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.) </