. For example, even if "All unicorns are white." is true, this does not ensure "There is at least one unicorn which is white."

Distribution

We call a subject or predicate distributed if it refers to the entire class under discussion.

Distribution of subject terms:

Subjects are considered distributed when a proposition makes reference to a relationship that all members of the subject class have to a predicate class.
In A and E statements, the subjects are always distributed; whereas in I and O, the subjects are always undistributed.
- A statements tell us that all members of the subject class also belong to the predicate class. That is, for any member of the subject class that we might choose, it will also be a member of the predicate class. Thus, A statements are have distributed subjects since they say something about each and every member of the subject class.
- E statements tell us that no members of the subject class also belong to the predicate class. That is, for any member of the subject class that we might choose, it will not be a member of the predicate class. Thus, E statements are have distributed subjects since they say something about each and every member of the subject class.

Distribution of predicate terms:

Predicates are considered distributed when a proposition makes reference to a relationship that all members of the predicate class have to a subject class.
In E and O statements, the predicates are always distributed; whereas in A and I, the predicates are always undistributed.
- E statements tell us that no members of the predicate class also belong to the subject class. That is, for any member of the predicate class that we might choose, it will not be a member of the subject class. Thus, E statements are have distributed predicates since they say something about each and every member of the predicate class.
- O statements tell us that there are some members of the subject class which are not identical to any members of the predicate class. Thus, their predicates are distributed, because we know that for a given set of members in the subject class, no member of the predicate class will overlap with them.

Note: “distributed” and “undistributed” only apply to subjects and predicates, not to propositions as a whole or statements.

Immediate Inferences

Immediate inferences are simple deductions that we can make. Given the knowledge that certain propositions in standard form are true, we can deduce (establish with absolute logical certainty) the truth of other standard formulations. The key to recognizing acceptable immediate inferences is to remember that any deductively valid operation must in all cases be truth preserving. That is, if we begin with a true proposition, it is impossible to apply a deductively valid operation and derive a false conclusion. Pay special attention to Summary Box 8.4 on page 220 of Kelley.

Conversion is a process whereby we switch the subject and predicate terms to create a new proposition. Taking the converse of a proposition does not alter the form of the proposition. It is always possible to generate the converse of E and I statements. This is not the case for A and O statements. This is because it can be demonstrated that for any true E or I statement, the converse of that statement will always be true. Thus, it is only acceptable to apply conversion to E and I statements.
Obversion is a process whereby we replace the predicate statement with its compliment, and we change the quality of the proposition from affirmative to negative, or negative to affirmative. This changes A statements to equivalent E statements and visa versa. It also changes I statements to O statements and visa versa. It is acceptable to generate the obverse of any standard form categorical imperative.
Contraposition is a process whereby we switch the subject and the predicate and replace each with its compliment. Contraposition is only acceptable for A and O statements, it is not acceptable for E and I statements. In natural language, contraposition will rearley be used with O statements.

Taken from David Kelley's The Art of Reasoning, Third Edition, page 220.

Venn Diagrams

Venn diagrams are visual representations of formal arguments. We use circles to represent the classes of subjects and predicates, then use shading to block off areas where nothing can exist, and put an “X” in any area where we know that some (at least one) thing must exist.
To create a Venn diagram, we must first draw two overlapping circles in a box. Label one circle as the “subject” domain and the other as the “predicate” domain. The area of the box outside the circles represents everything that is not in one or both of the circles. It should look something like the diagram below, except that the words “subject” and “predicate” should be replaced by some shorthand for your specific subject and predicate.

Now, we want to shade off any areas where we know that nothing can exist. Consider the following diagram of an A statement.

We have shaded off the area that is in our subject class, but outside our predicate class. This represents the fact that there is nothing in the subject class which isn't also in the predicate class. In other words, all S are P.

We can also place X's in places where we know something must exist. Consider the following I statement:

This diagram tells us that there is at least one thing in the area that is within both the subject and predicate domains. In other words, some S are P.

Reading Assignment: 3.1 and 3.2 (pp. 113-134)

Click here to skip the following discussion and go straight to the assignments.

In Chapter 3 we will be dealing with the content of the argument and not with the form. We will be identifying various types of defects in argument that can't be recognized by merely looking at the form. We call these defects informal fallacies. (Defects that have to do with the form are called formal fallacies, not surprisingly.) Each section will introduce a number of new fallacies and then you will be asked to identify the types of fallacies in particular arguments. You will need to consider carefully the relation between the conclusion and the reasons given to support the conclusion, i.e., the inferential relationship. Try to determine what is really going on in the argument.

The way the fallacies are presented in this textbook is a bit problematic, in my opinion. Grouping them together in their own chapter encourages us to forget that these are all merely problematic instances of otherwise good reasoning forms. Sometimes students get so excited about "fallacy hunting" that they consider any analogy a false analogy, or any generalization a hasty one; or they reject any reference to authority, forgetting that these are the main ways we get new information.

"Fallacy hunting" can also lead us to forget that only intended arguments can be fallacies. No matter how dire my threats, I am not committing the fallacy of appeal to force if I am not presenting the threat as a premise in an argument. I'm simply being a bully. Similarly with attacking someone's character. It may be a bad thing to do, but it isn't ad hominem abusive unless the character smear is given as a reason to disregard the smeeree's argument.

This said... it's fallacy season--happy hunting!!

Summary of Section 3.2

All of these fallacies lack a logical connection between the premises and the conclusion, though there might be a psychological or emotional connection.

1. Appeal to Force - This argument uses a physical or psychological threat as a reason to believe a conclusion. Obviously if someone uses force to argue that you should do something, self-preservation may give you a strong motive to do what they say, But even if you comply, you wouldn't say that they have used good reasoning to get you to do it.

2. Appeal to Pity -This argument uses pity; tries to act on sympathy, rather than reason.

3. Appeal to the people- This group of arguments uses our desire to be admired or accepted by others to convince us of something, rather than logically relevant reasons.

A - Direct approach - uses loaded or charged language to evoke a mob mentality either in speech or writing

B-D Indirect approach - used frequently by advertising industry

B Bandwagon - everyone else is doing it, so you should too

C Appeal to vanity - usually associated with a celebrity. Don't you want to be like this celebrity or model?

D Appeal to snobbery - somewhat the reverse of bandwagon, i.e., not everyone else is doing it, but only very special people like you

4. Argument against the person, or ad hominem argument- This argument always involves two arguers. the first arguer is attacked, rather than the argument she presented. To give a sound argument in response to an argument you must deal with the argument itself.

A - Abusive - a second arguer attacks the first arguer with verbal abuse

and personal criticism instead of attacking the argument

B - Circumstantial - a second arguer tries to show that first arguer has

ulterior motives instead of attacking the argument itself

C - You too - (tu quoque) - In a nutshell this fallacy calls the first arguer a hypocrite. This may be true, but is still attacking the arguer and not the argument.

5. Accident - A general rule is wrongly applied to a specific case where an exception should obviously be allowed.

6.*Straw Man - This involves two arguers. The second arguer misinterprets or restates in a weaker, easier to refute version the argument of the first arguer and then attacks the new argument, rather than the real one. In other words, the second arguer creates a figure made of straw that is much easier to knock down than a real man.

7. *Missing the point - The argument seems to be leading to a particular conclusion, but then a different conclusion is submitted, which is irrelevant or too strong to be warranted by the premises. In these arguments the premises are usually coherent and work together, distinguishing them from red herring.

8. *Red Herring - The arguer tries to divert the attention of the reader/listener in a different direction than that in which the argument began. Arguer tries to draw you off track. The new subject is relevant to the original subject, but only mildly so. These arguments tend to "meander"

*These three fallacies are the most difficult to distinguish. Read carefully the last two paragraphs of 3.2 on p. 128-129.

Assignment: (10 points each)

Identify the following arguments as one of the 13 different fallacies listed above or as no fallacy. If you have any doubts about your answer, explain how you got it--you might get me to agree with you.

1. The boss's business practices aren't shady; because if you think they are she might find out and fire you.

2. My client is innocent and deserves to go free. He has a wife and five children who will become destitute without his income to support them.

3. Everybody who is anybody wears Levis.

4. Don't vote for her for governor, she's immoral. She divorced her husband and ran off with a younger man.

5. (Bumper sticker) As long as there are tests, there will be prayer in school. (This one leaves a lot of the argument unstated)

6. Without checks to corporate power, capitalism fosters greed and corruption. therefore we should be anarchists.

7. When you say you oppose affirmative action programs, you obviously mean that you are against any efforts to improve the lot of minorities and women and to give them the opportunities they have long been denied. This view is unconscionable and unsupportable. You are, therefore, a bigot.

8. Candidate: "In response to your questions on the death penalty let me just say this about that. I have always been a strong advocate of making the punishment fit the crime. That's why I support longer sentences for convicted criminals and stricter controls of those who are paroled. The parole board has been far too lax in its duties."

9. Dr. Harrison has argued that the open position in the mathematics department should be given to Dr. George. But Harrison's arguments should be discounted since Harrison and George are good friends.

10. Stealing is wrong. Therefore, it would be wrong for me to steal the spark plug wires from that getaway car while the driver is busy robbing the First National Bank.

Problem 1

Decide whether each subset of $\mathbb{R}^3$ is linearly dependent or linearly independent.

$\{\begin{pmatrix} 1 \\ -3 \\ 5 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 4 \end{pmatrix}, \begin{pmatrix} 4 \\ -4 \\ 14 \end{pmatrix} \}$
$\{\begin{pmatrix} 1 \\ 7 \\ 7 \end{pmatrix}, \begin{pmatrix} 2 \\ 7 \\ 7 \end{pmatrix}, \begin{pmatrix} 3 \\ 7 \\ 7 \end{pmatrix} \}$
$\{\begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 4 \end{pmatrix} \}$
$\{\begin{pmatrix} 9 \\ 9 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 3 \\ 5 \\ -4 \end{pmatrix}, \begin{pmatrix} 12 \\ 12 \\ -1 \end{pmatrix} \}$

Answer

For each of these, when the subset is independent it must be proved, and when the subset is dependent an example of a dependence must be given.

It is dependent. Considering $c_1\begin{pmatrix} 1 \\ -3 \\ 5 \end{pmatrix} +c_2\begin{pmatrix} 2 \\ 2 \\ 4 \end{pmatrix} +c_3\begin{pmatrix} 4 \\ -4 \\ 14 \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ gives rise to this linear system. $\begin{array}{*{3}{rc}r} c_1 &+ &2c_2 &+ &4c_3 &= &0 \\ -3c_1&+ &2c_2 &- &4c_3 &= &0 \\ 5c_1 &+ &4c_2 &+ &14c_3 &= &0 \end{array}$ Gauss' method $\left(\begin{array}{*{3}{c}|c} 1 &2 &4 &0 \\ -3 &2 &-4 &0 \\ 5 &4 &14 &0 \end{array}\right) \xrightarrow[-5\rho_1+\rho_3]{3\rho_1+\rho_2} \;\xrightarrow[]{(3/4)\rho_2+\rho_3} \left(\begin{array}{*{3}{c}|c} 1 &2 &4 &0 \\ 0 &8 &8 &0 \\ 0 &0 &0 &0 \end{array}\right)$ yields a free variable, so there are infinitely many solutions. For an example of a particular dependence we can set $c_3$ to be, say, $1$ . Then we get $c_2=-1$ and $c_1=-2$ .
It is dependent. The linear system that arises here $\left(\begin{array}{*{3}{c}|c} 1 &2 &3 &0 \\ 7 &7 &7 &0 \\ 7 &7 &7 &0 \end{array}\right) \;\xrightarrow[-7\rho_1+\rho_3]{-7\rho_1+\rho_2} \;\xrightarrow[]{-\rho_2+\rho_3}\; \left(\begin{array}{*{3}{c}|c} 1 &2 &3 &0 \\ 0 &-7 &-14 &0 \\ 0 &0 &0 &0 \end{array}\right)$ has infinitely many solutions. We can get a particular solution by taking $c_3$ to be, say, $1$ , and back-substituting to get the resulting $c_2$ and $c_1$ .
It is linearly independent. The system $\left(\begin{array}{*{2}{c}|c} 0 &1 &0 \\ 0 &0 &0 \\ -1 &4 &0 \end{array}\right) \;\xrightarrow[]{\rho_1\leftrightarrow\rho_2} \;\xrightarrow[]{\rho_3\leftrightarrow\rho_1}\; \left(\begin{array}{*{2}{c}|c} -1 &4 &0 \\ 0 &1 &0 \\ 0 &0 &0 \end{array}\right)$ has only the solution $c_1=0$ and $c_2=0$ . (We could also have gotten the answer by inspection— the second vector is obviously not a multiple of the first, and vice versa.)
It is linearly dependent. The linear system $\left(\begin{array}{*{4}{c}|c} 9 &2 &3 &12 &0 \\ 9 &0 &5 &12 &0 \\ 0 &1 &-4 &-1 &0 \end{array}\right)$ has more unknowns than equations, and so Gauss' method must end with at least one variable free (there can't be a contradictory equation because the system is homogeneous, and so has at least the solution of all zeroes). To exhibit a combination, we can do the reduction $\xrightarrow[]{-\rho_1+\rho_2} \;\xrightarrow[]{(1/2)\rho_2+\rho_3}\; \left(\begin{array}{*{4}{c}|c} 9 &2 &3 &12 &0 \\ 0 &-2 &2 &0 &0 \\ 0 &0 &-3 &-1 &0 \end{array}\right)$ and take, say, $c_4=1$ . Then we have that $c_3=-1/3$ , $c_2=-1/3$ , and $c_1=-31/27$ .

This exercise is recommended for all readers.

Problem 2

Which of these subsets of $\mathcal{P}_3$ are linearly dependent and which are independent?

$\{3-x+9x^2,5-6x+3x^2,1+1x-5x^2\}$
$\{-x^2,1+4x^2\}$
$\{2+x+7x^2,3-x+2x^2,4-3x^2\}$
$\{8+3x+3x^2,x+2x^2,2+2x+2x^2,8-2x+5x^2\}$

Answer

In the cases of independence, that must be proved. Otherwise, a specific dependence must be produced. (Of course, dependences other than the ones exhibited here are possible.)

This set is independent. Setting up the relation $c_1(3-x+9x^2)+c_2(5-6x+3x^2)+c_3(1+1x-5x^2)=0+0x+0x^2$ gives a linear system $\left(\begin{array}{*{3}{c}|c} 3 &5 &1 &0 \\ -1 &-6 &1 &0 \\ 9 &3 &-5 &0 \end{array}\right) \;\xrightarrow[-3\rho_1+\rho_3]{(1/3)\rho_1+\rho_2} \;\xrightarrow[]{3\rho_2} \;\xrightarrow[]{-(12/13)\rho_2+\rho_3}\; \left(\begin{array}{*{3}{c}|c} 3 &5 &1 &0 \\ 0 &-13 &4 &0 \\ 0 &0 &-128/13 &0 \end{array}\right)$ with only one solution: $c_1=0$ , $c_2=0$ , and $c_3=0$ .
This set is independent. We can see this by inspection, straight from the definition of linear independence. Obviously neither is a multiple of the other.
This set is linearly independent. The linear system reduces in this way $\left(\begin{array}{*{3}{c}|c} 2 &3 &4 &0 \\ 1 &-1 &0 &0 \\ 7 &2 &-3 &0 \end{array}\right) \;\xrightarrow[-(7/2)\rho_1+\rho_3]{-(1/2)\rho_1+\rho_2} \;\xrightarrow[]{-(17/5)\rho_2+\rho_3}\; \left(\begin{array}{*{3}{c}|c} 2 &3 &4 &0 \\ 0 &-5/2 &-2 &0 \\ 0 &0 &-51/5 &0 \end{array}\right)$ to show that there is only the solution $c_1=0$ , $c_2=0$ , and $c_3=0$ .
This set is linearly dependent. The linear system $\left(\begin{array}{*{4}{c}|c} 8 &0 &2 &8 &0 \\ 3 &1 &2 &-2 &0 \\ 3 &2 &2 &5 &0 \end{array}\right)$ must, after reduction, end with at least one variable free (there are more variables than equations, and there is no possibility of a contradictory equation because the system is homogeneous). We can take the free variables as parameters to describe the solution set. We can then set the parameter to a nonzero value to get a nontrivial linear relation.

This exercise is recommended for all readers.

Problem 3

Prove that each set $\{f,g\}$ is linearly independent in the vector space of all functions from $\mathbb{R}^+$ to $\mathbb{R}$ .

$f(x)=x$ and $g(x)=1/x$
$f(x)=\cos(x)$ and $g(x)=\sin(x)$
$f(x)=e^x$ and $g(x)=\ln(x)$

Answer

Let $Z$ be the zero function $Z(x)=0$ , which is the additive identity in the vector space under discussion.

This set is linearly independent. Consider $c_1\cdot f(x)+c_2\cdot g(x)=Z(x)$ . Plugging in $x=1$ and $x=2$ gives a linear system $\begin{array}{*{2}{rc}r} c_1\cdot 1 &+ &c_2\cdot 1 &= &0 \\ c_1\cdot 2 &+ &c_2\cdot (1/2) &= &0 \end{array}$ with the unique solution $c_1=0$ , $c_2=0$ .
This set is linearly independent. Consider $c_1\cdot f(x)+c_2\cdot g(x)=Z(x)$ and plug in $x=0$ and $x=\pi/2$ to get $\begin{array}{*{2}{rc}r} c_1\cdot 1 &+ &c_2\cdot 0 &= &0 \\ c_1\cdot 0 &+ &c_2\cdot 1 &= &0 \end{array}$ which obviously gives that $c_1=0$ , $c_2=0$ .
This set is also linearly independent. Considering $c_1\cdot f(x)+c_2\cdot g(x)=Z(x)$ and plugging in $x=1$ and $x=e$ $\begin{array}{*{2}{rc}r} c_1\cdot e &+ &c_2\cdot 0 &= &0 \\ c_1\cdot e^e &+ &c_2\cdot 1 &= &0 \end{array}$ gives that $c_1=0$ and $c_2=0$ .

This exercise is recommended for all readers.

Problem 4

Which of these subsets of the space of real-valued functions of one real variable is linearly dependent and which is linearly independent? (Note that we have abbreviated some constant functions; e.g., in the first item, the " $2$ " stands for the constant function $f(x)=2$ .)

$\{2,4\sin^2(x),\cos^2(x)\}$
$\{1,\sin(x),\sin(2x)\}$
$\{x,\cos(x)\}$
$\{(1+x)^2,x^2+2x,3\}$
$\{\cos(2x),\sin^2(x),\cos^2(x)\}$
$\{0,x,x^2\}$

Answer

In each case, that the set is independent must be proved, and that it is dependent must be shown by exhibiting a specific dependence.

This set is dependent. The familiar relation $\sin^2(x)+\cos^2(x)=1$ shows that $2=c_1\cdot(4\sin^2(x))+c_2\cdot(\cos^2(x))$ is satisfied by $c_1=1/2$ and $c_2=2$ .
This set is independent. Consider the relationship $c_1\cdot 1+c_2\cdot\sin(x)+c_3\cdot\sin(2x)=0$ (that " $0$ " is the zero function). Taking $x=0$ , $x=\pi/2$ and $x=\pi/4$ gives this system. $\begin{array}{*{3}{rc}r} c_1 & & & & &= &0 \\ c_1 &+ &c_2 & & &= &0 \\ c_1 &+ &(\sqrt{2}/2)c_2 &+ &c_3 &= &0 \end{array}$ whose only solution is $c_1=0$ , $c_2=0$ , and $c_3=0$ .
By inspection, this set is independent. Any dependence $\cos(x)=c\cdot x$ is not possible since the cosine function is not a multiple of the identity function (we are applying Corollary 1.17).
By inspection, we spot that there is a dependence. Because $(1+x)^2=x^2+2x+1$ , we get that $c_1\cdot(1+x)^2+c_2\cdot(x^2+2x)=3$ is satisfied by $c_1=3$ and $c_2=-3$ .
This set is dependent. The easiest way to see that is to recall the trigonometric relationship $\cos^2(x)-\sin^2(x)=\cos(2x)$ . (Remark. A person who doesn't recall this, and tries some $x$ 's, simply never gets a system leading to a unique solution, and never gets to conclude that the set is independent. Of course, this person might wonder if they simply never tried the right set of $x$ 's, but a few tries will lead most people to look instead for a dependence.)
This set is dependent, because it contains the zero object in the vector space, the zero polynomial.

Problem 5

Does the equation $\sin^2(x)/\cos^2(x)=\tan^2(x)$ show that this set of functions $\{\sin^2(x),\cos^2(x),\tan^2(x)\}$ is a linearly dependent subset of the set of all real-valued functions with domain the interval $(-\pi/2..\pi/2)$ of real numbers between $-\pi/2$ and $\pi/2)$ ?

Answer

No, that equation is not a linear relationship. In fact this set is independent, as the system arising from taking $x$ to be $0$ , $\pi/6$ and $\pi/4$ shows.

Problem 6

Why does Lemma 1.4 say "distinct"?

Answer

To emphasize that the equation $1\cdot\vec{s}+(-1)\cdot\vec{s}=\vec{0}$ does not make the set dependent.

This exercise is recommended for all readers.

Problem 7

Show that the nonzero rows of an echelon form matrix form a linearly independent set.

Answer

We have already showed this: the Linear Combination Lemma and its corollary state that in an echelon form matrix, no nonzero row is a linear combination of the others.

This exercise is recommended for all readers.

Problem 8

Show that if the set $\{\vec{u},\vec{v},\vec{w}\}$ is linearly independent set then so is the set $\{\vec{u},\vec{u}+\vec{v},\vec{u}+\vec{v}+\vec{w}\}$ .
What is the relationship between the linear independence or dependence of the set $\{\vec{u},\vec{v},\vec{w}\}$ and the independence or dependence of $\{\vec{u}-\vec{v},\vec{v}-\vec{w},\vec{w}-\vec{u}\}$ ?

Answer

Assume that the set $\{\vec{u},\vec{v},\vec{w}\}$ is linearly independent, so that any relationship $d_0\vec{u}+d_1\vec{v}+d_2\vec{w}=\vec{0}$ leads to the conclusion that $d_0=0$ , $d_1=0$ , and $d_2=0$ . Consider the relationship $c_1(\vec{u})+c_2(\vec{u}+\vec{v})+c_3(\vec{u}+\vec{v}+\vec{w}) =\vec{0}$ . Rewrite it to get $(c_1+c_2+c_3)\vec{u}+(c_2+c_3)\vec{v}+(c_3)\vec{w}=\vec{0}$ . Taking $d_0$ to be $c_1+c_2+c_3$ , taking $d_1$ to be $c_2+c_3$ , and taking $d_2$ to be $c_3$ we have this system. $\begin{array}{*{3}{rc}r} c_1 &+ &c_2 &+ &c_3 &= &0 \\ & &c_2 &+ &c_3 &= &0 \\ & & & &c_3 &= &0 \end{array}$ Conclusion: the $c$ 's are all zero, and so the set is linearly independent.
The second set is dependent $1\cdot(\vec{u}-\vec{v}) +1\cdot(\vec{v}-\vec{w}) +1\cdot(\vec{w}-\vec{u}) =\vec{0}$ whether or not the first set is independent.

Problem 9

Example 1.10 shows that the empty set is linearly independent.

When is a one-element set linearly independent?
How about a set with two elements?

Answer

A singleton set $\{\vec{v}\}$ is linearly independent if and only if $\vec{v}\neq\vec{0}$ . For the "if" direction, with $\vec{v}\neq\vec{0}$ , we can apply Lemma 1.4 by considering the relationship $c\cdot\vec{v}=\vec{0}$ and noting that the only solution is the trivial one: $c=0$ . For the "only if" direction, just recall that Example 1.11 shows that $\{\vec{0}\}$ is linearly dependent, and so if the set $\{\vec{v}\}$ is linearly independent then $\vec{v}\neq\vec{0}$ . (Remark. Another answer is to say that this is the special case of Lemma 1.16 where $S=\varnothing$ .)
A set with two elements is linearly independent if and only if neither member is a multiple of the other (note that if one is the zero vector then it is a multiple of the other, so this case is covered). This is an equivalent statement: a set is linearly dependent if and only if one element is a multiple of the other. The proof is easy. A set $\{\vec{v}_1,\vec{v}_2\}$ is linearly dependent if and only if there is a relationship $c_1\vec{v}_1+c_2\vec{v}_2=\vec{0}$ with either $c_1\neq 0$ or $c_2\neq 0$ (or both). That holds if and only if $\vec{v}_1=(-c_2/c_1)\vec{v}_2$ or $\vec{v}_2=(-c_1/c_2)\vec{v}_1$ (or both).

Problem 10

In any vector space $V$ , the empty set is linearly independent. What about all of $V$ ?

Answer

This set is linearly dependent set because it contains the zero vector.

Problem 11

Show that if $\{\vec{x},\vec{y},\vec{z}\}$ is linearly independent then so are all of its proper subsets: $\{\vec{x},\vec{y}\}$ , $\{\vec{x},\vec{z}\}$ , $\{\vec{y},\vec{z}\}$ , $\{\vec{x}\}$ , $\{\vec{y}\}$ , $\{\vec{z}\}$ , and $\{\}$ . Is that "only if" also?

Answer

The "if" half is given by Lemma 1.14. The converse (the "only if" statement) does not hold. An example is to consider the vector space $\mathbb{R}^2$ and these vectors.

$\vec{x}=\begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad \vec{y}=\begin{pmatrix} 0 \\ 1 \end{pmatrix},\quad \vec{z}=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$

Problem 12

Show that this $S=\{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix},\begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}\}$ is a linearly independent subset of $\mathbb{R}^3$ .
Show that $\begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix}$ is in the span of $S$ by finding $c_1$ and $c_2$ giving a linear relationship. $c_1\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} +c_2\begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix} =\begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix}$ Show that the pair $c_1,c_2$ is unique.
Assume that $S$ is a subset of a vector space and that $\vec{v}$ is in $[S]$ , so that $\vec{v}$ is a linear combination of vectors from $S$ . Prove that if $S$ is linearly independent then a linear combination of vectors from $S$ adding to $\vec{v}$ is unique (that is, unique up to reordering and adding or taking away terms of the form $0\cdot\vec{s}$ ). Thus $S$ as a spanning set is minimal in this strong sense: each vector in $[S]$ is "hit" a minimum number of times— only once.
Prove that it can happen when $S$ is not linearly independent that distinct linear combinations sum to the same vector.

Answer

The linear system arising from $c_1\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} +c_2\begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ has the unique solution $c_1=0$ and $c_2=0$ .
The linear system arising from $c_1\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} +c_2\begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix} =\begin{pmatrix} 3 \\ 2 \\ 0 \end{pmatrix}$ has the unique solution $c_1=8/3$ and $c_2=-1/3$ .
Suppose that $S$ is linearly independent. Suppose that we have both $\vec{v}=c_1\vec{s}_1+\dots+c_n\vec{s}_n$ and $\vec{v}=d_1\vec{t}_1+\dots+d_m\vec{t}_m$ (where the vectors are members of $S$ ). Now, $c_1\vec{s}_1+\dots+c_n\vec{s}_n =\vec{v} =d_1\vec{t}_1+\dots+d_m\vec{t}_m$ can be rewritten in this way. $c_1\vec{s}_1+\dots+c_n\vec{s}_n -d_1\vec{t}_1-\dots-d_m\vec{t}_m =\vec{0}$ Possibly some of the $\vec{s}\,$ 's equal some of the $\vec{t}\,$ 's; we can combine the associated coefficients (i.e., if $\vec{s}_i=\vec{t}_j$ then $\cdots+c_i\vec{s}_i+\dots-d_j\vec{t}_j-\cdots$ can be rewritten as $\cdots+(c_i-d_j)\vec{s}_i+\cdots$ ). That equation is a linear relationship among distinct (after the combining is done) members of the set $S$ . We've assumed that $S$ is linearly independent, so all of the coefficients are zero. If $i$ is such that $\vec{s}_i$ does not equal any $\vec{t}_j$ then $c_i$ is zero. If $j$ is such that $\vec{t}_j$ does not equal any $\vec{s}_i$ then $d_j$ is zero. In the final case, we have that $c_i-d_j=0$ and so $c_i=d_j$ . Therefore, the original two sums are the same, except perhaps for some $0\cdot\vec{s}_i$ or $0\cdot\vec{t}_j$ terms that we can neglect.
This set is not linearly independent: $S=\{\begin{pmatrix} 1 \\ 0 \end{pmatrix},\begin{pmatrix} 2 \\ 0 \end{pmatrix}\}\subset\mathbb{R}^2$ and these two linear combinations give the same result $\begin{pmatrix} 0 \\ 0 \end{pmatrix}=2\cdot\begin{pmatrix} 1 \\ 0 \end{pmatrix}-1\cdot\begin{pmatrix} 2 \\ 0 \end{pmatrix} =4\cdot\begin{pmatrix} 1 \\ 0 \end{pmatrix}-2\cdot\begin{pmatrix} 2 \\ 0 \end{pmatrix}$ Thus, a linearly dependent set might have indistinct sums. In fact, this stronger statement holds: if a set is linearly dependent then it must have the property that there are two distinct linear combinations that sum to the same vector. Briefly, where $c_1\vec{s}_1+\dots+c_n\vec{s}_n=\vec{0}$ then multiplying both sides of the relationship by two gives another relationship. If the first relationship is nontrivial then the second is also.

Problem 13

Prove that a polynomial gives rise to the zero function if and only if it is the zero polynomial. (Comment. This question is not a Linear Algebra matter, but we often use the result. A polynomial gives rise to a function in the obvious way: $x\mapsto c_nx^n+\dots+c_1x+c_0$ .)

Answer

In this "if and only if" statement, the "if" half is clear— if the polynomial is the zero polynomial then the function that arises from the action of the polynomial must be the zero function $x\mapsto 0$ . For "only if" we write $p(x)=c_nx^n+\dots+c_0$ . Plugging in zero $p(0)=0$ gives that $c_0=0$ . Taking the derivative and plugging in zero $p^\prime(0)=0$ gives that $c_1=0$ . Similarly we get that each $c_i$ is zero, and $p$ is the zero polynomial.

Problem 14

Return to Section 1.2 and redefine point, line, plane, and other linear surfaces to avoid degenerate cases.

Answer

The work in this section suggests that an $n$ -dimensional non-degenerate linear surface should be defined as the span of a linearly independent set of $n$ vectors.

Problem 15

Show that any set of four vectors in $\mathbb{R}^2$ is linearly dependent.
Is this true for any set of five? Any set of three?
What is the most number of elements that a linearly independent subset of $\mathbb{R}^2$ can have?

Answer

For any $a_{1,1}$ , ..., $a_{2,4}$ , $c_1\begin{pmatrix} a_{1,1} \\ a_{2,1} \end{pmatrix} +c_2\begin{pmatrix} a_{1,2} \\ a_{2,2} \end{pmatrix} +c_3\begin{pmatrix} a_{1,3} \\ a_{2,3} \end{pmatrix} +c_4\begin{pmatrix} a_{1,4} \\ a_{2,4} \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ yields a linear system $\begin{array}{*{4}{rc}r} a_{1,1}c_1 &+ &a_{1,2}c_2 &+ &a_{1,3}c_3 &+ &a_{1,4}c_4 &= &0 \\ a_{2,1}c_1 &+ &a_{2,2}c_2 &+ &a_{2,3}c_3 &+ &a_{2,4}c_4 &= &0 \end{array}$ that has infinitely many solutions (Gauss' method leaves at least two variables free). Hence there are nontrivial linear relationships among the given members of $\mathbb{R}^2$ .
Any set five vectors is a superset of a set of four vectors, and so is linearly dependent. With three vectors from $\mathbb{R}^2$ , the argument from the prior item still applies, with the slight change that Gauss' method now only leaves at least one variable free (but that still gives infinitely many solutions).
The prior item shows that no three-element subset of $\mathbb{R}^2$ is independent. We know that there are two-element subsets of $\mathbb{R}^2$ that are independent— one is $\{\begin{pmatrix} 1 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \end{pmatrix}\}$ and so the answer is two.

This exercise is recommended for all readers.

Problem 16

Is there a set of four vectors in $\mathbb{R}^3$ , any three of which form a linearly independent set?

Answer

Yes; here is one.

$\{\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \}$

Problem 17

Must every linearly dependent set have a subset that is dependent and a subset that is independent?

Answer

Yes. The two improper subsets, the entire set and the empty subset, serve as examples.

Problem 18

In $\mathbb{R}^4$ , what is the biggest linearly independent set you can find? The smallest? The biggest linearly dependent set? The smallest? ("Biggest" and "smallest" mean that there are no supersets or subsets with the same property.)

Answer

In $\mathbb{R}^4$ the biggest linearly independent set has four vectors. There are many examples of such sets, this is one.

$\{\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \}$

To see that no set with five or more vectors can be independent, set up

$c_1\begin{pmatrix} a_{1,1} \\ a_{2,1} \\ a_{3,1} \\ a_{4,1} \end{pmatrix} +c_2\begin{pmatrix} a_{1,2} \\ a_{2,2} \\ a_{3,2} \\ a_{4,2} \end{pmatrix} +c_3\begin{pmatrix} a_{1,3} \\ a_{2,3} \\ a_{3,3} \\ a_{4,3} \end{pmatrix} +c_4\begin{pmatrix} a_{1,4} \\ a_{2,4} \\ a_{3,4} \\ a_{4,4} \end{pmatrix} +c_5\begin{pmatrix} a_{1,5} \\ a_{2,5} \\ a_{3,5} \\ a_{4,5} \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$

and note that the resulting linear system

$\begin{array}{*{5}{rc}r} a_{1,1}c_1 &+ &a_{1,2}c_2 &+ &a_{1,3}c_3 &+ &a_{1,4}c_4 &+ &a_{1,5}c_5 &= &0 \\ a_{2,1}c_1 &+ &a_{2,2}c_2 &+ &a_{2,3}c_3 &+ &a_{2,4}c_4 &+ &a_{2,5}c_5 &= &0 \\ a_{3,1}c_1 &+ &a_{3,2}c_2 &+ &a_{3,3}c_3 &+ &a_{3,4}c_4 &+ &a_{3,5}c_5 &= &0 \\ a_{4,1}c_1 &+ &a_{4,2}c_2 &+ &a_{4,3}c_3 &+ &a_{4,4}c_4 &+ &a_{4,5}c_5 &= &0 \end{array}$

has four equations and five unknowns, so Gauss' method must end with at least one $c$ variable free, so there are infinitely many solutions, and so the above linear relationship among the four-tall vectors has more solutions than just the trivial solution.

The smallest linearly independent set is the empty set.

The biggest linearly dependent set is $\mathbb{R}^4$ . The smallest is $\{\vec{0}\}$ .

This exercise is recommended for all readers.

Problem 19

Linear independence and linear dependence are properties of sets. We can thus naturally ask how those properties act with respect to the familiar elementary set relations and operations. In this body of this subsection we have covered the subset and superset relations. We can also consider the operations of intersection, complementation, and union.

How does linear independence relate to intersection: can an intersection of linearly independent sets be independent? Must it be?
How does linear independence relate to complementation?
Show that the union of two linearly independent sets need not be linearly independent.
Characterize when the union of two linearly independent sets is linearly independent, in terms of the intersection of the span of each.

Answer

The intersection of two linearly independent sets $S\cap T$ must be linearly independent as it is a subset of the linearly independent set $S$ (as well as the linearly independent set $T$ also, of course).
The complement of a linearly independent set is linearly dependent as it contains the zero vector.
We must produce an example. One, in $\mathbb{R}^2$ , is $S=\{\begin{pmatrix} 1 \\ 0 \end{pmatrix}\} \quad\text{and}\quad T=\{\begin{pmatrix} 2 \\ 0 \end{pmatrix}\}$ since the linear dependence of $S_1\cup S_2$ is easily seen.
The union of two linearly independent sets $S\cup T$ is linearly independent if and only if their spans have a trivial intersection $[S]\cap [T]=\{\vec{0}\}$ . To prove that, assume that $S$ and $T$ are linearly independent subsets of some vector space. For the "only if" direction, assume that the intersection of the spans is trivial $[S]\cap [T]=\{\vec{0}\}$ . Consider the set $S\cup T$ . Any linear relationship $c_1\vec{s}_1+\dots+c_n\vec{s}_n +d_1\vec{t}_1+\dots+d_m\vec{t}_m=\vec{0}$ gives $c_1\vec{s}_1+\dots+c_n\vec{s}_n= -d_1\vec{t}_1-\dots-d_m\vec{t}_m$ . The left side of that equation sums to a vector in $[S]$ , and the right side is a vector in $[T]$ . Therefore, since the intersection of the spans is trivial, both sides equal the zero vector. Because $S$ is linearly independent, all of the $c$ 's are zero. Because $T$ is linearly independent, all of the $d$ 's are zero. Thus, the original linear relationship among members of $S\cup T$ only holds if all of the coefficients are zero. That shows that $S\cup T$ is linearly independent. For the "if" half we can make the same argument in reverse. If the union $S\cup T$ is linearly independent, that is, if the only solution to $c_1\vec{s}_1+\cdots+c_n\vec{s}_n +d_1\vec{t}_1+\cdots+d_m\vec{t}_m =\vec{0}$ is the trivial solution $c_1=0$ , ..., $d_m=0$ , then any vector $\vec{v}$ in the intersection of the spans $\vec{v}=c_1\vec{s}_1+\cdots+c_n\vec{s}_n =-d_1\vec{t}_1-\cdots=d_m\vec{t}_m$ must be the zero vector because each scalar is zero.

This exercise is recommended for all readers.

Problem 20

For Theorem 1.12,

fill in the induction for the proof;
give an alternate proof that starts with the empty set and builds a sequence of linearly independent subsets of the given finite set until one appears with the same span as the given set.

Answer

We do induction on the number of vectors in the finite set $S$ . The base case is that $S$ has no elements. In this case $S$ is linearly independent and there is nothing to check— a subset of $S$ that has the same span as $S$ is $S$ itself. For the inductive step assume that the theorem is true for all sets of size $n=0$ , $n=1$ , ..., $n=k$ in order to prove that it holds when $S$ has $n=k+1$ elements. If the $k+1$ -element set $S=\{\vec{s}_0,\dots,\vec{s}_{k}\}$ is linearly independent then the theorem is trivial, so assume that it is dependent. By Corollary 1.17 there is an $\vec{s}_i$ that is a linear combination of other vectors in $S$ . Define $S_1=S-\{\vec{s}_i\}$ and note that $S_1$ has the same span as $S$ by Lemma 1.1. The set $S_1$ has $k$ elements and so the inductive hypothesis applies to give that it has a linearly independent subset with the same span. That subset of $S_1$ is the desired subset of $S$ .
Here is a sketch of the argument. The induction argument details have been left out. If the finite set $S$ is empty then there is nothing to prove. If $S=\{\vec{0}\}$ then the empty subset will do. Otherwise, take some nonzero vector $\vec{s}_1\in S$ and define $S_1=\{\vec{s}_1\}$ . If $[S_1]=[S]$ then this proof is finished by noting that $S_1$ is linearly independent. If not, then there is a nonzero vector $\vec{s}_2\in S-[S_1]$ (if every $\vec{s}\in S$ is in $[S_1]$ then $[S_1]=[S]$ ). Define $S_2=S_1\cup\{\vec{s}_2\}$ . If $[S_2]=[S]$ then this proof is finished by using Theorem 1.17 to show that $S_2$ is linearly independent. Repeat the last paragraph until a set with a big enough span appears. That must eventually happen because $S$ is finite, and $[S]$ will be reached at worst when every vector from $S$ has been used.

Problem 21

With a little calculation we can get formulas to determine whether or not a set of vectors is linearly independent.

Show that this subset of $\mathbb{R}^2$ $\{\begin{pmatrix} a \\ c \end{pmatrix},\begin{pmatrix} b \\ d \end{pmatrix}\}$ is linearly independent if and only if $ad-bc\neq 0$ .
Show that this subset of $\mathbb{R}^3$ $\{\begin{pmatrix} a \\ d \\ g \end{pmatrix}, \begin{pmatrix} b \\ e \\ h \end{pmatrix}, \begin{pmatrix} c \\ f \\ i \end{pmatrix} \}$ is linearly independent iff $aei+bfg+cdh-hfa-idb-gec \neq 0$ .
When is this subset of $\mathbb{R}^3$ $\{\begin{pmatrix} a \\ d \\ g \end{pmatrix}, \begin{pmatrix} b \\ e \\ h \end{pmatrix} \}$ linearly independent?
This is an opinion question: for a set of four vectors from $\mathbb{R}^4$ , must there be a formula involving the sixteen entries that determines independence of the set? (You needn't produce such a formula, just decide if one exists.)

Answer

Assuming first that $a\neq 0$ , $x\begin{pmatrix} a \\ c \end{pmatrix} +y\begin{pmatrix} b \\ d \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ gives $\begin{array}{*{2}{rc}r} ax &+ &by &= &0 \\ cx &+ &dy &= &0 \end{array} \;\xrightarrow[]{-(c/a)\rho_1+\rho_2}\; \begin{array}{*{2}{rc}r} ax &+ &by &= &0 \\ & &(-(c/a)b+d)y &= &0 \end{array}$ which has a solution if and only if $0\neq-(c/a)b+d=(-cb+ad)/d$ (we've assumed in this case that $a\neq 0$ , and so back substitution yields a unique solution). The $a=0$ case is also not hard— break it into the $c\neq 0$ and $c=0$ subcases and note that in these cases $ad-bc=0\cdot d-bc$ . Comment. An earlier exercise showed that a two-vector set is linearly dependent if and only if either vector is a scalar multiple of the other. That can also be used to make the calculation.
The equation $c_1\begin{pmatrix} a \\ d \\ g \end{pmatrix} +c_2\begin{pmatrix} b \\ e \\ h \end{pmatrix} +c_3\begin{pmatrix} c \\ f \\ i \end{pmatrix} =\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ gives rise to a homogeneous linear system. We proceed by writing it in matrix form and applying Gauss' method. We first reduce the matrix to upper-triangular. Assume that $a\neq 0$ . $\begin{array}{rcl} \xrightarrow[]{(1/a)\rho_1} \left(\begin{array}{*{3}{c}|c} 1 &b/a &c/a &0 \\ d &e &f &0 \\ g &h &i &0 \end{array}\right) &\xrightarrow[-g\rho_1+\rho_3]{-d\rho_1+\rho_2} &\left(\begin{array}{*{3}{c}|c} 1 &b/a &c/a &0 \\ 0 &(ae-bd)/a &(af-cd)/a &0 \\ 0 &(ah-bg)/a &(ai-cg)/a &0 \end{array}\right) \\ &\xrightarrow[]{(a/(ae-bd))\rho_2} &\left(\begin{array}{*{3}{c}|c} 1 &b/a &c/a &0 \\ 0 &1 &(af-cd)/(ae-bd) &0 \\ 0 &(ah-bg)/a &(ai-cg)/a &0 \end{array}\right) \end{array}$ (where we've assumed for the moment that $ae-bd\neq 0$ in order to do the row reduction step). Then, under the assumptions, we get this. $\begin{array}{rcl} &\xrightarrow[]{((ah-bg)/a)\rho_2+\rho_3} &\left(\begin{array}{*{3}{c}|c} 1 &\frac{b}{a} &\frac{c}{a} &0 \\ 0 &1 &\frac{af-cd}{ae-bd} &0 \\ 0 &0 &\frac{aei+bgf+cdh-hfa-idb-gec}{ae-bd} &0 \end{array}\right) \end{array}$ shows that the original system is nonsingular if and only if the $3,3$ entry is nonzero. This fraction is defined because of the $ae-bd\neq 0$ assumption, and it will equal zero if and only if its numerator equals zero. We next worry about the assumptions. First, if $a\neq 0$ but $ae-bd=0$ then we swap $\begin{array}{rcl} \left(\begin{array}{*{3}{c}|c} 1 &b/a &c/a &0 \\ 0 &0 &(af-cd)/a &0 \\ 0 &(ah-bg)/a &(ai-cg)/a &0 \end{array}\right) &\xrightarrow[]{\rho_2\leftrightarrow\rho_3} &\left(\begin{array}{*{3}{c}|c} 1 &b/a &c/a &0 \\ 0 &(ah-bg)/a &(ai-cg)/a &0 \\ 0 &0 &(af-cd)/a &0 \end{array}\right) \end{array}$ and conclude that the system is nonsingular if and only if either $ah-bg=0$ or $af-cd=0$ . That's the same as asking that their product be zero: $\begin{array}{rl} ahaf-ahcd-bgaf+bgcd &=0 \\ ahaf-ahcd-bgaf+aegc &=0 \\ a(haf-hcd-bgf+egc) &=0 \end{array}$ (in going from the first line to the second we've applied the case assumption that $ae-bd=0$ by substituting $ae$ for $bd$ ). Since we are assuming that $a\neq 0$ , we have that $haf-hcd-bgf+egc=0$ . With $ae-bd=0$ we can rewrite this to fit the form we need: in this $a\neq 0$ and $ae-bd=0$ case, the given system is nonsingular when $haf-hcd-bgf+egc-i(ae-bd)=0$ , as required. The remaining cases have the same character. Do the $a=0$ but $d\neq 0$ case and the $a=0$ and $d=0$ but $g\neq 0$ case by first swapping rows and then going on as above. The $a=0$ , $d=0$ , and $g=0$ case is easy— a set with a zero vector is linearly dependent, and the formula comes out to equal zero.
It is linearly dependent if and only if either vector is a multiple of the other. That is, it is not independent iff $\begin{pmatrix} a \\ d \\ g \end{pmatrix}=r\cdot\begin{pmatrix} b \\ e \\ h \end{pmatrix} \quad\text{or}\quad \begin{pmatrix} b \\ e \\ h \end{pmatrix}=s\cdot\begin{pmatrix} a \\ d \\ g \end{pmatrix}$ (or both) for some scalars $r$ and $s$ . Eliminating $r$ and $s$ in order to restate this condition only in terms of the given letters $a$ , $b$ , $d$ , $e$ , $g$ , $h$ , we have that it is not independent— it is dependent— iff $ae-bd=ah-gb=dh-ge$ .
Dependence or independence is a function of the indices, so there is indeed a formula (although at first glance a person might think the formula involves cases: "if the first component of the first vector is zero then ...", this guess turns out not to be correct).

This exercise is recommended for all readers.

Problem 22

Prove that a set of two perpendicular nonzero vectors from $\mathbb{R}^n$ is linearly independent when $n>1$ .
What if $n=1$ ? $n=0$ ?
Generalize to more than two vectors.

Answer

Recall that two vectors from $\mathbb{R}^n$ are perpendicular if and only if their dot product is zero.

Assume that $\vec{v}$ and $\vec{w}$ are perpendicular nonzero vectors in $\mathbb{R}^n$ , with $n>1$ . With the linear relationship $c\vec{v}+d\vec{w}=\vec{0}$ , apply $\vec{v}$ to both sides to conclude that $c\cdot|\vec{v}|^2+d\cdot 0=0$ . Because $\vec{v}\neq\vec{0}$ we have that $c=0$ . A similar application of $\vec{w}$ shows that $d=0$ .
Two vectors in $\mathbb{R}^1$ are perpendicular if and only if at least one of them is zero. We define $\mathbb{R}^0$ to be a trivial space, and so both $\vec{v}$ and $\vec{w}$ are the zero vector.
The right generalization is to look at a set $\{\vec{v}_1,\dots,\vec{v}_n\}\subseteq\mathbb{R}^k$ of vectors that are mutually orthogonal (also called pairwise perpendicular): if $i\neq j$ then $\vec{v}_i$ is perpendicular to $\vec{v}_j$ . Mimicking the proof of the first item above shows that such a set of nonzero vectors is linearly independent.

Problem 23

Consider the set of functions from the open interval $(-1..1)$ to $\mathbb{R}$ .

Show that this set is a vector space under the usual operations.
Recall the formula for the sum of an infinite geometric series: $1+x+x^2+\cdots=1/(1-x)$ for all $x\in(-1..1)$ . Why does this not express a dependence inside of the set $\{g(x)=1/(1-x),f_0(x)=1,f_1(x)=x,f_2(x)=x^2,\ldots\}$ (in the vector space that we are considering)? (Hint. Review the definition of linear combination.)
Show that the set in the prior item is linearly independent.

This shows that some vector spaces exist with linearly independent subsets that are infinite.

Answer

This check is routine.
The summation is infinite (has infinitely many summands). The definition of linear combination involves only finite sums.
No nontrivial finite sum of members of $\{g,f_0,f_1,\ldots\}$ adds to the zero object: assume that $c_0\cdot (1/(1-x))+c_1\cdot 1+\dots+c_n\cdot x^n=0$ (any finite sum uses a highest power, here $n$ ). Multiply both sides by $1-x$ to conclude that each coefficient is zero, because a polynomial describes the zero function only when it is the zero polynomial.

Problem 24

Show that, where $S$ is a subspace of $V$ , if a subset $T$ of $S$ is linearly independent in $S$ then $T$ is also linearly independent in $V$ . Is that "only if"?

Answer

It is both "if" and "only if".

Let $T$ be a subset of the subspace $S$ of the vector space $V$ . The assertion that any linear relationship $c_1\vec{t}_1+\dots+c_n\vec{t}_n=\vec{0}$ among members of $T$ must be the trivial relationship $c_1=0$ , ..., $c_n=0$ is a statement that holds in $S$ if and only if it holds in $V$ , because the subspace $S$ inherits its addition and scalar multiplication operations from $V$ .

Neoliberal international relations thinkers often employ game theory to explain why states do or do not cooperate; since their approach tends to emphasize the possibility of mutual wins, they are interested in institutions which can arrange jointly profitable arrangements and compromises.

Neoliberalism is a response to Neorealism; while not denying the anarchic nature of the international system, neoliberals argue that its importance and effect has been exaggerated. The neoliberal argument is focused on the neorealists' underestimation of "the varieties of cooperative behavior possible within ... a decentralized system."[2] Both theories, however, consider the state and its interests as the central subject of analysis; neoliberalism may have a wider conception of what those interests are.

Neoliberalism argues that even in an anarchic system of autonomous rational states, cooperation can emerge through the building of norms, regimes and institutions.

. The main points of neo-liberalism include:

THE RULE OF THE MARKET. Liberating "free" enterprise or private enterprise from any bonds imposed by the government (the state) no matter how much social damage this causes. Greater openness to international trade and investment, as in NAFTA. Reduce wages by de-unionizing workers and eliminating workers' rights that had been won over many years of struggle. No more price controls. All in all, total freedom of movement for capital, goods and services. To convince us this is good for us, they say "an unregulated market is the best way to increase economic growth, which will ultimately benefit everyone." It's like Reagan's "supply-side" and "trickle-down" economics -- but somehow the wealth didn't trickle down very much.
CUTTING PUBLIC EXPENDITURE FOR SOCIAL SERVICES like education and health care. REDUCING THE SAFETY-NET FOR THE POOR, and even maintenance of roads, bridges, water supply -- again in the name of reducing government's role. Of course, they don't oppose government subsidies and tax benefits for business.
DEREGULATION. Reduce government regulation of everything that could diminsh profits, including protecting the environmentand safety on the job.
PRIVATIZATION. Sell state-owned enterprises, goods and services to private investors. This includes banks, key industries, railroads, toll highways, electricity, schools, hospitals and even fresh water. Although usually done in the name of greater efficiency, which is often needed, privatization has mainly had the effect of concentrating wealth even more in a few hands and making the public pay even more for its needs.
ELIMINATING THE CONCEPT OF "THE PUBLIC GOOD" or "COMMUNITY" and replacing it with "individual responsibility." Pressuring the poorest people in a society to find solutions to their lack of health care, education and social security all by themselves -- then blaming them, if they fail, as "lazy."
Around the world, neo-liberalism has been imposed by powerful financial institutions like the International Monetary Fund (IMF), the World Bank and the Inter-American Development Bank. It is raging all over Latin America. The first clear example of neo-liberalism at work came in Chile (with thanks to University of Chicago economist Milton Friedman), after the CIA-supported coup against the popularly elected Allende regime in 1973. Other countries followed, with some of the worst effects in Mexico where wages declined 40 to 50% in the first year of NAFTA while the cost of living rose by 80%. Over 20,000 small and medium businesses have failed and more than 1,000 state-owned enterprises have been privatized in Mexico. As one scholar said, "Neoliberalism means the neo-colonization of Latin America."
In the United States neo-liberalism is destroying welfare programs; attacking the rights of labor (including all immigrant workers); and cutbacking social programs. The Republican "Contract" on America is pure neo-liberalism. Its supporters are working hard to deny protection to children, youth, women, the planet itself -- and trying to trick us into acceptance by saying this will "get government off my back." The beneficiaries of neo-liberalism are a minority of the world's people. For the vast majority it brings even more suffering than before: suffering without the small, hard-won gains of the last 60 years, suffering without end.
Despite the fact that there are some similarities between neo-realism and neo-liberalism, it shall be the differences between these theories that will be the focus of my attention, as it will help me to determine more rigorously which of the arguments is the more convincing. The points of comparison shall be the effects of the anarchical international system, and thus, the extent to which cooperation can be achieved, the importance of relative and absolute gains, the conflict between state capabilities and interests, and finally the importance of institutions and regimes. It is important to note that neo-realism is often also called structural realism, and neo-liberalism neo-liberal institutionalism. As the question prefers to call the theories neo-realism and neo-liberalism, this is what I shall do throughout.
The first thing to consider is the effects of the international system, and what this means for the prospects of cooperation. Lamy asserts that “both [neo-realism and neo-liberalism] agree that the international system is anarchic” (Baylis and Smith 2001, p. 190). I question that the international system is continuously anarchic, and agree with Alexander Wendt, who would claim that there is nothing inevitable about the international system. No matter, the important thing is that neo-realists and neo-liberals agree that the international system is anarchic. The major point of contention is that neo-liberals are sure such a system will not constrain the foreign policy options of the state to simple survival, with neo-realists essentially believing the opposite (ibid). As neo-realists have the idea that man by nature has a restless desire for power (Keohane 1986 pp. 211-212), cooperation becomes more difficult to achieve, because in trying to gain power a state will upset another state in doing so. Neo-liberals agree that states act in their own interests, but have a greater belief in cooperation, for the very reason that “it is in the self-interest of each [actor] to cooperate” (Mingst 2004, p. 64). The situation the neo-liberals put forward is the prisoner’s dilemma, a tale of two prisoners who are questioned after committing an alleged crime. Neither prisoner knows what is being said by the other, but if they both cooperate and confess to the crime, their time in prison will be shortened, where if neither confesses the sentence length will be even shorter (Mingst 2004, pp 63-64). However, if one confesses and the other does not, then the one who confessed will be set free and the one who did not will receive a lengthy jail term. This risk is why both will confess, and thus get a reduced sentence (ibid). However, it was in the interest of each to “have cooperated with each other by remaining silent” (Mingst 2004, p. 64), and thus have received the shortest sentence. This is a scenario neo-liberals use to explain why states would wish to cooperate with each other, with the implication that there is a great risk for states if they do not.
However, neo-realists are certain that cooperation “will not happen unless states make it happen” (Baylis and Smith 2001, p. 190). It is unlikely that states will often “make it happen” because, from the viewpoint of the state, involvement in international cooperation and a strengthening of your own position rarely go hand in hand. For example, in summer 2003Iranrejected the chance to sign an international agreement with 168 states that would “eliminate discrimination against women” because “it was against Sharia law and the constitution” (www.bbc.co.uk). As aforementioned, neo-liberals believe states can cooperate with one another, but “especially with the assistance of international institutions” (Baldwin 1993, p. 117). Neo-liberals believe in the effectiveness of organisations such as the UN Commission on Human Rights, to give an example, as even countries such as China and Russia have wished “to avoid the sting of criticism by the foremost human rights body” (www.un-globalsecurity.org). However, I am deeply sceptical about the overall neo-liberal position on cooperation. Why does theUSadministration refuse to cooperate with theKyototreaty? The answer, to my mind, is because it is not within their interests to do so. This is hardly a unique view, as it is acknowledged elsewhere that George W. Bush “said he wouldn’t be ratifying theKyotoprotocol because it could significantly damage the country financially” (www.bbc.co.uk). So we have a clear case of it not being within state interest to cooperate, in this case because it would weaken their state economically. I think neo-liberals lack conviction in claiming that it is always within a states’ interests to cooperate, and would have to question whether the prisoner’s dilemma is flexible enough to be applied to any given situation, for it does not appear to account for the US policy on Kyoto. They might say that the US are simply not intelligent enough to realise that it’s in their interests to cooperate, but the neo-realist focus on power and thus the argument that cooperation is more difficult to achieve appears to have greater credence.
Additionally, following cooperation, we have the debate between the two theories about the importance of absolute and relative gains, with focus on the latter proving a considerable obstruction to cooperation. Neo-realists speak of how vital relative gains are in state considerations, or to put it another way, states “are compelled to ask not “will both of us gain?” but “who will gain more?”” (Mingst 2004, p. 69) As highlighted by Tim Dunne in a recent lecture, this neo-realist philosophy can be seen in mercantilist economics. To illustrate this point, if a state adopted a mercantilist approach they would rather that their own economic growth stood at 3% and rival states at 3%, than that their own economy grew by 5% and their rivals’ by 6%. In other words, under neo-realist thought states simply want to gain a comparative advantage, whatever that may be, ahead of performing very well along with other states. Another example of relative gains in practice sees economic relations being influenced by relative gains, such as “America’s forty-year effort to use export controls to weaken soviet military, and at times economic, capabilities” (Baldwin 1993, p. 256). Neo-liberals, on the other hand, are sure states are happy to have any gain for themselves. This is a focus on absolute gains, i.e. “state leaders will accept any accord that makes the state better off regardless of the gain achieved by any other state” (www.ssc.upenn.edu). For example, the Iraqi interim Prime Minister Iyad Allawi happily accepted that “we are better off… without Saddam Hussein”, yet at the same timeAmericaprobably gains more from the democratisation ofIraqas “terrorists will suffer a dramatic defeat” (www.bbc.co.uk). It is my belief that neo-realists are right to stress the importance of relative gains; it is true that some states usually gain more than others. At the recent G7 “100% debt relief summit” America could not support the plans to boost aid for developing countries because “it does not work for the United States”, with the acknowledgement that “it works for other countries” (www.bbc.co.uk). So there is a clear suggestion thatAmericafelt the plan was not worthwhile because other countries would gain more than themselves, a fine reflection of the neo-realist belief in the importance of relative gains.
The next area of debate is of the respective importance of capabilities (essentially power) and interests. Neo-realists “are likely to emphasise capabilities more than intentions” (Baldwin 1993, p. 7). Clearly, power is of the utmost importance to neo-realists, as the fact that theUSand theUSSRwere the two cold war superpowers “explains the similarity in their behaviour [at the time]” (Baylis and Smith 2001, p. 185). Neo-realists are probably right to point out that you can be more certain of the capabilities of the state than their intentions and interests. When France promised to “veto a second resolution, “whatever the circumstances” Tony Blair “could not at first believe that Chirac had said this” (Seldon 2004, p. 592), such was his surprise that this was the French intention. On the other hand, any assertion about capability “begs two vital questions-“capability to get whom to do what?”” (Baldwin 1993, p. 17) If these questions cannot be answered satisfactorily, then a dent in the crucial neo-realist argument that capabilities help shape state behaviour can be found. Neo-liberals are keen to say how vital intentions and interests are to state behaviour. The notable neo-liberal Stein argues that “capabilities count only insofar as they affect the preferences and intentions of states” (Baldwin 1993, p. 8). I feel that it is essential to point out that it is rather dangerous to assume that states always know exactly what their capabilities are. Mingst describes power as “the ability to influence others” (Mingst 2004, p. 321). Tony Blair being confident of UN support for theIraqwar is a case of a leader not accurately understanding their states’ capabilities. His intentions were not adjusted in this case, but it certainly made Blair less certain that committing to war was the right think to do. This clear potential for capabilities to affect the intentions of states supports this argument of states and the neo-liberals. This conviction that capabilities are not important in themselves, only for intentions and interests, is perhaps the most convincing of all the neo-liberal arguments, but I am still convinced more by the neo-realist position. It is true that nobody can ever be entirely certain of neither intentions nor capabilities, but I certainly feel it is easier to gain an accurate estimation of the latter. This is so because the ability to influence others depends on matters such as economic power, something that is easy to establish in this modern, technological world. Thus, as states will find it more difficult to establish intentions and interests of other states, they will look at other states’ capabilities as the crucial component shaping their own behaviour.
Finally, we have somewhat different views on international institutions and regimes. With both of these, there are a clear set of rules for state behaviour (Baylis and Smith 2001, p. 189). While neo-realists have not exactly derided institutions and regimes, they believe that neo-liberals have overestimated their importance and believe themselves that states only “work to establish these regimes and institutions if they serve their interests” (Baylis and Smith 2001, p. 192), not if they do not. With regard to this neo-realist argument, it is worth bearing in mind that the ordering principle of the international system is anarchy (Baylis and Smith 2001, p. 185). Anarchy is defined as “a situation where nobody obeys laws or rules” (Hanks 1993). This may be a somewhat simplistic definition, but it is clear nonetheless that institutions and regimes cannot get round this dominance of anarchy in the international system. The fact that an institution such as the UN could not stop theIraqwar would appear to support the neo-realist argument very well. As I have touched upon earlier, neo-liberals believe institutions and regimes facilitate cooperation, and that institutions “moderate state behaviour” (Mingst 2004, p. 85). Furthermore, neo-liberals believe institutions “make it easier to punish cheaters” (Baldwin 1993, p. 124). This may be true when they are effective, but unfortunately a lot of the time they are not. Not only were they ineffective overIraq, but going back in history, the League of Nations failed “to take assertive action against Japanese, Italian, and German aggression in the 1930s” (Baylis and Smith 2001, p. 56). Any neo-liberal ideas that institutions such as Greenpeace will check the actions of states should be approached with caution. In 1944, “attempts were made to ‘de-politicise the international economy,” but by 1971 President Nixon abandoned such principles to “strengthen his political position at home” (Brown 2001, pgs 161 and 165). It is far from impossible that institutions such as Greenpeace will operate at the behest of the leading states, and some might say they do currently. I am far more convinced by the neo-realist argument on institutions, because there is nothing that makes me think they will determine state behaviour consistently in the future, for states in the past have dismissed institutions contemptuously.
My overwhelming feeling is that neo-realism is a more convincing argument than neo-liberalism. I feel they are right to assert that cooperation will only occur if sates want it to, and do not agree that it is always within state interest to cooperate. I am certain that relative gains are an important consideration for certain states and cannot be ignored. It appears that states focus on the capabilities of other states because these are easier to measure than intentions. Nor am I convinced that institutions will continue to play a vital role in checking the actions of states. To put it another way, maybe, as all liberalisms are prone to do, the problem with neo-liberalism is that it focuses on how the world “ought” to be, rather than how it is. In the fragile international climate following September 11th, I happen to believe that neo-realist focus on anarchy as the dominating force in the international system is a more accurate reflection of the world we live in.

Neorealism is microeconomic and Neoliberalism the efficiency of private enterprise. One keeping things small and one expanding.

First, the similarities: Both schools of thought believe the international world to be anarchic. There is no world government that regulates states actions.

The differences.

Neorealism:

Because the world is anarchic, war is inevitable and is going to be around forever. They favour a bipolar world (think USA and USSR), because it is easier to balance and appease the other side. When there are many sides, it becomes very difficult to hear everyone's issues and deal with them. So, a bi-polar world for neorealists is ideal because it leads to less war. Further, and this is the big thing that makes it realism, is that they believe states are self interested only. If a state intervenes in the affairs of another country, then they are not doing it out of the goodness of their own heart. They have a personal interest, whether it is visible or not. But, the implications of neorealism are usually non-intervention, period. Instead, they do advocate a large military, but it is for defense purpouses only.

The difference between neorealism and classical realism is that classical realists made claims that it was human nature to be self interested. Neorealists do not believe this.

Neoliberalism:

War is preventable, for neoliberals. They believe that it is possible for states to co-operate, and that we do have a duty to come to the aid of other countries (whether it is military aid or not). I had a teacher who used to say "liberalism will get you killed." That's because there is no interventionism in realism, but there is in liberalism. A crucial notion in neoliberal philosophy is "complex interdependence." It is exactly what it sounds like. States are all so interdependent and connected that it makes more sense to co-operate than to wage war.

In the real world, liberalism is more common. There aren't really any states that have realist foreign policy. It is mostly liberal, even places like China. In the era of globalization, realism doesn't really work that well.

The debate between neorealism and neoliberal institutionalism has dominated International Relations (IR) theory, particularly in the United States. The ‘neo-neo’ debate has brought much contention between the scholars of IR, equally the two schools of thought have been considered by many to be remarkably similar. In the first part of this essay I will outline the framework of the ‘neo-neo’ debate, discussing the fundamental points of contention between neorealism and neoliberal institutionalism. Two prominent Institutionalists, Robert Keohane and Lisa Martin (1995), have suggested that “for better or for worse institutional theory is a half-sibling of neo-realism” (Keohane & Martin, 1995, cited in Lamy, 2005, p.215). The study of IR has experienced dramatic change as the foundational epistemology has been criticized by post-modern theorists who attack the underlying assumptions of positivism. As post-positivists are simply united through their opposition to the positivist movement, it is not a clean two-sided debate.

In the second part of this essay I will highlight the fundamental similarities that bring neorealism and neoliberal institutionalism under the theoretical umbrella of rationalism, whilst comparing the rationalist position to the recently surfaced reflectivism. As the debate has evolved the common assumptions of neorealism and neoliberal institutionalism have become increasingly obvious. In the final part of the essay I will analyse the similar assumptions of the international system held by rationalist theories, with particularly close attention paid to ’anarchy’, ’self-help’, and ’collective security’. To some extent the ‘Great Debate’ was an artificially constructed ‘debate’, invented for “specific presentational purposes, teaching and self-reflection of the discipline” (Waever, 1996, p.161). Moreover, the debate has highlighted the comparable paradigm positions of neorealism and neoliberal institutionalism, giving rise to a ‘neo-neo synthesis’ (Waever, 1996, p. 163), further consolidating the idea that the two approaches are simply manifestation of the same approach.

The ‘neo-neo’ debate

The debate between neorealism and neoliberal institutionalism has dominated IR debate for decades. The two schools of thought have jostled over views of the international system in an attempt to define the world of international politics. These two paradigms have been important to defining policymaking and the research within international relations (Lamy, 2005, p.207). The debate is characterized by their disagreement over specific issues such as: the nature and consequences of anarchy, international cooperation, relative versus absolute gains, intentions versus capabilities, institutions and regimes, and priority of state goals.

Kenneth Waltz is one of many scholars responsible for expanding the ideas of traditional realists such as Hans Morganthau, who looked at the actions and interactions between states in the system, in an attempt to explain international politics (Lamy, 2005, p.208). Neorealism looks to separate the internal factors of the international political systems from the external. This separation isolates one realm from another, allowing theorists to deal with each at an intellectual level. Neorealists focus on the structure of the system, analysing the variations, how they affect the interacting units, and the outcomes they produce (Waltz, 1990, p.29).

Waltz (1986) claims that the anarchic international system was a force that fashioned the states which constitute the system. The structure of the anarchic system compelled states to worry about security and take adequate measures achieve it. The preferences of states could not explain international outcomes, rather, Waltz argued that “state behaviour varies more with differences of power than with difference in ideology, in internal structure of property relations or in governmental form” (Waltz, 1986, cited in Walt, 2002, p.202-203).

Where neorealists were seen to focus on security measures, neoliberal institutionalists are believed to have placed greater emphasis upon environmental and economic issues, with a specific focus on the latter. Keohane and Nye (2001) argue that interdependence, particularly economic interdependence, is now an important feature of world politics. Furthermore, Keohane and Nye argue that states are dominant actors in international relations; equally there is an assumption that hierarchy exists within international politics and force can be used as an effective instrument of policy. Globalization represents an increase in interconnectedness and linkages; this mutual interdependence between states positively affects behavioural patterns and changes the way states cooperate (Keohane and Nye, 2001).

The realist view on international cooperation is rather more pessimistic. As man by nature has a restless desire for power and self-interest (Keohane, 1986, p.211-212), cooperation becomes difficult to achieve as this strive for power is likely to upset the status-quo. According to Mearsheimer (1995), the two main obstructions to international cooperation are relative gains considerations and cheating, both of which stem from the logic of anarchy (Mearsheimer, 1995, p.12). Grieco (1988) argues that realists find that states are positional, not atomistic, in character; therefore as well as being anxious about cheating, states are primarily concerned with how their partners might benefit from any cooperative arrangements (Grieco, 1988, p.487). Since international relations are a zero-sum game, states compete with each other to ensure their own benefits outweigh that of others.

For realists, survival within the anarchic international system is paramount. The intentions of states are unknown and subsequently state actors are cautious about the gains of others when cooperating; a friend may gain from cooperation one day and use it as a threat the next. Waltz (1979) argues, under global anarchy, “when faced with the possibility of cooperating for mutual gains, states that feel insecure must ask how the gain will be divided. They are compelled to ask not ‘Will both of us gain?’ but ‘Who will gain more?’” (Waltz, 1979, cited in Kegley, 2008, p.30). For neorealist’s, balance of power is essential to understanding world politics; when states have such concerns about the balance of power cooperation is much more difficult to achieve.

Neoliberals show more concern as to how a state benefits overall, as opposed to how a state will benefit in comparison to others; it is suggested that policy makers will consider absolute gains to be made from an agreement, including potential longer-term gains. Neoliberals argue that to focus on relative gains is misguided as economic interdependence ensures that neither side can effectively exploit the economic relationship and take advantage of the other politically. Mastanduno (1991) suggests that relative gains can be destructive as they are conducive to the twin evils of protectionism and nationalism (Mastanduno, 1991, p. 76).To focus on distribution of benefit could affect the total benefit overall.

Neoliberal institutionalists agree that states act in their own interests, yet hold a much more optimistic view on cooperation. Keohane (1984) recognized that cooperation is not an easy feat and can lead to tension, but states could potentially benefit from cooperative strategies (Keohane, 1984). Duncan Snidal (1991) believes that if absolute gains from cooperation are considerable then relative gains are likely to have minimal effect on cooperation (Snidal, 1991, cited in Keohane & Martin, 1995, p.44). Like realists, institutionalists are concerned about cheating, but unlike neorealists, they place great faith in institutions themselves. Institutions provide a coordinating mechanism to help states capture potential gains from cooperation; this “constructed focal point” increases the opportunity of cooperative outcomes (Keohane & Martin, 1995, p.45). Furthermore, institutions provide an arbitrary body that is able to provide states with information preventing states from cheating. As explained in the game theory, more specifically Prisoners dilemma, states seek to maximize individual pay-offs, and so institutions offer a platform through which greater coordination and cooperation can be executed, subsequently benefitting both parties.

In Mearsheimer’s article The False Promise of International Institutions, he purports that institutions reflect the distribution of power in the world; moreover, institutions have little influence on state behavior and offer diminutive opportunity for holding stability in a post Cold War period. Where neoliberals believe there to be strong correlation between institutions, economic cooperation and peace, neorealists doubt the link made between cooperation and stability as neoliberal theorists avoid military issues (Mearsheimer, 1995).

Mearsheimer (1995) argues that absolute gains logic can only apply to the economic realm, whereas relative gains apply to the security realm. Neoliberal institutionalists attempt to divide a line between the economic and security realm, yet there is correlation between economic might and military might. If neoliberals accept this realist claim that states act in accordance to self interest in an anarchic system where military powers matter, then according to Mearsheimer they must deal with the issue of relative gains (Mearsheimer, 1995, p.20). Keohane and Martin (1995) recognize that there is not a clear analytical line between security and economic issues, but institutionalist theory has placed an importance on the role of institutions providing information removing the problem of uncertainty (Keohane & Martin, 1995, p.43).

Driven by survival, neorealists are sensitive to any erosion of their relative capabilities as these factors are the basis for security and independence (Grieco, 1988, p.498). Similarly, Krasner (1991) criticizes the neoliberal school of thought for placing too much emphasis upon intentions, interests, and information, paying little attention to the distribution of capabilities (Krasner, 1991, cited in Baldwin, 1993, p.7) Againinstitutionalists envisage the issue of capabilities being amended through security institutions signaling governments’ intentions by providing others with adequate information. Institutions reflect advancing principles and norms of community standards lowering the costs of multilateral enforcement strategies (Kay, 2011, p.60).

The ‘inter-paradigm’ debate between neorealists and neoliberal institutionalists lasted for decades as scholars continued to pick flaws in position of the opposing approach, in an attempt highlight problems with the causal logic. It was not until the emergence of alternative approaches to international theory did the axis of the debate change.

Positivism/rationalism & post-positivism /reflectivism

The ‘inter-paradigm’ debate that has taken place within IR fails to illuminate the ongoing controversies in the discipline; the ‘neo-neo’ debate is not the story today (Waever, 1996, p.149). The debate between neorealism and neoliberal institutionalism has been sidelined as a thing of the past, as these two theoretical approaches essentially share similar views of the social world. The fourth debate between positivism and post-positivism, or rationalism and reflectivism, emerged in the late 1980s. This emerging debate is centred as much on epistemological and ontological basis of IR as on theoretical claims and methodologies (Doherty, 2000, p.235). In the following section I plan to illustrate how neorealism and neoliberal institutionalism fall under the umbrella of positivism or rationalism, and how they differ to the reflectivist approach.

According to Waever (1996) both neorealism and neoliberal institutionalism underwent a “self-limiting redefinition towards an anti-metaphysical, theoretical minimalism, and they became increasingly compatible”; a dominant neo-neo synthesis became the research programme in the 1980’s (Waever, 1996, p.163). In a presidential address for ISA in 1988, Keohane clearly brought both neorealism and neoliberalism under the umbrella of rationalism. The ‘inter-paradigm’ debate had been diluted as the two approaches share a ‘rationalist’ research programme, a conception for science, a similar approach anarchy and willingness to assess the evolution of co-operation and whether institutions matter (Waever, 1996, p.163). This ‘redefinition’ ultimately changes the axis of debate within study of IR.

Neoliberals and neorealists are two views of the same approach. Both assume similar positions regarding the international system: states are main actors, they act rationally, and international anarchy shapes their behaviour. Most notably, neorealism and neoliberal share similar methodology, epistemology and ontology. The methods by which neorealists and neoliberals study the world are analogous. Crucially, they agree that the acquisition of knowledge is based on the liberal notion of power and politics, which under-problematises the use of empirical material (Smith, 1997). Simon (1985) argues that rationalism is contextual, much depending of the presuppositions before the analysis. The principle of rationality is to formulate hypotheses about the real human behaviour, but must have combined additional assumptions about the structure of utility functions and the formation of expectations (Simon, 1985, cited in Keohane, 1988, p.381).

Positivism is the epistemological approach taken by rationalist theorists. Positivism holds onto the idea that there international system is essentially the same as the systems in the natural world. The scientific approach of positivism views both the social and political world as having patterns and regularities, a type of naturalism, suggesting that observation and experience is crucial to formulating and reviewing scientific theory. Positivist IR scholars draw a basic distinction between empirical theory and normative theory, and therefore remain neutral between theories. In philosophical terms this is an objectivist position, one that recognizes that observations may be subjective, yet objective knowledge in the world is possible (Smith, 1996, p.16). Rather than spend time on debates about what the world should look like, positivists prefer to look at the way thing ‘really’ are (Smith, 2005).

Positivism has been a methodological commitment, tied to an empiricist epistemology, which undeniably restricts the range of permissible ontological claims (Smith, 1996, p.17). Neorealism and neoliberalism share a similar materialist ontological approach to theoretical analysis. For rationalists, reality is comprised of tangible and palpable objects; therefore the theory of knowledge is interlinked with materialism. This materialist approach reduces everything to matter and what is observable. Social processes (culture, values and norms) between state actors are an indirect function of the material dimension.

Positivism has been the dominant epistemology of IR theorists throughout history. Elias and Sutch (2007) go as far to suggest that positivists have acted as gatekeepers by setting strong parameters as to what would count as a fact in the discipline, using this to prevent non-positivist forms of knowledge from being examined (Elias & Sutch, 2007, p.14) The ‘fourth debate’ debate according to Lapid moved away from positivist assumptions and stimulated self-reflection and pointed towards new measures of objectivity (Waever, 1996, p. 156). Scholars of reflectivism take a more sociological approach to understanding world politics. Reflectivism has given birth to a number of adherent sub-discipline theories of IR such as feminist theory, critical theory, normative theory, historical sociology, and post-modernism. These opposing theories cannot simply be merged together as a counter to the ‘neo-neo synthesis’ approach as each theory differs enormously and vary as to how they construct knowledge. What unites these theoretical perspectives is how each of these one of these theories reject one or more key assumptions of the rationalist accounts, constituting the birth of post-positivism.

On the other hand reflectivists maintain an idealist approach to ontology. Rather than being concerned with materialism, they argue that the social world is constructed by the ideas and values; language, ideas and concepts are at the basis of the reflectivist approach. This subjectivist position suggests that international relations is at its most basic form an idea or concept that people share about how states should organize themselves and relate to each other politically (Jackson & Sorensen, 2007, p.300).

Unlike the mainstream theorists of IR, reflectivist theorists adopt a post-positivist epistemological approach, rejecting the idea that social sciences can adopt the empiricist observation of the natural sciences. For post-positivists reality is a subjective creation of people; reflectivist theory looks to understand political phenomena through asking relevant questions that help determine what contributes to certain outcomes within the international arena (Jackson & Sorensen, 2007, p.300). Rather than merely focussing on high politics of the state, post-positivism looks to promote a normative approach to IR and go ‘beyond’ the Wesphalian model, opening the debate towards issues such as poverty, disease, migration, religious and cultural pluralism, gender issues, environmentalism, human rights and humanitarian intervention (Elias & Sutch, 2007, p.14).

In a movement away from the objective, value-free, universal knowledge that characterised the rationalist and positivist movement, post-positivism looked to interpret and explain why things are the way they are, as opposed to merely describing what they were (McNabb, 2010, p.19). Cox (1981) argued that ‘theory is always for someone, and for some purpose’ (Cox, 1981, p.128), suggesting that the time or context play a role in developing social knowledge; therefore contrary to positivist belief, the facts that constitute this knowledge cannot be objective and must reflect some aspect of the value of its origin. Similarly normative theory, a sub-discipline of the reflectivist approach, takes two issues with this idea that facts are not value-laden. First, it is a very narrow definition of what politics is about, with too much focus on the politics that ‘really’ already exists in the social world.

The second problems is that all theories reflect the values of what the theorist chooses to focus on and explain as the ‘facts’, through the methods they use to study these ’facts’, down to the policy prescriptions they suggest (Owens & Smith, 2005, p.279). Rationalism ignores the social processes that lead to changes in the outlook of world politics. Preferences are assumed to be fixed, which prohibits research from understanding how interests and beliefs change over time, whereas reflective theorists look to understand how politics has changed based on post hoc observation of values or ideology.

Reflectivism posits an emphasis upon interpretation, or “inter-subjective meanings” of international institutional activity; Keohane appropriately labelled these theorists as “interpretive scholars” (Kratochwill & Ruggie, 1986, cited in Keohane, 1988, p.381). Reflective theorists have a different understanding as to what institutions constitute and represent. Unlike rationalists who believe that institutions echo the power and preferences of unit constituting them, reflectivists argue that the preferences of individuals are not treated as exogenous; values, norms and practices differ across international society and so effect the formation of institutions. This approach is “a critical process of inquiry that goes beyond surface illusions to uncover the real structures in the material world in order to help people change conditions and build a better world for themselves” (Neuman, 2000, cited in McNabb, 2010, p.20).

Common assumptions of both neorealism and neoliberal institutionalism

As discussed, we can see that neorealism and neoliberalism have their differences, yet equally they share similar analytical premises. Both are state-centric structural theories, using state actors as basic units of theoretical analysis. Through the state-centric approach both theories try to explain the behaviour of states with reference to the material structure of the international system (Thomas, 2001, p.10). Whether concerned with relative-gains or absolute-gains, there is common agreement that states act within the rational choice model.

Grieco (1988) recognizes that for both realists and neoliberals there is a common understanding of international anarchy, an absence of a common inter-state government (Grieco, 1988, p.497). In his ground breaking book Theory of International Politics (1979) Kenneth Waltz focussed on the ‘structure’ of the international system and the ramifications of the structure of international relations. A defining feature of the international system is that it is anarchic, with no overarching power governing states (Jackson & Sorensen, 2007, p.46). For neoliberals, international order is defined by the state of anarchy, but contrary to realists, this absence of an overarching authority does not mean that we are in a constant state of war. Charles Lipson (1984) believes that anarchy is the “rosetta stone of international relations”, although neorealists slightly exaggerate the importance of anarchy at the expense of interdependence; nevertheless both neorealists and neoliberals recognize anarchy as fundamental to shaping the future of world politics (Lipson, 1984, cited in Baldwin, 1993, p.4).

Although neorealists were primarily concerned with security, and neoliberals focussed on the economy, rationalist theories share a common analytical starting point: i.e states are in the self-interested main actors within the anarchic international system (Baldwin, 1993). Regardless of their slight differences, this self-help approach to anarchy held by rationalists generates a competitive notion to security and creates an issue for collective action. The logic of self-help encourages states to adapt to the system. Although neoliberals have conceded to neorealist the causal powers of the anarchic structure, they argue that this process of self-help can spawn cooperative behaviour between states, even in an exogenously given, self-help system (Wendt, 1992, p.392).

The rationalist approach provides analytical debate for notable issues within the study of IR, such as cooperation among great powers, but offer little guidance in situations where their basic ontological assumption that states are autonomous actors is violated. If decisions made within a state are constrained by external factors, the autonomy is not demonstrated. For both neorealism and neoliberalism, the Westphalian model presents a logical paradox as both theories assume autonomy and self-help. A logical contradiction between self-help and autonomy is purported through focussing on wars between great powers or economic bargaining between major powers, where autonomy is rarely an issue; these actions are not consistent with the rules and principles of Westphalian and international legal sovereignty (Krasner, 1999).

Arguments of collective security recognise the importance of military force as a characteristic of international life, but similarly advocates of this theoretical approach believe that there are realistic opportunities to move beyond the self-help world of realism. In order to accept collective security one must adhere to three main principles. First, states must surrender the use of military force to alter status quo. Second, in order to take in the interests of the international community states must broaden their conception of national. Finally, states must look past the fear that encapsulates world politics and begin to trust one another (Baylis, 2005, p.310). The preservation of NATO, even since the end of the Cold War and the Soviet threat, appeared as confirmation that international cooperation could outlast the initial realist-inspired conditions for that institution (Dannreuther, 2007, p.39). Since the end of the Cold War collective security theorists believe that the international environment is more conducive for states to cooperate, sharing values and interests. Neither neorealism nor neoliberalism is able to account for the variability of states willingness to take part in collective security institutions as both theoretical approaches choose to ignore the role of domestic politics in shaping the interests and, hence, the behaviour of states (Spiezio, 1997, p. 112).

Conclusion

As debate over international relations has evolved over the years, it could certainly be claimed that both neorealism and neoliberal institutionalism are simply manifestations of the same approach. The idea that these two theoretical approaches made up the great debate has been challenged; Inis L. Claude (1981) suggested that neorealism and neoliberalism were complementary as opposed competitive approaches to international relations (Claude, 1981, cited in Baldwin, 1993, p.24). The term ‘neo-neo’ mentioned by Waever does not suggest that there has been a reformulation of either approach; rather it refers all of the synthesis between realism and liberalism that became possible through emergence of neo-realism and neo-liberal institutionalism (Neumann & Waever, 1997, p.19). This essay has shown that new theoretical approaches that look at world politics through a different lens have become the opponents to the ‘neo-neo’ debate; uniting through their opposition to positivism and rationalism.

Both ‘neo’ theoretical approaches have their differences, neorealists focus primarily on high politics and neoliberal institutionalists focus on low politics, but regardless of this, they both share similar worldviews. They share a comparable epistemology and ontology, focus on similar questions, and have a number of assumptions about world politics, solidifying the IR mainstream against reflectivist attacks. The assumptions shared by neo-neo purport that there is no common authority and states are unitary and interest-maximizing actors. Furthermore, the research platform for which these theories focus on behavioural regularities, and the state-centric empirical focus addressing issues that disrupt the status-quo, show clear evidence of synthesis. To conclude, I firmly believe that the evolution of both neorealism and neoliberal institutionalism has resulted in these theories falling under one header, and has subsequently together come under fire from positivist attacks.

Procedural democracy is a democracy in which the people or citizens of the state have less influence than in traditional liberal democracies. This type of democracy is characterized by voters choosing to elect representatives in free elections.

Procedural democracy assumes that the electoral process is at the core of the authority placed in elected officials and ensures that all procedures of elections are duly complied with (or at least appear so). It could be described as a republic (i.e., people voting for representatives) wherein only the basic structures and institutions are in place. Commonly, the previously elected representatives use electoral procedures to maintain themselves in power against the common wish of the people (to some varying extent), thus thwarting the establishment of a full-fledged democracy.

Procedural democracy is quite different from substantive democracy, which is manifested by equal participation of all groups in society in the political process.

Certain southern African countries such as Namibia, Angola, and Mozambique, where procedural elections are conducted through international assistance, are possible examples of procedural democracies.

Substantive democracy is a form of democracy in which the outcome of elections is representative of the people. In other words, substantive democracy is a form of democracy that functions in the interest of the governed.[1] Although a country may allow all citizens of age to vote, this characteristic does not necessarily qualify it as a substantive democracy.

In a substantive democracy, the general population plays a real role in carrying out its political affairs, i.e., the state is not merely set up as a democracy but it functions as one as well. This type of democracy can also be referred to as a functional democracy. There is no good example of an objectively substantive democracy.

The opposite of a substantive democracy is a formal democracy, which is where the relevant forms of democracy exist but are not actually managed democratically. The former Soviet Union can be characterized in as such, since its constitution was essentially democratic but in actuality the state was managed by a bureaucratic élite.

As I understood, the procedural part of a democracy is sort of like the frame; it's the procedures that has to be fulfilled for us to even consider a society as democratic. This includes the ability to be candidates in politics, the ability to easily access information, everybody having the same importance when speaking etc. Substantive is probably the thing I can't quite understand. I would be very happy if someone would try to explain it. Thanks in advance!

Substantive democracy means that the state reflects the will of the governed in practice.

For example: North Korea's official name is the Democratic People's Republic of Korea. While its constitution might pay lip service to elections and the importance of the people, it has no substantive democracy. This is common in many dictatorships where "elections" are held and the dictator is conveniently elected by 99% of the people for the 50th year in a row.

It's easy to have laws and a constitution (the framework or procedures) which proclaim democracy as the guiding principle of government, but that doesn't mean a county actually operates democratically (that the country is democratic in substance).

Similarly, you might have countries like the US and South Africa whose constitutions say certain things and certain procedures are followed, but in practice are not substantive. This can be seen, for example, anytime a national poll shows the general public support X by a huge margin, yet the elected officials oppose it or ignore the will of the people. (They were elected democratically but are not acting democratically).

Procedural democracy is a democracy in which the people or citizens of the state have less influence than in traditional liberal democracies. This type of democracy is characterized by voters choosing to elect representatives in free elections.

Procedural democracy assumes that the electoral process is at the core of the authority placed in elected officials and ensures that all procedures of elections are duly complied with (or at least appear so). It could be described as a republic (i.e., people voting for representatives) wherein only the basic structures and institutions are in place. Commonly, the previously elected representatives use electoral procedures to maintain themselves in power against the common wish of the people (to some varying extent), thus thwarting the establishment of a full-fledged democracy.

Procedural democracy is quite different from substantive democracy, which is manifested by equal participation of all groups in society in the political process.

Certain southern African countries such as Namibia, Angola, and Mozambique, where procedural elections are conducted through international assistance, are possible examples of procedural democracies.

Substantive democracy is a form of democracy in which the outcome of elections is representative of the people. In other words, substantive democracy is a form of democracy that functions in the interest of the governed.[1] Although a country may allow all citizens of age to vote, this characteristic does not necessarily qualify it as a substantive democracy.

In a substantive democracy, the general population plays a real role in carrying out its political affairs, i.e., the state is not merely set up as a democracy but it functions as one as well. This type of democracy can also be referred to as a functional democracy. There is no good example of an objectively substantive democracy.

The opposite of a substantive democracy is a formal democracy, which is where the relevant forms of democracy exist but are not actually managed democratically. The former Soviet Union can be characterized in as such, since its constitution was essentially democratic but in actuality the state was managed by a bureaucratic élite.

Nowadays, we know about some perspectives which can be learned in international studies. Those perspectives are very useful for us to identify relations between some states or among the other international relation’s actors. Nevertheless, it can be said that the relations of states are marred by divisions between its primary units, states. It is rare to find states that are more than just superficially similar. These differences, though, do not mean that any group of people are more or less important than another, simply they reflect the diversity that exists in the world. Many of these differences are self-imposed or in other words, created by the social distinctions that we place on ourselves. One such example is the apparent distinction between developed and underdeveloped states.

Currently there is a massive divide between rich and powerful states and those that make up the remainder, who have weak economies and little influence globally. If we make a comparative analysis with the economic or political global situation at this time, actually there is any dominantly state which appear as the one who has privilege to take the role as decision-maker for some problems which happened in the world wide, it refers to the superpower country exactly. Though, the regimes as the solution that related with mutual self-interest among the states especially are being ready to solve those problems within the states who involved in one of consideration or international organization like United Nations.

When we are talking about that case certainly related with any perspective which emerged dominantly until we entered the globalisation era. Many people said that liberalism is the specific way towards the globalisation era, moreover until the neoliberalism exists right now as the complete-act of liberalism. Indeed, there are some points between liberalism and neoliberalism which quietly same. But actually, neoliberalism included some substances that differ with liberalism. Firstly, we can concern our topic about liberalism. According to ‘Steans and Pettiford, 2005, on Introduction to International Relations: Perspectives and Themes’, actually the liberal view of the nature state is similar to realism, in so far as liberals accept that the defining characteristic of the state is sovereignty. Liberals would also agree with realists that the basic characteristics of the state are that it has a territory, a people, and a government. Liberals regard the state as, at best, a ‘necessary evil’.

The main basic assumption between liberalism and neoliberalism is totally different. The reason about emergence of liberal theory why until it does arise is identified as a response the inability of states to control and limit war in their international relations after World War I, because the important thing is states cannot be able to walk their function as the highest sovereignty owner in international level. Hence, most who view liberal theory find it limited at best in its ability to cover all aspects of world politics. According to ‘Liberal Theory as told by Andrew Moravcsik’, Moravcsik's Three Assumptions of liberal international relations theory are the primacy of societal Actors, representation and state preferences, and also interdependence and the international system.

While, neoliberalism as the one of ‘ism’ is not only political system, but also as a point of view about the relations between human and the civil society, which in liberal view state and civil society are clearly separated. Continuing the neoliberalism assumption, regarding to Friedrich von Hayek about this term, Hayek told that shaping the life order through authority would make the human cannot reach their own free life, so that the economic activities are needed to direct that human free life to shape that order. Furthermore, neoliberalism assumed that the price signal is used to get the own interest of each people in all sectors of civil society like economy, politic, social, culture, education, service, and the other goods. On this point, there is revolutionary meaning of neoliberalism from philosophical meaning into the one of term of economic sector. It can be inferred that there is existence of market which controlled by the price signalin all life sectors which can be called as market fundamentalism and automatically to be the main agenda of neoliberalism.

The difference between liberalism and neoliberalism eventually related with the actors of both of them. Regarding to ‘Steans and Pettiford, 2005, on Introduction to International Relations: Perspectives and Themes’, liberalism argue that sovereign states are important, but they are not the only significant actors in international relations. Just as the separation of powers implies that the essence of sovereignty is difficult to pin down or locate, contemporary liberals argue that the state can cede some element of their sovereignty to the other bodies, such as MNCs, NGOs, IGOs, etc. Whereas, neoliberalism seeks to update liberalism by accepting the neorealist presumption that states are the key actors in international relations, but still maintains that non-state actors. It implies that in neliberalism, the state actor and non-state actors are having the same role to reach the own welfare towards the world prosperity.

Liberalism vs Neo-liberalism

The word “liberal” carries strong connotations in modern political discussions. About as many self-identify as being liberal in their political views as those who adamantly avoid such a label. However, the historical roots of liberalism have produced a rich and diverse system of philosophical branches. In fact, many of these branches of liberalism stand diametrically opposed to one another on many political and economic issues. The word “liberal” doesn’t adequately capture the dexterity around this philosophical concept.

Liberalism was the product of Enlightenment thinking. John Locke is considered the godfather of liberal political thought, based on his prolific writing on the natural rights of individuals, separation of state and religion, social contract, and many other philosophical concepts – many of which were incorporated in the democratic revolutions that occurred decades after his death. What made liberalism unique was that it empowered the role of the individual and drastically challenged the absolutist foundation of the monarchies everywhere.

However, in the late 19th and early 20th century, liberalism morphed from an individualistic philosophy to one that is more communal in nature. Borrowing from John Stuart Mill’s utilitarian concept of providing “the greatest happiness for the greatest number,” liberalism sought to defend the “common good” – namely a political and economic system that maximized social progress for the group as a whole, and not benefitting a certain portion of individuals. Franklin D. Roosevelt best embodied this value with the “New Deal” in the 1930s. This body of legislation produced a large scale government infrastructure – characterized by public works projects, social welfare safety nets, and financial institution reforms – with a purpose to mitigate the effects of the rampant individualism that is commonly associated with the 1929 stock market crash and subsequent Great Depression.

Today, the modern interpretation of liberalism is associated with left wing causes. Borrowing from the New Deal, liberal economic thought strongly empowers public institutions as a means to support individuals who are adversely affected by the externalities – such as poverty and pollution – of free market capitalism. On issues of political rights, liberalism strives to secure civil liberties for minority groups, from the Civil Rights Movement for African-Americans in the 1960s to the current struggle for marriage equality for the LGBT community. Present day advocates of modern liberalism include individuals like consumer rights advocate Ralph Nader, current President of the United States Barack Obama, and Canadian Liberal Party leader Justin Trudeau.

Over the past few decades, a new form of liberalism – or rather a reinterpretation of the original merits of it – emerged in the form neo-liberalism. Not pleased with modern liberalism’s disempowerment of the individual in favor of the state, neo-liberal philosophers returned to the founding principles offered by Adam Smith’s Wealth of Nations. Considered to be the blueprints for free market capitalism, Smith described the need for human economic activity to be driven by the “invisible hand” of the marketplace, rather than by any governmental institution. To quote Smith,

“As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value; every individual necessarily labours to render the annual revenue of the society as great as he can.”

Allowing free individuals to trade in unfettered markets will produce the greatest amount of wealth and overall conditions for an affluent society in the eyes of neo-liberalism.

Neo-liberalism – also referred to as “classical liberalism” since it borrows from 18th century philosophical principles – was primarily an economic school of thought in its original form. Neo-liberalism highlighted the importance of deregulating markets and privatizing public institutions. The transition of this philosophy from economics to a political movement has gained momentum in recent years with the rise in libertarianism in the United States, popularized by individuals like Rep. Ron Paul and Governor Gary Johnson. Although modern libertarians may be equated with what is considered “modern conservativism” (although those ideas are liberal on some economic policies, they strongly disagree with policies that relate the role of the state in the private lives of citizens – more specifically, the rights of citizens to marry freely, not be the subject of government surveillance, and freely purchase and produce banned substances like marijuana. The individual is the true arbiter of a free society in both economic and political terms in the eyes of neo-liberals, classical liberals, and libertarians alike.
As one can deduce, the term “liberal” isn’t exactly a cookie cutter label that adequately describes the diverse nature of the philosophical tradition. The next time somebody attempts to use this term in conversation, be sure to challenge them by asking, “What kind of a liberal are talking about?”

classical liberalism is based on adam smith, jefferson, locke, etc which was mostly based on the philosophy of rights more than economic prosperity~ (obviously it has a bit to do with "the market works best when left alone" and "that government which governs least governs best", some good/relevant quotes I remember having to memorise) but mostly it's about liberty and all that jazz, not strictly the collectivistic outcome that is in neo-liberalism which was its argument against socialism in the cold war context where socialism was failing to deliver on a community level. so classical liberalism, is about limited government, constitutionalism, separation of powers, freedom, etc
neo-liberalism is based on new economic globalisation and "world market" competition mostly (and focusing on the prosperity aspect, at least that's how it appears from my point of view), e.g. the philosophies of milton friedman. so neo-liberalism is about the world economy and its relevance to the national economy, internationalism, free trade globally, supply side/trickle down theories, lower taxes, less tariffs/protectionism and that kind of thing in focus. it's basically milton friedman's views as an ideology, while classical liberalism isn't so much based on economics, but people like thatcher and reagan would be mentionable as well as just friedman. neoliberalism, therefore, is about the consequential aspects of economic liberalism/laissez faire, while classical liberalism is about the individualistic aspect of it and freedom as the goal in itself, not the means etc; neoliberalism isn't really a "new philosophy", it's simply (in my view) a new way of approaching/looking at classical liberalism