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 Boston patriots.

      c.    Discriminations; e.g., properly labeling federalist vs. anti-federalist 



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.



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.



            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          |                      |                      |                      |                         *





            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


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


                                                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



                        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 Case Building), Speculations, and Beliefs as Hypotheses to Test--to Verify or to Falsify.  For example, perhaps you want to test an alternative to the frustration-aggression hypothesis.  You suspect that frustration is not a sufficient condition.  Even if they are frustrated, persons are more likely to engage in aggression if they have a history of being reinforced for it; e.g., by getting what they want.

"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 Chile have little food.  They drink polluted water.  They have hardly any clothes.  Therefore, you should send us money every month to help these children."  [TV commerical.  Appeal to pity.]


      "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 United States have school-to-work programs.  So do some foreign countries.  There is substantial government funding for these programs.  Obviously, Smith's opposition goes against the trend."  (The implication is, "How can Smith argue against these programs?  How can Smith be right and so many other persons and groups be wrong?")


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 United States has been dominated by faddish ideas and methods that later proved worthless.


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)




Coles, G.  (1998).  "No end to the reading wars."  Education Week, December 2.

Copi, I.M. (1986).  Introduction to logic  (seventh edition).  New York: MacMillan.

DeVries, R., & Zan, B. (1994).  Moral classrooms, moral children.   New York:  Teachers College Press.

Downes, S (1996).  Stephan's guide to logical fallacies.  Assiniboine Community College.  Brandon, Manitoba, Canada.  On-line at


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.  Washington, DC:  NAEYC.