The Good Teacher

Chapter 26 Inductive Fallacy

Modern scientific inquiry is based on two schools of logic: deductive and inductive reasoning. Nearly all theories and models developed through the scientific process are based on these two lines of inquiry.

To define them using a simple analogy, consider a pyramid.

Deductive reasoning can be equated to a person moving from the bottom of the pyramid to its tip. It is commonly observed in mathematics and other systems built on axioms. The goal is to determine the nature of a specific case given a general conclusion.

For example, let us consider a situation where we want to determine whether a horse is a mammal. We start with a general statement that can properly describe a mammal, such as its ability to give birth to live young. We then observe that a horse can birth live foals. Therefore by comparison we can conclude that a horse is a mammal.

While this method is straightforward and effective, there are some glaring faults that can topple the integrity of the process. First off, deductive reasoning relies on a robust general conclusion. Without the major premise, it is impossible to draw a comparison with the specific case.

Secondly, if the major premise used as the basis of comparison is not properly defined, then we experience something known as a deductive fallacy. Reconsidering the above horse example, while the fact that "all mammals give birth to live young" is true, it is not holistic. For instance, platypus and echidnas are mammals that lay eggs. Evidently, these are isolated cases, but the fact remains that unless defined during the deductive process, we open ourselves to a false conclusion. Because of that, there needs to be greater rigour when picking a general conclusion prior to deductive reasoning.

However, in nearly all other cases, scientists resolve to use inductive reasoning. This is akin to starting from the top of the pyramid and moving downwards. We begin with a specific case and work towards building a general conclusion through controlled experimentation and surveys.

While this process is more measured and rigorous, it is prone to a common logical fallacy known as the inductive fallacy. In simple terms, it states that to disprove a general conclusion, all it takes is a single exception to the statement. For example, the statement that "all swans are white" can be easily disproven by finding a single black swan.

And this problem needs to be considered when drawing sweeping conclusions through inductive reasoning.

As a teacher, Guy was intimate with these fallacies and always made it a point to enlighten his students - in a way that was easy to comprehend - to be aware of them in their daily lives. His goal was to not just inculcate knowledge in his students but to impress upon them the skills and tools needed to become effective learners. This was so that when they step foot into the jungle-like modern world, they would be more independent in advancing themselves, and would be more critical in absorbing information from different sources.

Ironically, Guy himself fell into the trap of forming a false generalisation based on insufficient evidence. When he first developed this new modular spellmaking method, he found that Markus was able to cast them without a hitch. And so, Guy figured that as long as the spells pertained to the basic elements, Markus would be able to cast all of them similarly.

The mistake he made was that he didn't test his hypothesis with all the variations of spells that could be developed using this spellmaking method. As a result, there was a subset of spells that were beyond Markus' reach.

Therefore, after having a filling lunch, Guy sat down in the library and approached the issue with a fresh frame of mind.

Following inductive reasoning, Guy had to determine the cause of the exception. To do this, he needed to first find out the commonality between the spells. If he could define the similarity, then he could work towards alleviating the problem.

As he stared at the spells constructs in the sheets before him, he started to note down the characteristics that could be gleaned through a cursory glance. It was likely that the answer was staring him right in the face, just that he was too involved to notice it.

Guy realised that element-wise, the spells were diverse. There was an equal mix from each basic element that Guy had set out to cover. After eliminating the element of the spell as the cause of the problem, Guy went a level deeper and evaluated the number of modular components composing the spell.

He couldn't find a pattern there either. Some spells were concise and contained up to four components, while others were more intricate with around eight or ten components. However, in the process of investigating this avenue, Guy realised that although he built the new method on the concepts of programming, the former was much smarter as it performed a lot of garbage collection automatically.

Usually in programming, when feeding a variable into a function, unless specified, the program copies the variable and runs the function on that copy. This can very quickly lead to the system running out of memory. The modular spell, however, does not create unnecessary copies and is smarter about how it handles variables fed from Source Components.

But Guy didn't go further along this path and shelved it for later because it was a segue from his current predicament.

After testing out other hypotheses, Guy quickly realised that the issue might not be in the technicalities of the spell, or the way it was put together.

Taking a different approach, Guy started to physically cast the spells in an open field outside the library.

After cementing an efficient method to develop spells modularly, Guy had kind of gone off the rails and put together a ton of spells, most of which were similar in nature, but were sufficiently distinct to be considered novel.

For instance, he took the |Fire Ball| spell and starting to play around with the Action Components to change its shape, developing the |Fire Torus|, |Flame Disk|, |Fire Arrow| and so on. Although they had a different visual, 90% of the components used in the spell were the exact same. He only fiddled with the components that defined the shape of the spell.

With that, he started to cast each spell sequentially.

The first spell produced a burgeoning fireball in his hand and propelled itself in the direction his palm was facing. The following spells were similar, except it wasn't a fireball anymore and the shape of the flaming projectile varied.

The second set of spells produced a ball of whirling wind in his palm, which then propelled itself. The following spells were also wind projectiles, except with different shapes.

The third set was similar, except the element was earth. Since earth was a generally static element, Guy couldn't produce many variations in its offensive spells. However, there were a lot of defensive solutions such as a spell that made a set of levitating condensed earth disk that could intercept opposing projectiles or a spell that formed sharp projectiles out of sand particles that could penetrate through the application of friction.

The final set comprised of water spells that worked both offensively and defensively such as |Water Jet| or |Water Shackles|.

As Guy progressed through the stack of spells, he finally figured out the reason why Markus failed to activate these spells.

"They're all range based!"

As defined previously, every mage has a mana domain that demarcates a region of space centred around them, which the mage has full control over. As the mage advances in their cultivation realm, this domain expands proportionally. Since Markus was only at the Early Stage of the Mana Condensation realm, his mana domain was a measly two metres.

As long as the spell's effect remained within this two metres limit, Markus didn't have to expend excess mana to sustain the spell. But as soon as the spell is required to leave this domain, additional mana is required to maintain the integrity of the spell.

This additional expenditure follows a square relationship. That is, the amount of mana is proportional to the square of the extra distance beyond the mage's mana domain. This is because the mana domain forms a sphere, whose surface area is a function of the square of its radius. And when the spell exceeds the mana domain, the mage is forced to control the spell by establishing a pseudo mana domain that has a larger radius.

Guy noted that this correction was another one of those automatic fixes implemented to ensure that the spell activated.

'This world seems to provide a lot of assistance to mages in its own way.'

He figured that given more time, he could optimise the modular spellmaking method, and streamline the mana use so that instead of making a pseudo mana domain that is a complete sphere, it only forms a spherical sector. But considering the time-cost-to-mana-performance relationship, Guy concluded that it would be more efficient to seek out alternate solutions.

Ultimately, it was predictable that Markus was unable of activating these ranged spells because he was physically incapable of doing so.

So Guy was now faced with a new problem, and that was to determine a solution to facilitate these ranged spells without the need for expending an astronomical amount of mana. To do this, he decided to seek assistance from a veritable information powerhouse, Al Jeeves.