Mill’s Inductive Methods - Amateur Logician

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Mill’s Inductive Methods

**John Stuart Mill**
https://en.wikipedia.org/wiki/John_Stuart_Mill

I. Method of Agreement
II. Method of Difference
III. Joint Method of Agreement & Difference
IV. Method of Residues
V. Method of Concomitant Variation

It’s not as if your “traditional” neo-Scholastic textbook on logic doesn’t cover modern developments! Whatever the flaws of the philosopher John Stuart Mill (1806 – 1873), he surely did make important contributions to logic.

(Mill also, alas, engaged in great errors about deduction.)

Seeking a Cause. . .

Chapter eight in Mill’s A System of Logic is “Of the Four Methods of Experimental Inquiry.” It might now be the book’s most famous chapter. There Mill explored ways in which we can discover causes.

No logician today would claim Mill’s methods are “perfect” or “exhaustive.” These methods can only yield probable, not certain results. Yet, they are helpful!

A cause brings something about.
– It can be thought of as the “independent variable.”
An effect is that which is brought about.
– It can be thought of as the “dependent variable.”

Generally, we start from observed effects and attempt to trace those effects back to its cause or causes. However, scientists often will predict certain effects from a cause. That prediction is based on a hypothesis. And, if that prediction turned out to be correct, it lends support (not absolute proof) to that hypothesis. This is the “scientific method.”

Causes may be more-or-less remote. A cause can be proximate or ultimate. There are often causal chains. A bowling ball knocks down a pin, but the bowling ball caused that effect by you throwing the ball. Note that philosophy is about “ultimate” causes. Physics is about “secondary” or “proximate” causes.

Philosophy and physics do not study the same thing or in the same way! For example, God may be the “ultimate” cause of gravity being what it is, but gravity (or better put, the mass of material objects) is a “proximate” or “secondary” cause of somebody falling down to the ground after jumping into the air.

There are huge metaphysical implications in unpacking what all of this means and entails. What exactly is causality? What’s the difference between a pre-condition, a principle, causal activity, and an actual causer?

And a cause (or condition) may be sufficient or necessary. A sufficient cause will produce its effects. If the sufficient cause is produced, then the effects will be produced. A necessary cause is that which when it is absent an effect will not be produced. If the necessary cause is absent, then the effect will be absent. In addition, a cause sometimes can be both sufficient and necessary.

Recall a conditional’s distinction between sufficient vs. necessary conditions!
“If it is raining, then it is cloudy.”
— Clouds are necessary for it to be raining. No clouds, then no rain.
— Raining is sufficient for it to be cloudy. No rain doesn’t mean it’s not cloudy.

Let‘s get to Mill. . .!

I. Method of Agreement

J. S. Mill writes: “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” (p. 454).

With the “Method of Agreement,” we may be led to discover a single factor that’s present among a number of separate circumstances, where all of those separate circumstances lead to a particular phenomenon. This single factor, if found, will be a necessary condition.

Example. . .

Let’s pretend you have an allergy. You show the symptoms of sneezing, having swollen eyes, and swollen lips. Your diet is radically simple. We record what you ate and didn’t. For each occurrence when a certain food was absent, we disregard that food as a necessary condition. Hence, e.g., occurrence “Number 1” did not have green beans, but, because the phenomenon occurred, green beans cannot be a necessary condition.

Looking through the data, eggs are present in each case.
Hence, eggs are a probable cause of the apparent allergy.

Recall that Q is necessary for P when if P is present, then Q is present.
Think of it this way: “If the phenomenon occurs, then there are eggs.”

As far as the “Method of Agreement” goes, it’s not bad as a way to engage in inductive reasoning. It just has to be used with great care! Maybe there is another factor that you somehow neglected, yet was present in each occurrence? (Perhaps each time you ate near a pet, for example. Maybe the pet is causing the phenomenon, not the eating?)

Even to construct that table requires an intuition about what to record and not to record. It takes some understanding of what can be causally relevant and what cannot be. There might as well be, theoretically, an infinite number of possible necessary conditions. It would be impossible, in practice, to eliminate all but one. Moreover, some factors might be always associated with the phenomenon but not be causally related. A relationship between things doesn’t automatically imply that one thing must be the cause of the other.

II. Method of Difference

J. S. Mill writes: “If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance save one in common, that one occurring only in the former; the circumstance in which alone the two instances differ, is the effect, or cause, of a necessary part of the cause, of the phenomenon” (p. 455).

Mill notes that these methods are based on elimination. In the “Method of Agreement” we disregard circumstances to finally arrive at the circumstance that suggests a possible answer. The same thing, basically, happens with the “Method of Difference.”

Presently, we’re looking for a sufficient condition. We compare two occurrences. Look for what they don’t have in common. Each occurrence has almost everything in common. The goal is to find a condition where (1) its presence is associated with the phenomenon and (2) its absence is associated with the absence of the phenomenon.

Example. . .

Let’s pretend we are making a dish together. It’s a dessert to satisfy our sweet tooths! We made the dish two times. The first time it had sugar but the second time (due to my stupidity?) it had no sugar. A condition is not sufficient for the phenomenon of being “sweet” if it is present when that phenomenon is absent.

Recall that P is sufficient for Q when if P is present, then Q is present.
So, if it happens that P is present and Q is not, P cannot be sufficient for Q.

Considering what it takes for the dish to be “sweet,” we must disregard each food as a sufficient condition if it is present when that phenomenon is absent. Hence, occurrence “Number 2” reveals to us that we should disregard eggs, vanilla extract, peanut butter, and nuts.

Looking through the data, only sugar’s absence makes it not sweet.
Hence, sugar is a probable cause for it being sweet.

This example is rather simplistic.
We should all admit that a more complicated situation like this could be of great aid to a chef!

Still, this method has to be used with great care! It is based on the different occurrences being identical save one factor. That’s impossible in practice. The trick is to make sure any differences are insignificant, though determining that is a matter of intuition. While this method searches for the presence of something to be a cause, it may also be that it is the absence of something that is relevant.

III. Joint Method of Agreement & Difference

J. S. Mill writes: “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” (p. 463).

Combining the two previous methods results in attempts to find a condition as both necessary and sufficient. We can start with the “Method of Agreement” to find a probable cause. To increase its likelihood to be a cause, we can then try the “Method of Difference.”

We’ll (1) find out if there is a single condition present in at least two occurrences when the phenomenon is present and (2) find out if that single condition is absent in at least two occurrences when the phenomenon is absent.

Example. . .

This time, let’s return to the first example.
What can we do to increase the odds that eggs are the culprit?

The “Method of Agreement” suggests that eggs are the culprit.
To increase the odds, let’s now apply the “Method of Difference.”

**Joint Method of Agreement and Difference Example**

Now, we have increased the odds!
You ran an experiment. Each occurrence was re-done but without eggs present.

(Note that when you see an apostrophe after a number, you read it as “prime.”)

Compare the difference between “Number 1” and “Number 1 Prime.”
Likewise, with “Number 2” and “Number 2 Prime.” Etc.
Hence, eggs are a probable cause of the apparent allergy.

To be sure, you could have run the experiment using the Method of Agreement with a greater number of occurrences than four. Maybe you ran it with a hundred occurrences. That would have increased the odds (if it, e.g., likewise suggested eggs as the cause).

IV. Method of Residues

J. S. Mill writes: “Subduct [i.e., subtract] 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” (p. 465).

There are knowns and an unknown. The “Method of Residues” seeks the unknown. A range of effects is produced by several causes collectively. We know those several causes. All of those causes, save one, we know their associated effects. The remaining effect, not accounted by those known causes, must thereby be caused by the remaining cause.

We might have three causes working together – A, B, and C – and we have three effects – a, b, and c. We know that A causes the effect a and that B causes the effect b. Given all of that, since ABC causes the effect abc, it is probably the case that C causes c.

Example. . .

A lot of logic textbooks give mathematical examples.
If (1) a+b+c = net effect; and (2) we know net effect, a, and b; then (3) we can solve for c.

Pretend we have an 80,000 pound loaded truck. When that same truck is unloaded it weighs 35,000. It’s carrying goods to Walmart. How much do those goods weigh?

Simply, we have an algebra problem.
35,000 + x = 80,000.
x = 80,000 – 35,000
x = 45,000

45,000 pounds is the “residue.” It’s the weight of those goods.

Other Examples. . .

Another example might concern a utility bill. We figure out what contributed to that bill to then estimate one factor’s contribution.

A far more interesting example concerns the planet Neptune. Its existence was mathematically predicted by “residuary” phenomena from the motions of Uranus.

V. Method of Concomitant Variation

J. S. Mill writes: “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” (p. 470).

A changing value of X may be followed by a changing value of Y. So, it might be that X is a cause and Y its effect. That cause is the “independent variable” that results in change in the “dependent variable.” Something increases in strength, for example, and another thing, as a result, increases in strength. Or, maybe, as something increases in strength, another thing, as a result, decreases in strength.

“Directly Proportional” ≈ as one amount increases, another amount increases
“Inversely Proportional” ≈ as one amount decreases, another amount increases

We reside in a material world, which is accordingly highly quantitative.

Mathematical modeling becomes important.

Example. . .

Blaise Pascal discovered that the mercury in a barometer lowers as the altitude increases. Hence, there is a “concomitant variation.” It turns out that atmospheric pressure is decreased at higher altitudes, and so mercury doesn’t rise as high as it would otherwise.

Another Example. . . Considering food and drinks, increasing or decreasing particular foods or drinks may increase or decrease certain conditions. How much coffee does it take to make you jittery, for example?

Another good example is . . .Ohm’s Law!

(The details of physics are not our present concern. The data gathering is. Here we won’t define “voltage,” “current,” “resistance,” etc.)

Go to the PhET simulation environment.
We can attempt to derive this law.
– Resistance will be kept at a constant 10 omega.
– Voltage will will vary (as an independent variable – which we will control).
– This will help us see what happens to current (as a dependent variable).

**https://phet.colorado.edu/en/simulations/circuit-construction-kit-dc**

I’ve collected the following. . .
Voltage was changed. This resulted in a different current.
Resistance was held constant.

Look closely. There is a relation! A graph is helpful. . .

**Graph of Data**
Vertical Axis represents **Current**
Horizontal Axis represents **Voltage**

Voltage is proportional, by a multiple constant deriving from resistance, which is held constant, to current. In our data set: V=10*I, where V is for voltage, the 10 is the constant resistance of ten ohms, and I is current.

We can do the same “experiment” with a different amount of resistance to help verify that it acts as the “slope” in the above graph. It’s a linear line.

So, we have good reason to state this equation: V = I*R
Voltage is proportional to the product of current and resistance.

*A First Course in Mathematical Modeling*

Mathematical modeling, of course, is an enormous subject!

A dynamic system is anything that changes in time that is to be modeled mathematically. Sometimes we model such systems with a difference equation when the change in time is discrete rather than continuous. However, if the change in time is continuous, we ideally can use a differential equation.