What Teaching Is—Applied Logic
Teaching is in the branch of philosophy called epistemology. ἐπιστήμη [epistemeh]--knowledge. λόγος [logos]--study of. What branch of epistemology? Applied logic. What is logic?
A branch of philosophy and mathematics that deals with the formal principles, methods and criteria of
validity of inference, reasoning and knowledge. Let’s look at those three words.
1. Inference. Generalizations that are the product of reasoning.
a. Inductive reasoning. Start with events--specifics. The teacher holds up objects that differ in size and shape, but are the same in one way, and for each one she says, “This is kokkivo.” [Gr. v = nnn] Then she juxtaposes objects that are the same in size and shape, but that differ in one way---color. She calls one of them, “This is kokkivo,” and she calls the other one “This is not kokkivo.” At each step in the communication, the learning mechanism performs a logical operation---a reasoning step.
The learning mechanism:
(1) describes each event; (2) compares events all called “kokkivo” to find sameness (“These look the same in one way and are called kokkivo.”), (3) contrasts events to find difference (“This one looks different in only one way from all of them that were called “kokkivo,” and the teacher called this one ‘not kokkivo.’”), (4) draws a conclusion (generalization) about what the events reveal. “Kokkivo must mean the color that all the ones called kokkivo had and the ones called not kokkivo didn’t have.”
b. Deductive reasoning. Start with a general (a definition of a concept, a rule about how things are connected, a routine for solving a kind of problem). Examine a new event and see that it has the defining features of the concept, rule, or routine. Deduce how to treat the example (name it, make a prediction about it, solve it).
"Here's a new one. It has the same color as the ones called ‘kokkivo’---the very feature that the ones called ‘not kokkivo’ didn’t have. We know that kokkivo is the name of a color. This new one HAS that color. So, by deduction, this new one must be the color kokkivo.”
2. Reasoning. The sequence of logical operations (steps) in thinking from specifics to a generalization (inductive) or from a
general (e.g., rule) to a prediction about specifics. An inference (#1 above) is valid if and only if the proper steps were followed,
and the conclusion (inductive or deductive inference) clearly follows the steps in the argument/reasoning.
3. Knowledge is the conclusion, the inference, the generalization made on the basis of valid reasoning.
The person has learned a new concept by induction (kokkivo), has learned the color name of a new event by deduction from the
definition, or has performed a new routine (e.g., solved a new problem based on a deduction from a general problem-solving routine).
“Every time she said ‘I’ll sound out this word,’ she said the first sound on the left; then the next sound on the right; and then the last sound on the right; and she didn’t stop between sounds.”
ma, am, sa, sam, mam.
“Now she’s showing a new word. mas. I bet she does it the usual way.” [Or, “My turn to sound it out? No problem. I’ll just use the routine that I have internalized.”]
Here’s the logic of generalization via deductive reasoning.
1. “So far, you sound out words by following these steps…..” [First premise. Rule.]
2. “Here’s a word.” [Fact relevant to the first premise.]
3. “So, I’ll sound out this word by following the steps.” [Conclusion deduced from new fact and
Well, teaching students to acquire new knowledge (by inductive reasoning) and to use or apply knowledge (by deductive reasoning) is exactly what successful teachers do.
But it’s easy to confuse the learning mechanism. What if the teacher uses examples that don’t reveal the defining features of a concept? What if she uses too few examples for the learning mechanism to “get it” (to infer the proper generalization)? What if students don’t know what they are supposed to pay attention to? What if students don’t have the pre-skills needed to do what the teacher does?
So, instruction has to be isomorphic with the logical operations of induction and deduction that the learner is trying to perform. It has to support each operation/step.
What Logical Operations the Learning Mechanism is Prepared to Do.
How Instructional Communication Should be Isomorphic with the Operations of the Learning Mechanism.
1. Examines events--which will after a process of inductive reasoning be seen as examples of something larger---a concept, rule, or routine.
1a. Teach how to examine events. A pre-skill.
1c. Teach what a fact, concept, rule, and routine are. Teach the logical operations by which they are learned so that the students can do the steps with intention and in a planful way, and can talk themselves through the reasoning.
“Well, she called that one linear and this one not linear. She treated them differently, so they must BE different in some way. I’ll compare and contrast them to find the difference in features than made the difference in how she named them.”
2. Identifies features of events.
2. Present events that are examples of a concept, rule, or routine.
|3. Compares events to identify sameness of features and sameness of treatment of each event in the midst of differences.
@$!*^ “This is flerm.”
Q#)!=+ “This is flerm.”
3. Use a set of examples, one after another (acquisition set) that differ in nondefining ways but are the same in the defining features. Treat each of these “same” examples the same way---name them, sort them, list them, solve them.
Teach students to note and to identify the samenesses and the differences across events.
Teach students to note that events that have sameness are treated the same. Have them say how they are the same and how this sameness leads to how they are treated.
[The enables the learning mechanism to use the method of agreement.]
4. Contrasts events to identify differences in features and associated differences in treatment.
4. Present events whose nondefining features are the same but whose defining features are different. Treat them differently---different name, list, group, solution.
Teach students to note and to identify the samenesses and the differences in features across events; to note differences in how they are treated; and to identify the difference in features that is associated with differences in how they are treated.
This enables the learning mechanism to use the method of difference.Note: if you first use a set of examples (#3), and then juxtapose examples and nonexamples (#4), this enables students to use the very powerful (in terms of clarity and validity) joint method of agreement and difference.
5. Makes a generalization.
“! (how events called ‘flerm’ were the same) is flerm (the common treatment of events with !).”
“Every rock that she called granite consisted solely of mica, feldspar and quartz. No rock that did not consist of all three—mica, feldspar, and quartz---was named granite. Therefore, granite (the concept) is an igneous rock (genus) composed solely of mica, feldspar, and quartz (difference between granite, as an igneous rock, and other igneous rocks).” (Concept as a generalization)
“She performed steps 1-6 every with the 10 problems arranged like this---(a +/- b)(c +/- d). The numbers were different, but she treated them (the steps) the same way. When the numbers were arranged differently—a (a +/- b)(c +/- d)---she did NOT perform steps 1-6. So, steps 1-6 must be the general way to solve that kind of problem. (Routine as a generalization).
“Planet 1 is 1,000,000 miles from its sun, and its orbit is 4,000,000 miles.
Planet 2 is 10,000,000 miles from the sun, and its orbit is 35,000,000 miles.
Planet 3 is 50,000,000 miles from the sun, and its orbit is 160,000,000 miles.
Therefore, the greater the distance a planet is from its sun (at least with these examples), the greater is its orbit around the sun.” (Rule as a generalization)
5. Teach students to state a generalization that connects the samenesses and their associated treatment.
a. Teach students to use proper form for simple declarative statements of fact knowledge.
“James Madison (subject) was the fourth President of the United States (predicate).
b. Teach students to use proper form for verbal definitions of abstract concepts.
Subject is in the larger class of (genus) and it has the defining features a, b, c (in contrast to other classes in that larger genus= the difference).
c. Teach students to state rules in proper form that reflects the examples.
(2) Whenever X (happens, increases, decreases), Y (happens, decreases, increases).
(3) Whenever X (happens, increases, decreases), Y (happens, increases, decreases), but only if Z (happens, increases, decreases). –intervening variable.
(4) X is a necessary condition for Y. [Y doesn’t happen without X.]
(5) X is a sufficient condition for Y. [Whenever X happens, Y happens.]
(6) When A, B, C, and D happen (in a sequence of connections), W, X, and Y happen. State and diagram.
d. Teach students to state the steps in a routine, as well as the concepts and rules that govern each step. “Multiply the numbers in the ones column. If the product is 10 or more….”
What is Knowledge?
The human learning mechanism transforms waves (e.g., sound), particles (e.g., photons), and molecules (e.g., from cheeseburgers) that are captured by our sense organs, into a representation of reality consisting of objects, persons, groupings, places, and events, all happening and connected in time and space.
These representations of reality are what we call “knowledge.”
Because human learning mechanisms operate much the same way, human beings experience a world in common.
Human beings communicate our representations (knowledge) with language, music, dance, pictures, and sculpture.
Here’s how it works.
Here’s the big question. How does a kaleidoscope of sensations---color, touch, odor, sound---become (1) individual things and their features, (2) classes of things defined by common features (concepts), and (3) connections among things (connections understood as systems, causal relationships, and category relationships—such as all cats are in the class of felines.
Here’s the answer. Look at column (3) in the above diagram.
1. We are born with a “learning mechanism”—brain, eyes, ears, skin, and nose (Engelmann and Carnine, 1991).
2. This learning mechanism is set to perform a series of logical steps (“operations”) with the sensations of color, sound, odor, and touch made by our sense organs.
3. This series of logical steps (“operations”) transforms sensations (colors, odors, touch, and sounds) into things, kinds of things, and connections among kinds of things.
4. Connections are in space (near, far, on, next to), in time (first, next, before, after), in togetherness (red is a kind of color; poison is not a kind of food), in causation (When X happens, Y will happen.), and in a series of steps (routine) that gets something done (“If you do 1, 2, and 3, you produce effect 4.”).
5. We use language (as well as painting, music, dance, and sculpture) to represent WHAT the learning mechanism has accomplished when it transforms sensations into things, kinds of things, and connections—that is, transforms sensations into a world that seems to be really there---solid.
6. These representations are called “knowledge.”
7. There are different forms or kinds of knowledge---that is, representations: facts (knowledge of features of individual things), concepts (knowledge of features of kinds of things), rule-relationships and routines (knowledge of connections). Like this.
a. Knowledge of features of individual things: fact knowledge. Statements of subject (individual
thing) and predicate (feature). “This table is brown, has four legs, and is made of pine.”
b. Knowledge of features of classes of things: concept knowledge. Statements of definitions.
“Monarchy (class of things) is a political system that involves (1) rule by one person (2) usually on
the basis of hereditary ascension (the two features of all things that are political systems grouped
together and called ‘monarchy because they share those two features).”
c. Knowledge of connections (three kinds) among classes of things:
(1) Category-rule knowledge. Statements of category propositions. “This class of things is
(inside, partly inside, outside ) that class of things.” “All mushrooms are fungi.” Or “All things
in the class of mushrooms are also in the larger class of things that are fungi.” Also, “Some
fungi are poisonous.” And “No sedimentary rocks are igneous rocks.” Or, “No rocks in the
class of sedimentary rocks are in the class of rocks that are igneous rocks.”
(2) Causal (hypothetical or functional) rule knowledge. Statements of causal propositions.
“When this happens, then that happens.” Or, “When the temperature of water decreases to
32 degrees Fahrenheit, the molecules crystallize.” Or more precisely, “If and only if the
temperature of water decreases to 32 degrees Fahrenheit, do the molecules crystallize.”
And, “The higher the demand for a commodity, the higher the price of the commodity will
d. Routine knowledge. “When you do steps 1, 2, 3, and 4, you end up with a product of some
kind.” “To sound out a word, say the first letter-sound slowly, and don’t stop; then say the
next letter-sound slowly; and don’t stop; and then say the last letter-sound slowly.”
And that’s all the kinds of knowledge there are.
Therefore, everything you know or can know or can communicate boils down to these kinds of knowledge: facts, concepts (sensory and abstract), rules (categorical and causal), and routines.
Therefore, our reality IS of (and is ONLY of) facts, concepts (sensory and abstract), rules (categorical and causal), and routines.
Let’s take a look at each one.
8. Fact knowledge is what we say, dance, write, paint, and chisel about the features of individual things. Fact knowledge is best communicated with simple declarative statements of subject and predicate.
“This meat chunk (individual thing) is greasy (a feature of this meat chunk).” A fact statement about ONE meat chunk.
“Today (individual thing: subject of a fact statement) is Thursday (a feature of this day: predicate of a fact statement).
9. Concept knowledge is what we say, dance, write, paint, and chisel about features of kinds (categories, classes) of things. Imagine that the learning mechanism sees how the following individual things are the same in important ways (shape of body, teeth, claws, how eyes operate, how they hunt), despite many differences (size, fur color, where they live), and so it puts these individual things in the same place---a circle.
And the learning mechanism represents this ACT of categorizing all these beings on the basis of their
identified samenesses with a definition that tells the common features---the basis for the grouping.
Felines are a class of mammals (genus: the larger class in which felines and other mammals
exist) that have four legs, retractable claws, eyes with elliptical pupils and a large number of
rods that permit accurate vision in low light, long canine teeth, that hunt, and that make “the
motorcycle” with their hind leg if you tickle their tum-tum (difference between felines and other
animals in the class of mammals).
10. Category rule knowledge (knowledge of ONE kind of connection) is what we say, dance, write, paint, and chisel about how things are connected in togetherness. One class of things is all part of, partly part of, or not at all part of other classes of things. Here are examples. Notice kinds of togetherness---all of one class is (a) INSIDE, (b) partly inside and partly outside, or (c) all outside another class/category.
a. All (things in the class/concept of) maple trees are also in the larger class/concept of (things that
are) deciduous trees. Or, All maple trees are deciduous.
Here’s another example of category rule knowledge of togetherness among classes/concepts.
All forests are ecosystems, or All things that are in the class/concept of forests are also into the
larger class/concept of things that are ecosystems.
All (things that are in the class of) ecosystems are also (in the larger class of things that are)
systems of living beings. Or, all ecosystems are systems of living beings.
b. No (things in the class/concept) of apples are also in the class/concept of (things that are)
vegetables. Or, No apples are vegetables.
Okay, so we looked at fact knowledge, concept knowledge, and category rule knowledge. Now….
11. Causal rule knowledge (knowledge of another kind of connection) is what we say, dance, write, paint, and chisel about how things are connected through time----When X happens (earlier), Y happens (later).
We aren’t usually sure that one earlier thing MAKES another later thing happen. One thing---a change in X---is simply followed by a change in Y. An earlier KIND of thing may not MAKE another later kind of thing happen; it just PREDICTS that a later thing will happen. So, another way to name this kind of before-after CONNECTION is hypothetical or functional rule knowledge.
For instance, we can say that
In general, whenever you see lightning, you’ll soon hear thunder.
In other words, the learning mechanism has noticed that a whole bunch of lightning and thunder facts SHOW the same connection: first lightning and then thunder. You can represent this knowledge with a rule sentence (as above), or with hand gestures and noises (Kablooie!), or with a picture [ZZZZZZ boom].
Another example of knowledge of a causal, hypothetical, or functional connection in time is….
In general, if and only if the temperature of steel increases to 2700 degrees Fahrenheit, will the steel melt.
The learning mechanism has noticed a bunch of cases (facts) about steel and its temperature. It found that when the temperature of the steel rose to 2700 degrees, the steel always melted, and when the temperature was less than 2700 degrees, the steel never melted. And so the learning mechanism SUMMARIZES or makes a GENERALIZATION from these individual facts, that, in general, if and only if (and whenever) the temperature of steel increases to 2700 degrees Fahrenheit, will the steel melt.
This learning mechanism is one smart guy---or gal.
“You bet! Without me, Pilgrim, working like a dog behind the scenes to transform mere incoming energy, and then raw sensations, into things, kinds of things, and connections among kinds of things, you would not have a world.”
|http://www.byrnerobotics.com/forum/uploads/JohnByrne2/2011-02-07_181719_alfred_e_neuman.jpg#alfred%20e%20neuman%20einstein%20298x414||http://www.who2.com/sites/ default/files/ imagecache/blog-full/imagecache/blog-full/blog/inline/5/ mad_apes_backcover.jpg
We represent routine knowledge with pictures that show “first do this and then do that” and with
spoken and written lists of steps.
12. Routine knowledge (knowledge of another kind of connection) is what we say, dance, write, paint, and chisel about how things are connected by a series of steps that gets a job done. The series of steps is a routine. Examples.
a. Get fuel. Add oxygen. Heat the fuel. à You get ignition.
b. ram Say rrr, say aaa, say mmm à You sounded out a word.
c. (a +/- b)( c +/-d) = what? Multiply the first, then the outside, then the inside, and then the last
numbers, and you get the answer.
How Do Human Beings Acquire Knowledge? Inductive Reasoning
How do human beings DO this learning? How does the learning mechanism find things, kinds of things, and interconnections among kinds of things IN the kaleidoscope of sensations?
It performs a set of logical operations---called inductive reasoning---ON the sensations, until what emerges is generalizations.
“All these lines and colors are (to be called) triangles.”
“All these trees are to be understood as a forest system.”
“The fact that the prices of all these goods increased when persons wanted to buy more and more of these goods, is to be seen as a causal relationship, not just a bunch of coincidental happenings.”
Interestingly, the methods of reasoning that scientists use to draw conclusions from data, are exactly the methods of reasoning that everyone uses to draw conclusions from data. The methods are called agreement, difference, joint method of agreement and difference, concomitant variation, and residues.
These methods were described by John Stuart Mill in A system of logic (1843). All quotations are from Mill.
Method of agreement
"If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon."
Symbolically, the method of agreement can be represented as:
When events A B C D go with or are followed by events L M N O, and
When events A E F G go with or are followed by events L S T U, and
When events A Q V E go with or are followed by events L H I K….
The learning mechanism concludes that A and L are connected. For example, “A must be the definition of concept L (concept).” or “A must be the cause of L (rule).” The instances agree in one way. This method of inductive reasoning is the method of agreement.
Note: the more examples, the more confidently the learning mechanism can draw the conclusion. Why? Because 10 examples with A and W always together is pretty unlikely unless they are connected.
Design instruction that enables students to use the method of agreement to draw a conclusion from examples.
1. Make sure examples are DIFFERENT in irrelevant features, but are the SAME (they agree) in the relevant feature.
For example, show 4 (or more) examples of linear functions. Each example differs in the angle (A, B, C, D) and the scale of values (10’s, 100’s, 1000’s) on the X and Y axes (E, F, G, H), but each line is straight (L).
The learning mechanism says, “Therefore, L is what makes it linear; A-H are irrelevant features.”
More examples of how the teacher presents examples as above, and the learning mechanism uses the method of agreement (comparing examples to identity how certain features that are the same go with the same treatment.
Teacher: “This (red circle) is called roja.” “This (red square) is called roja.” “This (red apple) is called roja.”
Roja is L, and the color red IN the examples is A.
Teacher: “This chunk of granite consists of mica, feldspar, and quartz. And THIS chunk of granite consists of mica, feldspar, and quartz. And notice that THIS chunk of granite also consists of mica, feldspar, and quartz. So, what do you think is the definition of the concept, granite?”
Method of difference
“If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.”
When events A B C D go with or are followed by events L M N O, and
When events B C D go with or are followed by events M N O…
The instances DISAGREE in one way. When A is there, L is there. When A isn’t there, L isn’t there.
Therefore A is the cause, or the effect, or a part of the cause of L (why events are named L or why events are solved with L).
This is the design of an experiment. The experimental and control groups are the same on the variables B, C, and D (for example, age, sex, pre-test scores) but differ in A (the intervention to increase scores). And the group with A (the experimental group) ends up with higher post test scores---W. So, A (the intervention) must be the feature of the group that makes the difference in achievement.
Design instruction that enables students to use the method of difference to draw a conclusion from examples.
Present events in this way (above and below) to show students the difference that makes the difference in whether something IS an example of a concept, rule, or routine.
“This (red circle) is called roja. This (blue circle) is called not roja.”
Or, “This is granite. It consists of mica, feldspar, and quartz. But THIS is NOT granite. It consists only of quartz. So, why is this one NOT granite?”
Or, “This graph shows the rule that when demand increases, price increases. This other graph does NOT show the rule that when demand increases, price increases.”
Or, “We read these REGULAR words (sad, ten) by sounding them out. We say the first sound, then the next sound, and then the last sound. sssaaad, teeehhhnn. And that is how we SAY the word---sad, ten. But we do NOT SAY these IRREGULAR WORDS (said, the) the way we sound them out. We sound them out---sahiiid, t/heeh---and then we SAY them THIS way---sehd, the.”
This technique is called “juxtaposing examples and nonexamples” to reveal difference.
Make sure examples are the SAME in irrelevant features, but are DIFFERENT (they DISAGREE) in the relevant feature. For instance,
Rome, 1st century, before 31 BC. Spoke Latin. Agricultural society. Representatives of the people---senate, tribunes. “This is a republic.”
Rome, 31 BC. Spoke Latin. Agricultural society. One ruler---emperor. “This is not a republic.”
The learning mechanism says, “Well, the one difference is representative (dispersed) rule vs. concentrated rule in one person, AND the first is called republic and the second is called not republic. So the concept---republic---must be defined by representative (dispersed) rule.”
Another----juxtapose a plot in which the line is straight (“This is linear.”) and a plot that is the SAME in the values of variables on the axes and in the direction of the line (up, for example), but the line is CURVED.
The learning mechanism says, “Linear CAN’T be the way the axes are labeled (by 10s), because they are the same, but one plot is called linear and the other not linear. So linear must be the ONE difference in the plots that goes along with calling one linear and the other not linear. It was called linear when the line was straight and not linear when it was curved (not straight). So, linear must mean straight.”
Joint method of agreement and difference
"If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance: the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon."
This method combines the methods of agreement and difference.
Symbolically, the Joint method of agreement and difference looks like this.
First set of juxtaposed examples that agree in one way.
A B C D occur together with x y z
E F G D occur together with s n z
Another set of juxtaposed examples that agree in one way.
L M N D occur together with w t z
I O H D occur together with s v z
So, by the method of agreement, I hypothesize that D goes with---is the definition of, is the way to solve---z.]
Now a set of juxtaposed examples that DIFFER in one way.
A E D occur together with x w z
A E occur together with x w
Another set of juxtaposed examples that DIFFER in one way.
F G D occur together with t u z
F G occur together with t u
The learning mechanism uses the method of agreement to conclude that D goes with z. Then it CONFIRMS that conclusion by applying the method of difference to the second two examples, to conclude that D goes with z]
Design instruction that enables students to use the joint method of agreement and difference to draw a conclusion from examples. For instance….
Method of agreement portion.
Show examples of republics (Rome, Switzerland, United States), and label them as such. Examples differ in size of country, time in history, and language, but are the same in a government in which representatives are elected and make decisions for the whole.
The learning mechanism concludes that the ways the examples differ can’t be what defines republic. It MUST be the way the examples are the same---representation.
Method of difference portion.
Now JUXTAPOSE an example of republic (Rome as a republic---“This is a republic.”) with Rome ruled by an emperor---“This is not a republic.”
The learning mechanism concludes that republic must be the feature whose presence or absence makes the difference in how the example is treated (named). “Republic must be the feature of representation.”
Method of Residues
"Deduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents."
If we know that certain factors go with one another, and we have matched them, and we find that one “effect” remains, then some other one factor that remains must be the cause (or the effect) of the other one that remains.
“It can’t be Professor Plum with a lead pipe. He was sick in bed, and the lead pipe was locked away.”
“It can’t be Colonel Mustard. He was dead drunk.”
“It can’t be Miss Primrose. She was busy dusting the ceiling.”
“The only remaining possibility is Miss Scarlet with a candlestick. Yup. It’s always the red heads!”
Symbolically, the Method of Residue can be represented as:
When A B C D happen, F G H I happen.
We know that G is caused by B.
When know that H is caused by C.
When know that I is caused by D.
So, F must be caused by A.
Present a situation in which several variables could have caused some other event. See if you can eliminate these causes until only one is left.
“So, what was the cause of the beginning of the decline of the Roman Empire? Various explanations have been given. (1) Overwhelmed by barbarians. (2) Weakened by the pacific religion of Christianity. (3) Overextended the empire and exhausted resources. (4) Replacement of republican government by monarchy (emperors). (5) Loss of martial spirit and patriotism. (6) Slavery. Let’s look at each of these to see when they might have affected the strength of the empire and how much of an effect they could have had on its decline… Well, it couldn’t be the barbarians; they invaded later….”
I have just given you the intellectual equipment to design instruction that will teach anything and everything---because everything COMES as an example of one of the above arrangements. Either…
a. All events are alike (agree) in some way.
b. There are some differences among events that make a difference in how they are to be treated (named, solved).
c. Two things change together.
d. It’s possible to eliminate everything but one variable, so that one must be the important one.
Method of concomitant variation
"Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation."
Symbolically, the method of concomitant variation can be represented as (with ± representing an increase or decrease):
When A B C D happen, F G H I happen
When A (+/-) B C D happen, F (+/-) G H I happen [+/- means increases or decreases]
[Let’s say A and F change in the same direction.]
Or A increases and F always decreases. [They change in the opposite direction.]
Since the only things that change together are A and F, the learning mechanism reasons that A and F must be connected in some way (perhaps a change in A causes a change in F). This is called the method of concomitant (both together) variation.
Therefore A and x are causally connected.
The more a government uses threats and punishment [jail, fines, SWAT teams breaking into homes without a search warrant, confiscation of property, violating the Constitution, recording internet and cell phone activity, police road block identification checks) to force compliance from citizens, the more the government comes to be seen as illegitimate by citizens, who form opposition groups and resist the government. [Peter Blau. 1965. Exchange and Power in Social Life. Also, the historical basis for the American Revolution.]
The less you exercise and the more carbs you eat, the more weight you will gain, the more likely you are to get diabetes, and the lower is your life expectancy. So, eat nothing and run all day.
Notice how these variables vary sometimes in the same and sometimes opposite directions.
Present examples in which everything else (as in B C D and G H I) above, is the same, but only two things vary (A and F).