A Brief Introduction to Logic
Martin A. Kozloff
October, 2003
I. Introduction
Four Forms of Knowledge
Virtually everything you teach your
students (in fact, virtually everything that is known and can be taught)
consists of:
1. Verbal associations, such as
a. Simple
facts. The Declaration of
Independence was signed in 1776.
b. Verbal
chains; e.g., listing seven
c.
Discriminations; e.g., properly labeling federalist vs.
anti-federalist
documents.
2. Concepts. Concepts are classes of things that share
certain features, such as red (lower order concept), color (higher order),
political system, democracy.
3. Propositions. Propositions, or rule relationships, are
statements that assert connections among concepts. Some
trees are deciduous. All mammals are
vertebrates. When leaders use violent
means to suppress dissent, it increases dissent and decreases the legitimacy of
the political relationship.
4. Cognitive
strategies. Cognitive strategies are sequences of rule
governed steps that accomplish some objective; e.g., solving math problems,
analyzing documents, writing papers, conducting experiments.
Logic is the Knowledge System That Studies Verbal
Associations, Concepts, Propositions, and Cognitive Strategies
Logic is the knowledge system that
addresses the nature and uses of verbal associations, concepts, propositions,
and cognitive strategies. For example, deductive reasoning strategies are used
to test (verify) hypotheses (one
kind of proposition) and to draw conclusions.
However, inductive reasoning
strategies are used to discover
(induce, construct) general categories (concepts) and general propositions
based on observations of specific events.
Both deductive and inductive reasoning are used in virtually everything
that counts as thinking. Following are
some examples.
1. Solving algebra problems involves making
deductions from general problem-solving strategies and applying them to
specific problems.
2. Reading historical examples and figuring out
what is common to them involves inducing (constructing) concepts (events with
common features) and generalizations (about how one thing affects
another). This is what Machiavelli did
in his book The discourses. Machiavelli
read the Roman historian Titus Livy's accounts of various events, and
then induced generalizations about what happens when a society follows a
certain course of action.
3 Determining whether the results of a
chemistry experiment support, refute, or expand on prior generalizations (a
form of proposition). Students use
methods of deductive reasoning to design an experiment. They use methods of inductive reasoning to
figure out what the data mean.
In summary, the more you know about verbal
associations, concepts, propositions, and cognitive strategies in logic and in
subject matter areas, the more you will understand the material and the better
able you will be to teach students and colleagues to think logically.
This paper is organized
as follows.
1. We begin with concepts, definitions, and propositions (rules, hypotheses, and
generalizations).
2. Then we examine cognitive strategies of two kinds:
a. Inductive reasoning strategies, or how
to begin with knowledge of specific
events and then induce (create, construct) generalizations (propositions,
hypotheses, or rules; e.g., about cause and effect). We will examine inductive strategies of
increasing complexity. These include:
(1) methods of inductive inference; (2)
methods for drawing casual inferences; and (3) the inductive style of research
and case building.
b. Deductive reasoning strategies, or how
to begin with generalizations (propositions, rules, hypotheses) and then deduce
conclusions about specific events (e.g., whether a hypothesis is
confirmed). We will examine deductive
strategies of increasing complexity: (1) deductive syllogisms; and (2) the
deductive style for conducting research (e.g., testing hypotheses).
The diagram below compares inductive and
deductive reasoning strategies.
Inductive Strategies Deductive
Strategies
[For discovering
relationships, [For verifying propositions
stated as propositions] about
relationships]
Start with facts, specific Start with
generalizations;
events, historical accounts, e.g., hypotheses.
statistical data.
| |
V V
Use strategies of inductive Use strategies of
deductive
reasoning; e.g., Mill's reasoning; e.g.,
syllogisms, and
methods of inductive inference; strategies for conducting
steps for inferring causal research; e.g., for
testing
relationships; strategies of hypotheses.
research and case building.
| |
V V
End with empirical End with conclusions
generalizations, stated as about specific events.
propositions.
Notice (by the
arrows above) that inductive strategies end with generalizations. These generalizations (when asserted as
hypotheses--predictions) can then be tested more formally with deductive
strategies, such as experimental research.
In other words, inductive and
deductive reasoning strategies can and often should be used together.
3. Finally, we examine fallacies of relevance and ambiguity; i.e., fallacies in ordinary
conversation, political speeches, advertisements, research, and
theorizing. This will increase skill at
attention to how words are used and at deciding whether and how conclusions are
reasonable.
This paper has
exercises to firm up the verbal associations, concepts, propositions, and cognitive
strategies presented. Please write your responses to these
exercises.
II. Concepts
and Definitions
What are Concepts?
Concepts are usually expressed as
nouns and adjectives (qualities of thingness).
The "stuff" (events) in a class or concept (e.g., blood
pressure and white cell count--events that define the concept
"health") may be objectively real.
However, the concept ("health") is conceptual; it exists as an
intellectual
synthesis . Health is pretty
much what we decide it is. For example,
the gray color and granite blocks of a cathedral are real; the window slits in
the high walls are real; the flying buttresses are real. But "Gothic style" is an idea, a
concept, an intellectual synthesis.
"Gothic style" does not exit "out there" in the same
way bricks do. Gothic style is a concept
that we create by noticing how certain things go together and then naming the
goes-with-ness. So, Gothic style (as
with most other "things" in our social world, such as
"achievement," "socialization," "proficiency,"
"intelligence") really only exists to the extent that we use the
words in a common way--to point to the same features of buildings,
relationships and actions. In other
words, the existence of some
"things" depends on how we create definitions, share the definitions,
and use the words defined.
Definitions
We
must distinguish between a concept and its name. For example, the concept aggression is a set
of events with certain features in common.
However, the word "aggression" is a name used to signify (point to) events in the category
aggression. Definitions are rules for using the words (names) that signify
concepts. For example, the
definition of aggression directs attention to certain events and away from
other events. And the definition, in a
way, permits us to call certain
events "aggression." But what
is it about these events that is
aggression? If we examine the events we
find certain things in common. Perhaps
we find that they involve intention to injure.
There may be different kinds of behavior (hitting, not providing help),
and they may be directed at humans and nonhumans. But they all involve intention to
injure. That feature becomes the core of
the definition, the core of the concept.
We state our definition in a form called "genus and difference."
The genus is the major category
for the thing defined. The difference is the way that thing
differs from other things in the same genus.
So, "Aggression is behavior (genus) that is intended to injure
humans or nonhumans (the difference between aggressive and nonaggressive
behavior).
Note: There is no such thing as a true
definition. Rather, some definitions
are better than other definitions; they are better at directing attention to
the right events. So, definitions are
better when:
1. All
of the terms have clear meaning;
that is, the words in the definition clearly point to the events named.
2. The range of events included by the
definition is not so broad that it
includes events that are part of many other definitions, too. For example, if aggression were defined, in
part, by falling on another person, that would also be a definition of
accident.
3. The range of events included in the
definition is not so narrow that the
definition excludes events that ought to be included. For example, if the definition of health
leaves out white blood cell count, then a person with leukemia (too many white
cells) would not be called unhealthy for that reason.
Note again: Definitions are not fixed. Further examination may suggest changes. For example, if we examine more examples of
behavior that involves the intention to injure, we discover something else
these behaviors share--namely, feelings
of antipathy towards the object of aggression. Therefore, we revise the earlier definition of aggression as follows: "Aggression is behavior that is intended
to injure humans or nonhumans, and is either preceded, accompanied by, or
followed by feelings of antipathy (hatred, disgust, anger) towards the object
of aggression."
Conceptual Definitions
The definition
of aggression, above, is a conceptual definition. This means it is abstract. The definition is
not of specific acts such as hitting, kicking, and insulting. What is defined was the class (the circle) that contains these examples. The conceptual definition of aggression
identified what is common to the specific acts.
EXERCISE
1. Examine material in your subject
matter areas. Find or develop conceptual
definitions for five concepts. Make sure
to recast the definitions in the form genus and difference. Here are examples: rational number, first
derivative, republic, telephase, gymnosperm, helium, conjugate, regular verb.
Operational Definitions
Operational
definitions of a concept are concrete and specific. An operational definition gives
examples. Examples of what? Examples of what is signified by the
conceptual definition. For example, if
the conceptual definition of aggression is "behavior that is intended to
injure humans or nonhumnans, and is either preceded, accompanied, or followed
by antipathy (hatred, disgust, anger) towards the object of aggression,"
then the operational definition of aggression would be a list of these events: intentional hitting, kicking, spitting at,
insulting, inhibiting promotion at work.
The length of the list (the
length of the operational definition) depends on how and where you want to use
it. If you are conducting research
on aggression through the life-span, you would want a comprehensive list. But if you were conducting research on
aggression in three year old girls, you would not need to include inhibiting
job promotion or smacking another person on the back of the head. In summary, operational definitions are derived (deduced) from conceptual
definitions and are then tailored to the way they will be used.
EXERCISE 2. What are the two kinds of definitions? How do
they differ?
EXERCISE 3.
What is the method by which conceptual definitions are stated?
EXERCISE
4. Derive operational definitions from
two of the five conceptual definitions that you created, above.
Concept__________________ Concept_____________________
III.
Propositions
Propositions assert relationships. Relationships among what? The answer is, relationships among concepts (classes
or families of specific events). There
are two kinds of propositions: categorical and hypothetical/causal. These two kinds of propositions assert two
kinds of relationships. Let's examine
categorical and causal propositions in more detail.
Categorical propositions
Following are
examples of categorical propositions.
1. All
fads in education are supported by flawed research. This categorical proposition asserts that one
category (fads in education) is completely within
another category (things that supported by flawed research).
2. Some
fads in education are recycled in about 20 years. This categorical proposition asserts that
part of one category (fads in education) is within another category (things that are recycled in about 20
years).
3. No
fad in education benefits children. This
proposition asserts that none of one category (fads in education) is within
another category (things that benefit children).
Notice that
categorical propositions use either all,
no, or some to describe the relationship of inclusion or exclusion
between two categories.
In summary,
categorical propositions assert that all (or part) of one class is included or
excluded from another class.
EXERCISE 5. Write three categorical propositions using all,
no, and some. Diagram the relationships.
Hypothetical or Causal Propositions
Hypothetical or causal propositions
assert that the existence or change in one set of events is determined by, is
contingent on, or is predicted by the existence or change in another set of
events. The proposition asserts
causation. If we merely believe that one sets of events is
determined or predicted by another set of events, then the causal proposition
is an hypothesis ("If X
happens, then Y will happen."). "Hypothetical" means we are
not confident that the proposition accurately describes what is the case. So
we must verify or test it. Below are
examples of causal propositions.
1. The more stressors that bear on people
during a year, the more illnesses they will have that year.
2. The more support persons have for moral
principles, the less likely they are to obey orders contradicting their moral
principles.
3. The larger
the dose of rhinovirus, the faster a
cold develops.
The events
asserted as causes are also called independent
variables. In proposition 3, above,
the size of viral dose is the independent variable. The events asserted as influenced by the
independent variable(s) are dependent
variables. In proposition 3, the
speed with which colds develop is the dependent variable.
EXERCISE
6. (1) Identify the independent and
dependent variables in propositions 1 and 2, above. (2) Write three causal propositions of your
own, and identify the dependent and independent variables.
What
Independent Variables Do. Causes
or independent variables can be causes in several ways. For example, independent variables may be
seen as necessary conditions, sufficient conditions, and intervening variables.
1. Necessary Condition. An independent variable is a necessary condition when
its existence, or when a change in it, is asserted to be required
for the existence or for a change in a dependent variable. For instance:
If and only if there are shared feelings of
exploitation among subjects (independent variable), will subjects mount
resistance against rulers whom they perceive to be exploiting them.
2.
Sufficient Condition.
An independent variable is a sufficient condition when its existence, or
when a change in it, is asserted to be enough
to bring about the existence or to change another (dependent)
variable. For example: Whenever
violence (independent variable) is used to punish dissent, it fosters even
more dissent (dependent variable). Generally, no one factor is a sufficient
condition. Instead, a set of necessary conditions (e.g.,
shared feelings of exploitation, plus an opposition ideology, plus opposition
leaders, plus opportunities to mount resistance) is usually asserted to make up
a sufficient condition (e.g., for revolution).
3.
Intervening Variable.
Some independent variables are neither necessary nor sufficient. Rather, they stand between main independent variable(s) and the dependent
variable(s). For example, it is
generally true that the larger the dose of cold virus (main independent
variable), the greater the likelihood that people will catch a cold. However, the relationship between viral dose
and the probability of catching cold is influenced by a third (in between)
variable--namely, the strength of the immune system. In other words, viruses are necessary conditions for catching colds, but they are generally not sufficient conditions.
Viruses cause colds only if the immune system is weak
enough. In a causal model of these relationships, the strength of the immune
system is a gatekeeper (intervening variable) standing between viruses and
colds, as shown.
Main Independent Intervening Dependent
Variable Variable Variable
Viral dose --------> [If Weak
Immune System] -------> Likelihood of Cold
EXERCISE
7. Give examples of propositions that
assert that the independent variable is necessary, sufficient, and intervening.
Co-variation. Co-variation has to do with how each variable changes
in relation to the other variables. Variables
can change in the same
direction--both increase or both decrease.
This is called a direct
relationship. Or variables can change in
opposite directions. One can increase and the other
decreases. This is called an indirect, or inverse relationship. Here
are examples.
1. The higher the rate of unemployment
(independent variable), the higher the rate of admissions to mental hospitals
(dependent variable). Both variables are
changing in the same direction (increasing).
Therefore, this is a direct relationship.
2. The stronger
the cohesion in a group (independent variable), the lower the rate of
deviant behavior (dependent variable).
These variables change in opposite directions. Therefore, this is an indirect, or inverse,
relationship.
3. The lower the number of cigarettes smoked
each day (independent variable), the longer it takes to get lung cancer
(dependent variable). These variables
change in opposite directions.
Therefore, this is an indirect, or inverse, relationship.
4. The lower the rate of interpersonal reward in
a group (independent variable), the weaker are sentiments of liking among
members (dependent variable). Both variables are changing in the same direction
(decreasing). Therefore, this is a direct
relationship.
EXERCISE
8. Write causal or hypothetical
propositions that assert direct and indirect (inverse) relationships.
Direction
of causal relationships. Causal propositions generally assert a causal
path or direction among the variables.
These paths are as follows.
1.
Unilateral. Unilateral relationships are in one
direction only. That is, change in
an independent variable effects change in the dependent variable, but the
change in the dependent variable does not then go backwards and affect the
independent variable. For example,
something about membership in different social classes affects the rate of
homicides performed by members in each social class.
Social Classes (Upper, Middle,
Lower) Homicide Rates
(Independent Variable) (Dependent
Variable)
However, the
causal flow does not also go the other direction; the rate of homicide does not
cause social class.
2.
Bilateral or reciprocal.
A bilateral relationship operates in both directions--back-and-forth.
Change in X engenders change in Y; the change in Y then effects a
further change in X. This relationship
is reciprocal (back-and-forth). It is called a feedback loop. There are two
kinds of feedback loops—positive and negative.
Here is an example of a positive
feedback loop.
The more often
teachers correct students' errors immediately, the more proficient students
become. The more proficient students
become, the more often teachers correct errors immediately in future lessons. This results in even higher student
proficiency. Eventually a limit is
reached; students cannot learn any faster and/or teachers correct every
error.
This is a positive feedback loop because each
variable is fostering an increase in the
other variable.
Here is another example of a positive
feedback loop.
The less often
Ms. Jones (the principal) provides direct instructional support to her
teachers, the less proficient her teachers become. The less proficient her teachers become, the
less Ms. Jones wants to observe them and the less her teachers want to be
observed. The less Ms. Jones wants to
observe them and the less her teachers want to be observed, the less Ms. Jones
provides direct instructional support.
This is a
positive feedback loop because each variable is fostering a decrease in the
other variables.
EXERCISE
9. So, what do the two examples of
positive feedback loops have in common?
Feedback loops can be negative. That is, one variable increases, and when it
does, the other variable decreases,
and this makes the first variable decrease.
For instance, an increase in the rate of urban crime produces an
increase in the number of police in the city, which results in a decrease in the rate of crime. Of course, as the crime rate goes down, the
politicians may reduce the size of police force, and then crime rises
again. This would be an example of oscillation.
EXERCISE
10. State a negative feedback loop
consisting of the following variables:
the teacher's consistent enforcement of rules and procedures and
students' noncompliance.
Notice that
positive feedback loops look a lot like direct relationships, and negative
feedback loops look a lot like indirect, or inverse, relationships. That's because they are! The difference is that feedback loop implies
that the variables are actually influencing each other reciprocally; and
the causal relationship is ongoing.
However, you could have direct or inverse relationships in which there
is no ongoing change and there is no reciprocal influence. For example, the lower the social class, the
higher the rate of alcoholism. But the
relationship only goes one way—is unilateral.
3.
Dialectical. A
dialectical relationship involves reciprocal influencing (feedback), but with
one more feature. As each set of
variables influences the other set, the quantitative
changes eventually yield a change in the quality, type, or state of each
variable, and also perhaps in the nature of the relationship. For example, if kindergarten teachers
accidentally reward students for throwing tantrums and hitting, the children
will perform these behaviors more
often. The teachers then try harder to
stop the problematic behaviors in ways that, again, reward these
behaviors. At some point, the increasing
rate of children's problem behaviors results in a qualitative shift in how the children are perceived. They are no
longer seen as normal children who perform problematic behavior too often; they
are seen as children with behavior
problems. At the same time, the teachers
no longer see themselves as regular teachers, but as guards or victims. Finally, as the nature of children's and
teachers' participation in the relationship changes, the nature of their relationship
changes; e.g., from sweet children and loving teachers (a complementary
relationship) to an adversarial relationship.
EXERCISE
11. Give examples of propositions that
assert unilateral, bi-lateral, and dialectical relationships. [Hint on bi-lateral--the effects of anxiety
on performance. Hint on
dialectical--arguing in marriage.]
Proximity. Some causal relationships are "proximal." That is, there is little time lag or
there are few intervening variables between the main independent variable and
the main dependent variable. Other
causal relationships are "remote"
(or distal). Sometimes, remote causes are considered predisposing factors and proximal
causes are considered precipitating
factors.
EXERCISE 12, ON
CONCEPTS AND PROPOSITIONS. Rewrite the
excerpts below as propositions (categorical and causal) and/or as definitions
of concepts (using the method of genus and difference). An excerpt may contain more than one
proposition.
1. "...a state is a human community that
[successfully] claims the monopoly of the legitimate use of physical force
within a given territory." [Max
Weber. "Politics as a vocation." 1918]
2. "...suicides are found to be in
direct proportion to the number of Protestants and in inverse proportion to
that of Catholic's." [Emile Durkheim, Suicide. 1897]
Hint: The higher the...
3. "No living being can be happy or even
exist unless his needs are sufficiently proportioned to his means." [Emile
Durkheim, Suicide. 1897]
Hint: categorical.
4. "If the state is to exist, the
dominated must obey the authority claimed by the powers that be." [Max Weber. "Politics as a
vocation." 1918] Hint: If and only if...
5. "...the term suicide is applied to
all cases of death resulting directly or indirectly from a positive or negative
act of the victim himself, which he knows will produce this result. An attempt is an act thus defined but falling
short of actual death." [Emile
Durkheim, Suicide. 1897]
6. "If therefore industrial or financial
crises increase suicide, this is not because they cause poverty, since crises
of prosperity have the same result; it is because they are crises, that is,
disturbances of the collective order." [Emile Durkheim, Suicide.
1897] Several
propositions--causal and categorical.
7. "Where the State is the only environment
in which men can live communal lives, they inevitably lose contact, become
detached, and thus society disintegrates."
[Emile Durkheim. The Division of
Labor in Society. 1893] Hint:
If X, then Y. More than one
proposition.
8. "[N]o psychopathic state bears a
regular and indisputable relation to suicide." [Emile Durkheim, Suicide.
1897] Hint: Venn diagram.
9. "[A] religious society cannot exist
without a collective credo." [Emile Durkheim, Suicide. 1897] Hint:
If and only if....
10. "[T]he more extensive the credo the
more unified and strong is the
society." [Emile Durkheim, Suicide. 1897]
Hint: more than one proposition.
11. "[T]he desire for knowledge wakens
because religion becomes disorganized." [Emile Durkheim, Suicide.
1897]
12. "Every disturbance of equilibrium...is
an impulse to voluntary death." Emile Durkheim, Suicide. 1897] Hint:
Whenever X,...
13. "...more depressed and anxious pregnant
teenagers, who perceive their social relationships to be less satisfying, and
who have less knowledge of child development, have more negative expectations
for their infants." [J.M. Contreras et al. (1995.) Journal of Applied Developmental Psychology, 16, 283-295.] Hint: Note the intervening variables.
14. High mother support was associated with more
secure infant attachment only for those adolescents living with partners."
[S.J. Spieker (1994]). Developmental
Psychology, 30, 1, 102-111.]
15. "There is the authority of the extraordinary
and personal gift of grace [charisma], the absolutely personal devotion and
personal confidence in revelation, heroism, or other qualities of individual
leadership. This is charismatic
domination..." [Max Weber.
"Politics as a vocation." 1918]
16. "[H]e who lets himself in for politics,
that is, for power and force as means, contracts with diabolical powers and for
his action it is not true that good can follow only from good and
evil only from evil, but that often the very opposite is true. Anyone who fails
to see this is, indeed, a political infant..." [Max Weber. "Politics
as a vocation." 1918] Hint: definitions and propositions here.
We now begin to
examine cognitive (reasoning) strategies, beginning with inductive (specific-to-general) strategies.
IV. Inductive
Strategies
Induction is a logical strategy that
seeks to discover, create, induce, or infer what is general in a set of
specific events. Specific events might
be actions; statistics on rates of suicide, unemployment, and divorce;
historical documents that describe different societies; or changes within a
school (e.g., leadership, teacher satisfaction, student achievement).
1. We examine these events to find
patterns—kinds of things (concepts), relationships between kinds of things.
2. We state these patterns as: (a) definitions
of concepts; and (b) propositions (see section III, on propositions.) For example, we discover a direct
relationship between the rate of divorce and the rate of suicide in many
societies, and we infer that there may be a causal relationship.
This section
begins with simple inductive strategies (inferring that two variables are
connected), and moves to more complex inductive strategies (e.g., for
conducting research).
Mill's Methods for Inducing Rules (Generalizations)
About What is Related to What
We never see a relationship
directly; e.g., a certain curriculum causing an increase in the percentage of
children passing an achievement test. We
only see specific events (independent and dependent variables) that might occur
together in a consistent and proximal way.
And when a situation is complicated (lots of factors interacting over
time), it is very difficult to determine what is related to what. The best we can do is infer (induce,
construct) a rule or statement that ties several sets of events together--that
they go together somehow. But
unless you are the lucky recipient of Divine Inspiration, or unless you are blessed
with powers of intuition (which you probably aren't), then you must use a
cognitive strategy to induce (figure out) relationships. John Stuart Mill (1806-1873) in A system of logic, describes five methods of inductive inference. Humans naturally use many of these
methods. But Mill describes them in a
way that we can use them explicitly.
In fact, we can design research (experiments, literature reviews,
documentary research) so we can gather and analyze data using Mill's methods. The
methods of inductive inference are: concomitant variation, agreement,
difference, joint agreement and difference, and residues. Let's examine each method.
1. The Method of Concomitant Variation. If two
variables are changing with respect to one another (e.g., both increasing, both
decreasing, or one increasing and the other decreasing) while everything else stays at about the same, then we have
logical evidence that one variable is a cause or an effect of the other (or
they are both being changed by a third variable). For example, the more practice students
receive, the longer they retain what they learned. If nothing else in this situation is changing
systematically along with the independent and dependent variables, then (by the
method of concomitant variation), we infer that the variables are causally
connected.
EXERCISE
13. Give two examples of a causal
inference (in your subject matter areas or in schools) drawn by the method of
concomitant variation.
2. The
Method of agreement. If things in a many settings are very different, but
two sorts of events go together (agree), they may be connected causally. For example, Arnold Toynbee, in A study of history, examined
civilizations that had one thing in common; they were gone. Despite their differences (size, culture,
location, religion, economics) they had another thing in common (that is,
another way they agreed); namely, they had faced a crisis and responded by
making the crisis worse. So, Toynbee
concluded (drew the inference) that there is a causal connection between how a
civilization responds to crisis and whether it survives.
EXERCISE 14.
Give two examples of a causal inference (in your subject matter areas or in
schools) drawn by the method of agreement.
3. The Method
of Difference. Let's say we study 5 battles between the
ancient Greeks and Persians. The battles
are the same in virtually every way: relative size of the armies, weapons used,
quality of leaders, motivation of soldiers.
But armies that won differed in one major way from armies that
lost. Armies that fought in a phalanx
formation (Greeks) won. Whenever the
Greeks did not fight in a phalanx, they lost.
So the difference (battle formation) appears to make the difference
(winning or losing).
X X
X X X
X X X X X Rows
and columns of hoplites
X X
X X X
X X X
X X (heavily armed soldiers)
X X
X X X
X X X X X
X X
X X X
X X X
X X
X X
X X X
X X X
X X Attack
X X
X X X
X X X
X X |
X X
X X X
X X X
X X V
The method of
difference is the logical strategy used in the classical experiment. You have experimental and control groups that
are virtually equivalent. They differ in
one major way: the experimental group
receives the treatment (e.g., new drug for arthritis); the control group does
not. If there are significantly larger
beneficial differences between the pretest and posttest (measuring, for
example, the dependent variable of inflammation of the joints) in the
experimental group, then, by the method of difference, we draw the inference
that the drug made the difference.
EXERCISE
15. Give two examples of a causal
inference (in your subject matter areas or in schools) drawn by the method of
difference.
4. The Joint Method of Agreement and Difference. Recall that Toynbee studied civilizations that
differed in many ways--size, religion, language, economics. Despite their differences, they agreed in two
ways: (1) they were gone; (2) they
responded to crises in ways that made the crises worse. So, Toynbee induced a generalization about
the relationship between how civilizations respond to crisis and whether they
survive. This generalization (rule)
would be stronger if we also found civilizations that were the same in many ways (e.g., size,
economics), but differed in two ways
(some are gone and some still exist).
And the civilizations that are gone had responded to crisis in ways that
made them worse, while the civilizations that are still here responded to
crises in ways that reduced the
crises. In other words, the joint method
of agreement and difference has more power.
It suggests what happens when a factor is there (bad choices make crises
worse) and what happens when it is not there (good choices make crises
improve). The following illustrates the
joint method of agreement and difference.
Sample of
civilizations differing in many ways but agreeing (the same) in two ways: (1)
gone;
(2) made
crises worse. [Method of agreement]
Sample of
civilizations that are the same in many ways but differ in two ways:
(1) the ones
that are gone made crises worse; (2) the ones that still exist responded to crises
in ways that lessened the crises.
[Method of difference]
Use both
methods and you have the joint method of agreement and difference.
EXERCISE
16. Give one example of a causal
inference (in your subject matter areas
or in schools) drawn by the joint method of agreement and difference.
5. The Method
of Residues
Imagine a
situation in which some phenomenon (Y) might be explained by four factors. We may determine the main cause through a
process of elimination. If we know that
factor 1 is a cause of Q, factor 2 is a cause of R, and factor 3 is a cause of
S, then factor 4, the only one left, is likely to be the cause of Y. As Sherlock Holmes used to tell Dr. Watson,
when you have eliminated all of the other possible explanations, the one that
remains, improbable though it may seem, must be the correct explanation.
EXERCISE
17. Give one example of a causal
inference (in your subject matter areas
or schools) drawn by the method of residues.
Hint: What do teachers' expectations, instruction, principal's
leadership, teachers' beliefs about learning, and parental involvement in PTA
have to do with student achievement?
Now let's
examine a slightly more complex inductive strategy--inferring causal
relationships.
The Inductive Strategy for Drawing Causal Inferences
Mill's methods let us induce
generalizations that some events go together with other events --perhaps in a
causal way. However, Mill's methods are
not enough to verify that we know what causes what, and what doesn't
cause what. The strategy for drawing
causal inferences has four steps or requirements.
1. We have evidence that the alleged cause
preceded the alleged effect ("temporal priority").
2. We have empirical evidence that the alleged
cause and effect occur together ("contiguity").
3. We have inductive logical evidence that ties
them together (supplied by Mill's methods).
4. We have evidence that alternative
explanations are implausible.
Let us examine
each step in the strategy for drawing plausible causal inferences.
First
Step or Requirement: Ensuring that The
Alleged Cause Precedes The Alleged Effect.
Consider the following assertion. "An increase in teachers' authority
to make curricular decisions (independent variable) fosters an increase in
teachers' attachment to their school."
This seems plausible only if there is evidence that teachers' authority
to make curricular decisions came before
an increase in teachers' attachment to their school. Otherwise, one could argue that teachers'
attachment leads them to seek and to
obtain more authority. In other words, you must determine which
changed first. Evidence of temporal
priority might be supplied by observation, experimental control, and/or
commonsense reasoning (e.g., it is not likely that a house burned down and then
someone smoked in bed).
Second
Step or Requirement: Obtaining Empirical
Evidence of Association. The inference
that an increase in teachers' authority to make curricular decisions fosters an
increase in teachers' attachment to their school, is more compelling if we have
data showing that these two
variables changed in close succession,
and in the order asserted. Similarly, we can conclude that a family
training program produced beneficial effects only if we have evidence of change
in families and evidence that family members attended meetings, understood
what was presented during meetings, and read and understood materials.
Consider what happens if you do not
have empirical evidence of the occurrence of all variables. Researchers supply a new drug to persons with
arthritis. After 3 months, 60 percent of
the persons receiving the drug report
significant pain reduction. It
seems the drug worked. However, are we
sure all subjects took the drug as prescribed? Maybe only the persons who did not take the drug as prescribed felt
less pain. In other words, not only do
we need to determine whether the pain went down after people took the drug rather than before (evidence of temporal
priority), we also need to measure both: (1) how persons took the drug (the
independent variable); and (2) their pain.
Third
Step or Requirement: Obtaining Evidence
Provided by Inductive Logic (Mill's Methods). Evidence of temporal priority of the alleged cause and
empirical evidence of an association are not enough to draw a sound causal
inference. We also need logical
evidence. Logical evidence is obtained
by designing research, analyzing data, and interpreting findings so we can
apply one or more of Mill's methods, discussed earlier. Here are three examples.
Using
Mill's Method of Concomitant Variation. If two variables are changing with
respect to one another (e.g., both are increasing, both are decreasing, or one
is increasing and the other is decreasing) while everything else remains at
about the same level, then we have logical evidence that one variable is a
cause or an effect of the other (or they are both being changed by a third
variable.) For instance, an experiment on the causes of aggression was
conducted in a class of 20 grade school children. During the first experimental period (A1 or
Baseline), the teacher handled the children's aggression (operationally defined
as hitting, kicking, insulting, etc.) her usual way. She would stare at the "offender,"
remind the offender of the rules, tell the offender to stop, or even give the
offender an enjoyable activity to "distract him" or "settle him
down." In the next period (B1), the
teacher was coached to respond in a matter-of-fact way to aggression (e.g.,
remove the aggressor) and to comfort and reinforce other children who were
engaging in nonaggressive behavior at that time. In the third experimental period (A2), called
a "reversal," the teacher was asked to do what she used to do during
A1 (which, again, meant that she paid a lot of attention to aggression). And during the final period (B2), she was
asked to go back to not reinforcing aggression and instead reinforcing
nonaggression.
The graph below shows that when the
children received a lot of teacher contact following aggression (A1 and A2),
the rate of aggression was high, and when the amount of teacher contact
following aggression decreased (B1 and B2), the rate of aggression
decreased. Since nothing else in the
classroom was changing along with
changes in the teacher's responses to aggression, it is plausible to infer that
changes in the teacher's responses somehow caused changes in the children's
rate of aggression.
A1 B1 A2 B2
Aggressive | *
* *| * | |
20 |
* * | * * |
* * * | *
Acts |
* | * | * | *
15 | | | * | *
Per | | * | | *
*
10 | | * | *
|
Hour | | | | *
5 | | | | *
|_|_|_|_|_|_|_|_|_|_|_|_|__|_|_|_|_|_|__|_|_|_|_|_|_|_
Days
Using
Mill's Method of Agreement. Imagine that we study twenty school reform
efforts that failed. Each school and
each reform effort was a different
configuration of variables (e.g., size, socioeconomic status, location,
teacher-student ratio, speed of reform).
Despite these differences, however, all the schools and reform efforts
had one thing in common—staff were
not committed to the mission or the reform plans. Since nothing else in the schools and plans
was common across the schools, it is reasonable to infer that what they had
in common was the cause of the failed reform efforts.
Using
Mill's Method of Difference. Mill's method of difference is the form of
inductive logic used in the typical pre-test, post-test, experimental-group,
control-group study. Let us say that we
have a pool of 50 families whom we randomly assign to two comparison groups. One group receives written materials, ten
weekly group meetings, and weekly home visits to improve family interaction and
home teaching. The second group receives
written materials only. We compare
pre-test and post-test scores on the quality of family interaction and home
teaching. Families in the first group
have significantly larger pre-post-test differences. What can we infer? Since we randomly
assigned families to the groups, any
personal and family differences that might have accounted for improvement or
lack of improvement (e.g., religion, support network, expectations of success,
initial teaching skill) had an equal
chance of being in each group.
Therefore, we can assume that the groups were fairly similar on these extraneous factors. (Of course, we could also measure those factors
that we think are important and see how similar the two groups actually
are.) Since the only other systematic difference between the
two groups (which we know about) was group meetings and home visits, it seems
likely that these two features of the training made the difference in the
amounts of improvement.
Fourth
Step or Requirement: Ruling Out or Weakening Rival Hypotheses. Let's say we have evidence that the alleged
causes preceded the alleged effects; that the two variables changed in the way that
was asserted; and we have used Mill's methods to provide logical evidence of a
causal connection. We now must show that
rival explanations are implausible. For
example, could something else (besides the drug) have caused the reduction in
inflammation in persons with arthritis?
Other variables that could account for the findings are called extraneous variables. Here are rival explanations in the arthritis
research: All or part of the beneficial
changes between pretest and posttest in the experimental group were due to:
1. Measurement error in either group. For example, errors in post-test data gave
the control group low scores. They
really changed as much as the experimental group.
2. Demoralization in the control group. Realizing they were getting no treatment,
some members of the control group experienced depression, which worsened their
arthritis. This made between-group
differences larger.
3. Changes in diet, exercise, rest, etc., in
either group between the pretest and post test.
This is the extraneous variable of "history." Maybe these factors helped the experimental
group (or hurt the control group) more than drug/no drug.
4. Other differences in the composition in the
groups ("sample bias"). For
example, maybe there were more persons in the experimental group who received
social support, and social support had beneficial effects on the body.
In order for
the causal inference (the drug produces the beneficial changes) to be
plausible, we must weaken the above rival explanations. For example, we could interview participants
to try to measure some of the extraneous variables (e.g., exercise), and see if
the groups differed on these. And we
could take reliability checks to ensure accurate data.
EXERCISE
18. Show how you could apply the four
above steps or requirements of causal inference to school research. For example, how could you tell (within a
school) if a new math curriculum is effective? How could you determine (within
a state) whether National Board Certification makes or reflects a difference in
teacher quality?
The Full Inductive Strategy for Conducting Research
For for Building a Case
The last inductive strategy is used
for larger projects such as conducting research or making a case (e.g., analyzing
a text, writing an essay, or prosecuting a defendant). This large strategy uses everything we have
discussed. It has the following steps.
Step
1. Collect "Facts." The facts might
consist of:
1. Events in a play. For example, Hamlet does things that alert
King Claudius that Hamlet suspects Claudius.
Is this Hamlet's tragic flaw? Hamlet many times also fail to act (to
kill Claudius). Is this his tragic
flaw? Can we build a case leading to one
generalization?
2. Counting different behaviors; e.g., how often
students disrupt class and how often a teacher ignores, reprimands, threatens,
or tries to calm children right after disruptive behavior. Can we use these facts to build an
explanation of children's disruptive behavior?
3. Official statistics; e.g., the rates of
violent crime and the rates of nonviolent crime in different areas of the city,
the average years of education of the population, the average annual income,
the percentage of the population that attends church regularly, the ratio of
police to population. Can we use these
data to build an explanation of violent crime?
4. Descriptions of historical events; e.g.,
genocidal movements. Can we use these
data (along with data on economic factors, cultural factors, and political
factors) to create a theory of genocide?
Data collection
is guided by an interest or question; e.g., "What variables influence the
rates of children's disruptive behavior" "Under what conditions do
civilizations decline?" "Who murdered the butler?"
Step
2. Examine the Identified Events (Facts)
and Discover How Some Events Share Certain Features. After identifying common features, group the events. The group is a category and may be called an "empirical
concept." In other words, the
empirical concept is like a family. The
similarities among the events are "family resemblances." (See Ludwig
Wittgenstein's Philosophical
investigations.) For example, when I
worked in a mental hospital and at an institution for persons with disabilities
I observed the following.
1. One woman always sat leaning over with one
hand under her chin. It turned out that
she had had jaw cancer, and her lower jaw was removed.
2. Another women would moisten toilet paper,
roll it into thin tubes, and pack it in her mouth. It turned out that she had no teeth.
3. One very old woman would guide me to the
window and tell me that her children were going to visit her that day. It turned out that no one ever visited her.
4. One young man wore expensive red Nike's. It turned out that his legs were only about
18 inches long and his feet were tiny.
5. One young man (considered mentally retarded)
would put on his Detroit Tigers baseball cap and jacket, pick up his boom box
and perch it on his right shoulder, and then strut along the sidewalk. His boom box had no batteries.
6. One young woman wore a red bandana around her
forehead. It turned out that she had
heavy scars all around her head.
What is common
to these events? Not exactly the same
things. Mostly the same sorts of things.
1. Each person had one or another form of stigmata--body stigmata (tiny feet),
intellectual stigmata ("retarded"), social stigmata (no visitors).
2. Each person had a routine for "normalizing" the
stigmata. They hid the stigmata (the
woman holding up her jaw); or they drew attention to something valuable about
themselves (their cool clothes).
So, now we
create an empirical concept--"normalization"(see
Stigma by Erving Goffman)--and we
give it a conceptual definition.
EXERCISE 19. Using the method of genus and difference,
create a conceptual definition of normalization.
Step
3. Continue Collecting Events (Facts)---at Different Times, in Different Places, and with Different Persons. Revise
Definitions Accordingly. This enables us
to fill in, and to determine the limits of, the empirical concepts. For example, based on more examples, we
decided to revise the earlier definition of aggression to include feelings of
antipathy towards the object of aggression.
EXERCISE
20. Using the form genus and difference,
create empirical definitions for three concepts.
Step
4. Identify Relationships Between the
Empirical Concepts. For example, perhaps observations
(or the literature) show that interpersonal aggression is in general more
likely among persons who know each other; that physical aggression is more
likely to be used by males; that aggression in any form is more likely when
persons have a history of being reinforced for aggression; that women are more
likely to normalize physical stigmata than men are.
Step
5. State Empirical Generalizations
(Propositions) That Summarize the Relationships Discovered So Far. This means that we begin to see specific acts of
aggression as examples of more general relationships (perhaps
"laws'). For example, our research
might enable us to say:
1. Aggression
is not reflexive behavior. It is
intentional.
2. One of the conditions that controls aggressive
behavior is the consequences of aggression.
3. The more often aggressive behavior has been
reinforced, the higher is its frequency of use.
4. The more a person anticipates being punished
for aggression, the less likely is the person to use aggression.
We might
represent our theory with a flow diagram.
For example,
If Predisposing
factors If
Precipitating Factors Then
Consequences
1. Models of aggression in 1.
Frustration 1. Punishment/
childhood (vicarious 2. Envy reinforcement
learning) 3. Presence of victims 2. Shame/lack of
2. Models of antipathy shame
in childhood (whom to
dislike)
3. Early reinforcement
of aggressive behavior
4. Learning alternatives
to aggression
Note that the
above four propositions are part of a theory or explanation of aggression. By gathering more data (observations,
literature), we find more examples of what we want to call aggression; we might
revise the definition or even create a typology (of different kinds) of
aggression; we discover more factors that predict aggression; and our theory
becomes more complete (covers more).
EXERCISE
21. Add more propositions to the theory.
V. Deductive
Strategies
Deductive
reasoning is a form of thinking. Deductive
reasoning is not used only by
logicians and scientists. It is used
whenever anyone draws conclusions or makes predictions about specific things from general beliefs or rules. This section examines strategies of deductive
reasoning (thinking) beginning with simple strategies (syllogisms) and moving
to the deductive strategy for testing propositions (hypotheses).
Syllogisms: A Simple Form of Deductive Reasoning
Deductive arguments (syllogisms) have three statements. These statements are: (1) rule (general statement, or first
premise); (2) evidence (specific
statement, or second premise); and (3) conclusion. Even long and complex arguments (e.g., an
explanation of war or a case presented by a prosecuting attorney) can be broken
down into a sequence of syllogisms.
Syllogisms lead either to valid or
invalid conclusions. The validity or
invalidity of conclusions depends on what is said in the first two statements
(premises)--(1) the general rule, and (2) the specific evidence. For instance, I believe smoking increases the
chances of lung cancer (a general proposition, or rule). I see people smoking (evidence). I conclude (predict) that their risk of lung
cancer is higher than it is for persons who do not smoke. This is a valid conclusion. But not all deductive reasoning leads to valid (reasonable) conclusions. It depends on how words are arranged in each
statement of the argument. For example:
(1) All dogs
have canine teeth. (General statement:
rule, or first premise)
(2) Jack has
canine teeth. (Specific statement: evidence, or second premise)
(3) Therefore,
Jack is a dog. (Conclusion)
Actually, Jack
is a human being. The first statement
(the general rule, or first premise) says that all things in the category dogs
are also in the category things that have canine teeth. But this rule does not say that the only things with canine teeth are
dogs.
So, to conclude
(infer) that Jack is a dog just because he (as with dogs) has cannine teeth, is
invalid--and false.
This section
discusses valid and invalid forms of deductive reasoning. The first examples are followed by diagrams illustrating
what is stated by the rule and evidence.
You will be able to see how the conclusion is implied by the rule and
the evidence. Please create your own
diagrams to illustrate the remaining deductive arguments or syllogisms.
Valid
Syllogisms. Let's examine
three valid forms of deductive arguments, or syllogisms.
Valid
Form 1. Affirming the antecedent, or modus ponens: (From ponere, to affirm.)
(1) All
valid research is (in the category of things) derived from an extensive literature
review. (Rule)
(2) Jonas Salk (inventor of the polio vaccine)
conducted valid research. (Evidence)
(3) Therefore, Jonas Salk's research was derived from an extensive literature review. (Conclusion)
Things derived
from extensive lit
review
Valid
research
Salk's research
Galileo's research
Fermi's research
Now examine
this syllogism in light of the diagram, above.
The syllogism says that there is a category (things derived from extensive
literature reviews) that contains all valid research. Since Jonas Salk's research is inside the
category of valid research, his research must also be in the category of things derived from extensive literature
reviews. In other words, the conclusion
follows logically from the rule (first premise) and the evidence (second
premise).
EXERCISE
22. Make up and diagram another example
of affirming the antecedent, modus ponens.
Valid
Form 2. Denying the Consequent, or modus
tollens: (From tollere, to deny).
(1) All
effective curricula are (in the category of things) derived from valid
research. (Rule)
(2) Multiple intelligence curricula are not derived from valid research. (Evidence)
(3) Therefore,
multiple intelligence curricula are not effective curricula. (Conclusion)
Things derived
from Multiple
valid research Intelligence
curricula
Effective curricula
Open Court
Success for All
Reading Mastery
Distar Arithmetic
The above
syllogism says that all effective curricula are derived from valid
research. This implies that no effective curricula are derived from
invalid research. Multiple intelligence curricula are not in the category of things derived
from valid research. Therefore, it
follows that multiple intelligence curricula cannot be in the category of
effective curricula.
EXERCISE
23. Make up and diagram another example
of denying the consequent, or modus tollens.
Valid Form 3.
(1) No
research done in the service of a commercial curriculum can be trusted.
(2) Research on "Fantastic Phonics" is done in the service of a commercial curriculum.
(3) Therefore, "Fantastic Phonics"
research cannot be trusted.
EXERCISE
24. (1) Diagram the above
syllogism. (2) Make up and diagram
another example of a syllogism with the above form.
Invalid
Syllogisms. Conclusions are
invalid when they do not follow from--that is, they are not implied by--the
first two premises. Following are
examples of invalid deductive arguments.
Please diagram each one.
Invalid
Form 1. Denying the antecedent. Here is an example of the
fallacy of denying the antecedent.
(1) All (things in the category) organisms with
leaves are (things in the category) plants.
[Or, If an organism has leaves, then it is a plant.]
(2) The
barrel cactus does not have leaves
(denying the antecedent).
(3) Therefore the barrel cactus is not a plant.
The first premise
says that all organisms that are in the category of things with leaves are also
in the category of plants. The barrel
cactus is not in the category of things with leaves. The invalid (and false) conclusion is that
the barrel cactus is not a plant.
EXERCISE
25. Draw a diagram of this syllogism and
you will see that there is room in the plant category for other things besides
things with leaves.
Here is another example of the
invalid argument of denying the antecedent.
(1) No research
done in the service of a commercial curriculum can be trusted.
(2) Carl Crackpot's research is not done in the
service of a commercial curriculum
(denying the antecedent).
(3) Therefore, Carl Crackpot's research can be trusted.
This argument
is invalid. Let's see why. The first premise is that among the things
that make research untrustworthy is that the research is done in the service of
a commercial curriculum. But this does
not mean that the only thing that
makes research untrustworthy is that it is done in the service of a commercial
curriculum. Research is also not to be trusted if there are no
reliability checks on data collection, if tiny samples are used, and if
hypotheses are vague. Therefore, just
because Carl Crackpot's research is not done in the service of a commercial
curriculum does not mean that we can
trust it. Before we can trust it we also
have to rule out the other sources of untrustworthiness.
EXERCISE
26. Make up and diagram another example
of denying the antecedent.
Invalid
Form 2. Affirming the consequent. Here
is an example of affirming the consequent.
(1) If my hypothesis (adapting instruction to
students' learning styles fosters higher achievement) is correct (antecedent),
then students in the Adapted Instruction group will have higher test scores
(consequent).
(2) Students in the Adapted Instruction group did
receive higher test scores. (affirming
the consequent).
(3) Therefore, my hypothesis (adapting instruction
to students' learning styles fosters higher achievement) is correct.
Here, the
findings support the hypothesis, and the researcher concludes that the
hypothesis is correct. The logic seems
compelling, but the argument is invalid. The first premise does not say that the only
thing that causes higher achievement is adapting instruction to alleged
learning styles. For example, perhaps
more students in the Adapted Instruction group (in contrast to students in the
control group) received more practice, had more errors corrected, or received
more praise for success. Adapting
instruction may have had nothing to do with scores.
You can use an analogy to show that this argument--affirming the consequent--is
invalid.
(1) If it's a horse (antecedent), it has four legs
(consequent).
(2) It has four legs (affirming the consequent).
(3) Therefore, it's a horse.
Obviously, a
lot of things have four legs but are not horses--chairs and dogs. Also, horses are defined by more features
than four leggedness.
EXERCISE
26. Explain why the following syllogism
is invalid.
1. If multiple intelligence curricula are
effective, then some children in schools that use MI curricula will learn
reading, math, social, and emotional skills.
2. Some children in schools that use MI curricula
do learn reading, math, social, and emotional skills.
3. Therefore,
MI curricula are effective.
The fallacy of
affirming the consequent is always possible when we seek (and find) evidence to
support an assertion (e.g., hypothesis).
This is especially so when
findings support the assertion. The
implication of the fallacy of affirming the consequent is that hypotheses are never proved to be true
just because research findings support the hypotheses. The empirical support provided by findings
merely tells us that our hypothesis is at least not false. Our task is
to identify as many alternative
explanations as we can and then show that these other explanations are either
weak or false. By a process of
elimination (the method of residues--discussed earlier) we can have more
confidence in our hypothesis. We can't
be certain it is true, but it is safer to place a bet on it than on the other
explanations.
This section examined valid and
invalid forms of deductive reasoning; i.e., deductive arguments consisting of
three statements in a syllogism. It is
important to monitor your own thinking and writing, and to monitor the speaking
and writing of other persons, to assess the adequacy of arguments and conclusions. The next section examines a more complex
deductive strategy--the strategy used for testing propositions (e.g.,
hypotheses).
The Full Deductive Strategy for Conducting
Hypothesis-Testing Research or for Testing the Truth of a Case
Please review the part of section IV
that discusses the inductive strategy for conducting research or building a
case. Inductive research and case
building begin with facts (specifics) and gradually induce what is common or
general to them; e.g., kinds of events (concepts) and relationships (expressed
as propositions, rules--generalizations).
The deductive strategy is used in a different way. Instead of trying to discover concepts and
relationships, we are testing
them. For example, the literature might
suggest that frustration causes aggression.
Or perhaps we have observed in many settings and have induced the
generalization that most aggressive behavior follows some form of
frustration. But we may not be satisfied
that the literature or our inductive generalization is solid. We want to test the generalization more
formally; e.g., in an experiment. Or, we
may want to obtain a sample of historical documents (e.g., on war) or
statistical documents (on crime and economic well-being), to test (with these new materials) the hypothesis that
frustration is followed by aggression (e.g., that crime rises when unemployment
rises). To test hypotheses well, we
follow a deductive strategy with the following steps.
Step
1. Treat Empirical Generalizations
(e.g., From Inductive Research or
"The more
(often) aggression has been reinforced, the more likely it is to occur in the
future."
Notice that
this hypothetical proposition asserts a causal relationship not between particular events, but between classes (families, categories) of
events (frequency of aggression and frequency of reinforcement). These categories are called
"concepts" or "conceptual
variables." They are called
"variables" because each varies; that is, there can be more or less
past reinforcement for aggression and a higher or lower rate of
aggression. The hypothesis is called a
"conceptual hypothesis."
Step
2. Develop Conceptual Definitions For
Each Concept (Conceptual Variables).
We use the same method of genus and
difference to create definitions for each concept. What kind of thing is aggression and what
kind of things constitute reinforcement?
Step 3. Using
Conceptual Definitions as a Guide, We Derive or Deduce Operational Definitions. Following is an example. If aggression is defined conceptually as
"Behavior that is intended to injure humans or nonhumnans, and is either
preceded, accompanied by, or followed by feelings of antipathy (hatred,
disgust, anger) towards the object of aggression," then examples of aggression would be
hitting, kicking, preventing a person from getting a job they want, insulting,
and breaking a persons prized possessions, among other things. And if reinforcement is defined as "Any
consequent event that increases the frequency of the actions that it
follows," then examples would
include food (when hungry), water (when thirty), attention (when deprived of
attention), getting what one wants, etc.
Because the concepts aggression and reinforcement are now
operationalized as a set of specific examples, they now are called "operational variables" or
"operational concepts."
Notice that the above two
operational definitions were derived
from the conceptual definitions. If the
conceptual definitions are like circles of light that shine on portions of a
dark landscape, operational definitions
state what is inside the circles of light.
Step
4. Restate the Original Conceptual Hypothesis
as an Operational Hypothesis. We have
transformed the concepts in our original conceptual hypothesis into events we
can observe. Now we state that we expect
to find a relationship between the operationalized variables. Following is an example of an operational hypotheses.
"The more often people receive
attention, things they want, (etc.) for hitting, kicking, insulting (etc), the
more often they will use these behaviors in the future."
Step
5. Test the Operational Hypothesis (and
By Inference the Conceptual Hypothesis From Which It Was Derived). We do this by observing
the operational variables (described above) to determine whether they are associated as we hypothesized. For example, we might conduct an experiment in which the participants in
one experimental condition are sometimes put in frustrating circumstances and
receive a lot of reinforcement if they engage in aggression towards other
participants; the participants in an identically frustrating situation do not
receive reinforcement for aggression.
Step
6. We Use Methods of Inductive Inference
to See Whether the Two Sets of Variables are Connected as Hypothesized. This (Mill's
methods) was discussed in section IV.
Step
7. If the Findings Agree With Our
Prediction (From the Operational Hypothesis), We Tentatively Accept the
Conceptual Hypothesis From Which the Operational Hypothesis Was Deduced.
VI. Fallacies
of Relevance and Ambiguity
Fallacies of relevance and ambiguity
have to do with logical errors in everyday (and research) arguments. The errors may the result of sloppy thinking;
they may be unintentional slips; or they may be rhetorical tricks to sway
gullible audiences. By studying these fallacies,
you will become fluent at spotting errors in your own and in other persons'
arguments. Definitions of the fallacies
are from the work of Copi (1986) and Downes (1996). Useful
exercises involve spotting fallacies in articles and books, TV commercials,
political speeches, and everyday conversation.
For example:
"The children in this village in
"The famous basketball player, Judy Slamdunk,
believes that our vitamin supplements make her strong. Therefore, you should buy our vitamin
supplements." [Newspaper ad. Appeal
to authority.]
1. Arguing
Against the Person (argumentum ad hominem)
The fallacy of ad hominem is committed when an argument attacks an opponent (e.g.,
a person or group with a different view) rather than the opponent's evidence
and logic. Sometimes, the person or
group is said to have negative qualities; and therefore, the opponent's
argument should not be accepted. This is
the abusive version of the ad hominem argument. For example:
"You can’t
accept the implications of B.F. Skinner's research. After all, he was a behaviorist."
Sometimes the ad hominem argument is that a person's
or group's position should not be accepted because of their special
circumstances. This is the circumstantial version of the ad hominem argument. For example:
"Oglethorpe
is an engineer. Of course she advocates
focused and systematic math instruction based on solid research. She must be biased."
In other words,
it's implied that the opponent's argument is invalid because the opponent
benefits from the argument or because the opponent has to believe what he or she says, and therefore the argument
cannot be trusted. However, these
considerations are irrelevant to the
validity of the opponent's argument.
Ad
hominem arguments can be handled by: (1) determining whether the arguer
presents credible evidence in support of his own position and/or against the
opponent's position; (2) identifying the negative characterization of the
opponent and revealing how this characterization is used to invalidate the
opponent's argument or position; and (3) showing what sort of solid evidence is
needed to invalidate the opponent's position and/or support the arguer's
position.
EXERCISE
22. Make up (or find) an ad hominen
argument.
2. Prejudicial
Language
This invalid argument uses
emotionally loaded words to persuade an audience that the arguer's position,
conclusion, or suggestion is reasonable and acceptable because it seems morally good, or that an opponent's
position, conclusion, or suggestion is unacceptable because it seems morally
bad. The emotive words "pump
up" the audience, and give the audience the sense that it is on the side
of right. For example,
"In
contrast to the rigid, piece-meal and conformity-fostering curricula forced on
children by behaviorists, our child-centered curriculum provides children with
a seamless series of authentic and meaningful experiences that foster
self-esteem and enable children to develop their inner potentials."
Appealing as it
sounds, this argument gives no data
on what "behaviorist" vs. "child-centered" curricula actually do and what the curricula actually
yield--so that a reasoned comparison and choice can be made. Instead, the arguer uses words (not precisely
defined) appealing to the audience's negative sentiments about conformity and
piece-meal instruction, and positive sentiments for children, authenticity, and
development. The implication is that
anyone who disagrees with the arguer is against children, individual
development, and authenticity. This
argument can be handled by: (1) identifying prejudicial words and showing how
they are used to sway the audience; and (2) showing that the arguer has no
credible evidence for his or her position or against his or her opponent's
position.
EXERCISE
23. Make up (or find) an argument that
uses prejudicial language.
3. False
Dilemma
In this fallacy, an arguer makes it
seem as if there are two or three (and only two or three) opposing options;
e.g., two possible ways to understand things, two ways to interpret data, two
conclusions that can be drawn, or two responses to a problem. Then the arguer tries to discredit or
invalidate the position(s) he or she opposes.
This appears to leave only
the arguer's position--which, by elimination, the audience is logically bound
to accept--if the audience falls for the false dilemma. For example:
"There are
only two kinds of data--quantitative (numbers) and qualitative (meanings,
interpretations, narrations).
Quantitative data say nothing about how children make sense of their
school achievements. But this is just
the sort of information we need. Therefore,
we must choose qualitative over quantitative data."
The false
dilemma, here, is that research cannot
be divided neatly: (1) into qualitative vs. quantitative data, and (2) into data
that tell about persons' experiences vs. data that do not tell about persons'
experiences. In fact, quantitative data can speak to how persons see things (e.g., teachers can count the
number of times per day that they blame students for not getting the material);
and some qualitative data say nothing about how persons see things. So, the argument gives a false choice. The way out of this argument is to show that
the forced, limited choice is false and to suggest additional options.
EXERCISE
24. Make up (or find) an argument that
uses false dilemma.
4. Appeal to
Popularity (argumentum ad populum)
This invalid argument involves
persuading an audience to accept a speaker's or writer's conclusions because
other persons and groups already do so. For
example,
"Hundreds
of schools and businesses in the
Unfortunately,
this argument is often effective. For
instance, a study by Solomon Asch showed that at least one-third of the
participants in his experiments agreed with the majority's judgment about which
line was longer even when the group's judgment was obviously wrong. Subjects
went along to avoid being the lone nonconformist. Similarly, jurors in trials of teachers and
day care workers accused of child abuse said they went along with the majority
even though they believed the defendants were innocent; they could not see how they alone could be right when so many other persons had the opposite
opinion.
This fallacy can be handled by: (1)
showing how the arguer appeals to popularity to support his or her conclusions;
and (2) showing that the popularity of a position is not evidence for the
validity of the position. For example,
jurors have convicted innocent persons; our species long thought the sun
revolved around the earth; and education in the
EXERCISE
25. Make up (or find) an argument that
appeals to popularity.
5. Appeal to
Pity (argumentum ad misercordiam)
This fallacy is similar to the
appeal to the population; it, too, relies on emotion. An arguer implies that his or her
explanations, conclusions, positions, or suggestions should be accepted, and/or
that alternative explanations, conclusions, positions, or suggestions should be
rejected, because failure to agree would injure the arguer or some other
persons. For example, before there was
much research on whether inclusion did any good, many groups advocated full
inclusion of children with severe mental retardation in classes for typically
developing children by appealing to readers' sympathies.
"Will we
continue to keep these children in a shadowland--outside the circle of warmth
and protection with their fellow human beings?
Will we add even more misery to their lives? Or will we at last provide them with their
rightful place?"
The appeal to
pity can be handled by: (1) showing how the argument appeals to the audience's
sensibilities (e.g., about the difficult lives of many persons with
disabilities); (2) showing that the argument does not give direct evidence that supports the position
(e.g., that inclusion leads persons with disabilities out of a "shadowland,"
decreases "misery," or helps them achieve a "rightful
place") or that refutes opposing positions; and (3) that the argument
(e.g., advocating full inclusion) may be against the interests of persons whom
the argument claims to support.
EXERCISE
26. Make up (or find) an argument that
appeals to pity.
6. Fallacy of
Division
This fallacy is the argument that
the characteristics of a whole
(e.g., an automobile engine is heavy) apply to all of the elements (therefore all of its parts are heavy). An example of the fallacy of division in education would be an argument that the
average test scores in a school are high; therefore, all children in the school (or all classes in the school) got high
scores (or were proficient). In fact,
some classes may have gotten very high scores, which pulled up the average
(school) scores.
EXERCISE
27. Make up (or find) an argument that
commits the fallacy of division.
7. Fallacy of
Composition
This fallacy is the flip side of the
fallacy of division. It is the fallacy
of arguing that the characteristics of elements
(e.g., an engine's parts are light) apply to the whole (therefore the engine is
light). An example of the fallacy of composition is arguing that:
(1) since all of the children and teachers in a school improved their skills a
great deal, therefore, (2) the school as a whole improved a great deal. The problem is that the school is a social
organization; it has features (organizational features) that its elements
(individual human beings) do not have.
Teachers and students (elements) may have learned new skills, but: (1)
the school division of labor (a feature of the whole) may still involve a high
degree of specialization and little collaboration among teachers; (2) there may
have been no change in patterns of power; and (3) there may have been no change
in relationships with families.
EXERCISE
28. Make up (or find) an argument that
commits the fallacy of composition.
8. Argument
From Ignorance (argumentum ad ignorantiam)
There are two
forms of this fallacious argument:
1. There is no
solid evidence that supports a
position, conclusion, or suggestion. Therefore, the position, conclusion or
suggestion must be false, invalid,
or generally unacceptable. Or,
2. There is no
solid evidence that a position, conclusion, or suggestion is false, invalid, or unacceptable.
Therefore, the
position, conclusion or suggestion must be true,
valid or acceptable.
For example:
Ms. White: "You
say new teachers should be assessed for licensure by portfolios. But you don't
have evidence that portfolio assessment leads to the selection of better
teachers?"
Ms. Wong: "Maybe
not. But you don't have evidence that
portfolio assessment doesn’t lead to
selection of better teachers."
Ms. White is right;
Ms. Wong is wrong. Advocates for a
conclusion, technique, treatment, curriculum, or social policy are obliged to
provide positive evidence
(supporting data) for what they advocate.
In other words, lack of evidence
is not evidence. That's why prosecuting
attorneys must prove that defendants committed a crime; defendants don't have
to prove that they didn't.
EXERCISE
29. Make up (or find) an argument from
ignorance.
9. Slippery
Slope
In this argument, a person claims
that failure to accept his or her conclusions or suggestions and/or acceptance
of an opponent's conclusions or suggestions, will have increasingly bad
effects. For example:
"If state boards of education require
publishers to have empirical evidence that their textbooks or curricula are
effective, that will be the thin end of the wedge by which school boards take
away more teacher autonomy. Soon they
will require that we submit lesson plans to school boards for approval. Moreover, this policy will inhibit publishers
from developing new materials, and so we will have to use increasingly obsolete
materials."
This is an
appeal to fear. No evidence is presented
that adverse effects will occur or cannot be stopped.
EXERCISE 30.
Make up (or find) an argument that commits the fallacy of the slippery slope.
10. False Cause
(post hoc ergo propter hoc)
"Post hoc ergo propter
hoc" is Latin for "After this, therefore because of this." An argument commits this fallacy when a
person claims that because one event follows another event, it was caused by the prior event. However, the fact that one event follows
another event may be coincidence. There
may be no causal connection at all. Each
event may be caused by a separate chain of causation. Or two events may be caused by a third,
unknown event. Here is an example of post hoc ergo propter hoc.
"We pre-tested students' math
skills. Then we implemented the new
'Creative Calculus' math curriculum. And
then we gave students a post-test.
Post-test scores were much higher than the pre-test scores. Therefore, Creative Calculus is
effective."
After Creative Calculus, therefore because of Creative Calculus. Math scores may have changed, but not because of the new curriculum.
Perhaps teachers communicated to students that they had high
expectations that students would succeed (which they did not communicate with
the old curriculum); or perhaps the post-test was easier than the pre-test; or
perhaps some students with the lowest math aptitude dropped out after the
pre-test (and so their likely low post-test scores could not drag the average
down). Many other extraneous factors could account for the findings. We could more clearly show whether the
curriculum does or does not work by conducting an experiment with equivalent groups. One group gets the new math and the other
gets the old math. If the group that
receives the new math has higher pre-test to post-test differences, and if all
other extraneous variables are pretty much equal across the two groups, then we
can begin to suspect that the curriculum makes a difference.
EXERCISE
31. Make up (or find) an example of an
argument that commits the post hoc fallacy.
11. Wrong
Direction
In this fallacy, the direction of
cause and effect is reversed. For
example,
"The rates
of mental illness increase as we examine data from suburban to inner city
areas. Therefore inner city areas cause mental illness more than suburban
areas."
It is more
likely that as some persons who live in suburban areas become mentally ill, and
cannot hold their jobs, they lose income, abuse drugs and alcohol, lose their
families, and end up homeless in the inner city. In other words, moving into the inner city
does not cause mental illness. It’s the
other way around; some people move into the inner city because of impairments
resulting from mental illness. Here is
another example of the fallacy of wrong direction.
"When you
observe cooperative learning groups, you find that the high status students end up running discussions and learning
the most. Therefore, cooperative
learning groups produce social
inequality."
It is more
likely that causation runs in the opposite direction; students who enter
cooperative learning groups with high social status and skills control
discussions from the start. The
cooperative learning format may sustain inequality, but inequality was already there.
EXERCISE
32. Make up (or find) an argument that
commits the fallacy of wrong direction.
12. Begging the
Question (petitio principii)
In this fallacy, no empirical
evidence is given to support a conclusion.
Instead, the conclusion merely
restates the premise. For example,
(1) "God exists." (premise)
(2) "I
know God exits because if something exists I will know it." (evidence)
(3) "Since
I know God exists, God must exit." (conclusion)
Well, that
clears things up nicely! Here's another
example:
(1) "Children who are most disruptive
in class have ADD (conclusion) because
(2) Children
with ADD usually engage in a lot of disruptive behavior (premise)."
It may be true
that ADD is associated with (indeed is partly defined by) disruptive
behavior. However, this does not imply
that most disruptive behavior in classes can be traced to children with
ADD. Children without ADD also engage in disruptive behavior. Read the two statements again. Note that the conclusion and the premise say virtually the same thing. If you aren't careful, the premise seems to
provide good reason for the conclusion.
Here's still another example of petitio
principii (begging the question).
(1) "Whole
language is an effective way to teach children to read (conclusion) because
(2) Whole
language uses literature rich environments and authentic materials which are
conditions that foster reading skill (premise: evidence)."
Again, the
premise that is supposed to provide good reason (evidence) for the conclusion
merely restates the conclusion. However, no empirical evidence is given on,
for example, how many children in a group could read before and after whole
language, in contrast to a comparison group that received an explicit phonics
curriculum.
EXERCISE
33. Make up (or find) an example of an
argument that begs the question.
13. Converse
Accident (Hasty Generalization)
The fallacy of hasty generalization
is committed when a generalization is made from an exceptional case (or what later turns out to be an exceptional
case) to a larger population of events.
For example, I helped develop a curriculum for autistic children. This curriculum was quite successful. However, it would have been a hasty
generalization to imply that the curriculum would be as effective with other autistic children in the larger
population. Why? Because the sample of
35 children may have been exceptional
in some way--i.e., not representative of the larger population of autistic
children. The children we worked with
may have been younger; less impaired; living with parents who were more skilled
at teaching their children. What worked
with our sample of children may not have worked with other children. Therefore, before generalizations were made about
how the curriculum might be used with other children, it was replicated with more and more samples
of children--younger, older, more impaired, less impaired, single-parent
families, two-parent families, families with much social support, families with
less social support, etc. The more times
the curriculum was replicated with different
samples, and still shown to be effective, the more confident we could be
about making generalizations to the larger population of children with autism
and their families.
EXERCISE
34. Make up an example of an argument
using hasty generalization.
14.
Equivocation
This fallacy is committed when the
meaning of one or more important words changes during an argument to make the
conclusion seem valid. For example,
"Virtually
all of experience consists of constructs
such as time, space, objects and cause-effect (premise). Therefore, we may say that children construct their own experiences
(conclusion)."
The conclusion
that children construct experience seems plausible--because the meaning of
"construction" changes between the premise and the conclusion. In the premise, "construct" is a
noun. Constructs are things--tools--by
which children create experience. In the
conclusion, "construct" is used again--only now it is a verb
synonymous with "make." Since the
same word is used in the premise and conclusion, a reader may accept the
conclusion, just as one would accept the argument, "A rose is a rose is a
rose." Well, of course it is! However, just because experience consists of constructs, does not mean
that experience is constructed. Constructs (and experience) could be
transmitted through communication, shaped without a child's noticing as the
child interacts with her environment and learns language.
EXERCISE
35. Make up an example of an argument
using equivocation.
This section
examined informal fallacies--arguments
using words in a way that makes false or unsupported conclusions seem
reasonable--to an inattentive or naive audience.
Following are excerpts from various
publications. Each excerpt has one or
more logical fallacies. Try to identify
and correct them.
Is this 'war'
really about skills and how to teach them?
On the surface it is, but adequately understanding the conflict requires
addressing deeper issues ingrained in the arguments about teaching method. One concerns broad goals for children's
development. Accompanying the call for
the direct instruction of skills is a managerial, minimally democratic, predetermined,
do-as-you're-told-because-it-will-be-good-for-you form of instruction. Outcomes are narrowly instrumental, focusing
on test scores of skills, word identification, and delimited conceptions of
reading comprehension. It is a scripted
pedagogy for producing compliant, conformist, competitive students and
adults. (Gerald Coles. "No end to the reading wars." Education
Week, December 2, 1998.)
We see two major assumptions of the
behaviorist approach that contrast with the assumptions of the constructivist
approach. The first broad assumption of
the behaviorist approach is that environmental stimuli shape and control
individual behavior responses. This
assumption reflects the view that the child's interests and purposes are
irrelevant and leads to teacher-centered power assertion in relation to
children. This is in contrast to the
constuctivist view that the individual must actively construct knowledge,
including stimuli and responses. The
reader will recognize the practical implications of this behaviorist assumption
as contradictory to constructivist cooperation in relation to children.
(DeVries and Zan, 1994: p. 267)
The experiences
of three students, Mimi, Vivianne, and Hannah, who were members of two
different literature discussion groups, will be used to illustrate why we
believe issues regarding empowerment and voice need to be more closely
scrutinzed. (Evans, Alvermann, &
Anders, 1998. p. 109)
Mimi and
Vivianne's experiences also challenge the assumption that helping students
'find their voice' is an effective method of helping students excercise the
power. In the beginning of the group
[discussion of a book; three girls and three boys; fifth grade], Mimi and
Vivianne clearly had found their voices and were using them. Rather than resulting in empowerment however,
they were subjected to overt silencing attempts. While Mimi chose to continue to interrupt the
boys' efforts to silence her, Vivianne chose silence... Hence, rather than
simply view Vivianne's silence as a sign of oppression, the more important task
is to examine why she chose to be silent and the contextual factors that led to
her silence. Was Vivianne speaking more
loudly through her silence? (p. 113)...Hannah's silence raised several
questions for us. Can one have a silent
voice? Must one speak to show evidence of voice? (Evans, Alvermann, & Anders,
1998. p. 115)
Reform...is not
easy, but how we conceptualize things makes a difference. The viable alternative we have been exploring involves reconceptualizing the whole of education as inquiry. For us
and the teachers with whom we work, education-as-inquiry represents a real
shift in how we think about
education...We want to see reading
as inquiry, writing as inquiry, classroom discipline as inquiry, and both
teaching and learning as inquiry.
Instead of organizing curriculum around disciplines, we want to organize curriculum around
the personal and social inquiry questions of learners...Inquiry as we see it is
about unpacking issues for purposes of
creating a more just, a more equitable, a more thoughtful world...Theoretically, education-as-inquiry finds its roots in
whole language, sociopsycholinguistic, or, these days what we prefer to call
socio-semiotic theory or what others call cultural studies. (Harste & Leland, 1998. p. 192-3)
...when parents
and teachers plan children's environment and activities carefully so that literacy is an integral part of everything they do, then literacy learning becomes a natural and meaningful part of children's everyday lives. When you create this kind of environment, there
is no need to set aside time to teach formal lessons to children about reading
and writing. Children will learn about
written language because it is a part of their life. (Schickendanz, 1986. p.
125)
References
Coles, G. (1998).
"No end to the reading wars."
Education Week, December 2.
Copi, I.M.
(1986). Introduction to logic (seventh
edition).
DeVries, R.,
& Zan, B. (1994). Moral classrooms, moral children.
Downes, S
(1996). Stephan's guide to logical fallacies.
http://www.assiniboinec.mb.ca/user/downes/fallacy/index.htm
Evans, K.S.,
Alvermann, D., & Anders, P.L. (1998).
Literature discussion groups: An
examination of gender roles. Reading Research and Instruction, 37, 2,
107-122.
Harste, J.C.,
& Leland, C.H. (1998). No quick fix:
Education as inquiry. Reading Research and Instruction, 37, 3,
191-205.
Schickendanz,
J.A. (1986). More than the ABC's: The early
stages of reading and writing.