Necessity and Sufficiency

Even the most fervent critic of metaphysics must have pondered from time to time: what is the meaning of my existence to this world?

Feeling hopeless about her prospects in grades-obsessed South Korea on the day of the college entrance examination, mathematically challenged highschooler Jang Dan-bi jumps into a rain puddle transporting her to a drought-stricken Joseon, where Sejong the Great (King Sejong) and his ministers are praying for a timely, much-needed rain, also called “danbi” in Korean. Now in the 15th century, when modern multiplication tables are unheard of, her mediocre mathematics skills take on heightened importance as she teaches mathematics enthusiast King Sejong rudimentary arithmetic and science. Along the way, she befriends a man she identifies as Jang Yeong-sil, an actual historical figure credited for the invention of multiple meteorological and astronomical devices, and inadvertently takes the audience on a whirlwind tour of Joseon technological advances.

But the message Splish Splash Love conveyed to high school seniors, who received their examination results the week before the first broadcast, went deeper than that. Connecting King Sejong’s creation of the Korean alphabet to the way democratization of knowledge (and perhaps, technology) reduces the demand for qualified professionals, the 140-minute long mini-drama makes the point that human worth should not be tied to the ever-shifting concept of usefulness. There are even times when one has to be unafraid of turning himself redundant so that those he loves can stand on their own feet. The icing on the cake is King Sejong’s eventual romantic confession to Dan-bi that while he initially saw her as a talent the country needed, he now wants to hold on to her,

“Not because you are necessary, but because you are sufficient.

It is sufficient to have you by my side.”

Necessity and sufficiency are also important concepts in the field of logic. Saying that a condition X is necessary for a condition Y means that X must be true for Y to be true. This seems to be a very elementary concept but it is worth sounding a note of caution: X may not be sufficient for Y to be true; perhaps other conditions are essential too. Having a time portal, for instance, is necessary for time travel, yet it cannot make time travel happen by itself because you need someone who will take the trip.

Meanwhile, saying that a condition X is sufficient for a condition Y means that as long as X is true, Y is true, regardless of other conditions. Again, one has to be careful here: X may not be necessary for Y since there may be other conditions that can make Y true by themselves. Hence, that X is false does not imply Y is false. For example, a rain puddle may suffice as a time travel device, but it is not strictly necessary in a script as the writer can choose an automobile (Back to the Future), a cosmic treadmill (the DC Comics universe), an incense stick (Nine: Nine Time Travels), etc. instead. The limits to the power of human imagination are hard to discern.

Some pointers apply to both necessity and sufficiency. First, neither relationship always implies causality. The conditions may have co-existed as a result of sheer coincidence or a third condition. If Dan-bi could choose her time destination, it would be sufficient for her to pick a date between 1397 to 1450, the years King Sejong lived, to have a chance of meeting the greatest musician in Joseon, Park Yeon, who lived from 1378 to 1458. Yet this is certainly not because either man caused the other’s existence!

Second, notions of necessity and sufficiency often rely on background assumptions. The validity and stability of each relationship therefore depend on those of these assumptions. Going back to the first example, in saying that time portals are necessary for time travel, one is subconsciously conceptualizing time travel as sudden leaps through time while the traveler is awake. If, however, one were to abandon linguistic conventions, which are subject to change anyway, and think of any experience of passing through time as “time travel,” mere existence of the self would suffice for the activity. A more general assumption is that the conditions in each relationship do not participate in feedback mechanisms altering the relationship. Yet, if a sufficient but unnecessary object has an addictive property, it will become necessary to its user. And if we define sufficiency in terms of quantity, a quantity of the addictive object sufficient for a first-time user will turn insufficient as the addiction takes its toll.

Note that necessity and sufficiency are converse relations. This means that if X is necessary for Y, Y is sufficient for X, and vice versa. If a condition must be true for another condition to be true (necessity), knowing that that other condition is true is enough to know that the first condition is true (sufficiency). If a condition is adequate to make another condition true (sufficiency), the second condition must be true when the first condition is true (necessity). Hence, since Dan-bi’s presence by his side is presumably sufficient to make King Sejong happy, which implies that there is no way he will be miserable by her side, King Sejong must be happy for an outsider privy only to his mood, for example, to speculate about any possibility of him and Dan-bi being together at the particular time.

Confusion of logical necessity/sufficiency with causal necessity/sufficiency would have hampered understanding of the converse relations. While logical necessity/sufficiency does not preclude causal necessity/sufficiency, converse relations do not make any causal connection between a pair of conditions bidirectional by default. A water clock may be a sufficient cause for telling time, but the ability to tell time would be a logically necessary consequence of having a water clock, not the (necessary) cause of the clock.

In decision-making matrices, however, causation is indispensable. Decisions are made precisely because of their agency in creating possible impact. Although an individual’s worth should be independent of his utility value, the world probably benefits most from people who fulfill needs adequately, then make these targets achievable even without them and move on to fulfill other needs, just as Dan-bi returns to her 21-st century mother after imparting her knowledge to the king.

No able-bodied individual is necessarily excluded from opportunities to create such value because of his current limitations. To this end, Dan-bi’s adventure arguably symbolizes that great achievements often have humble beginnings. An inept young learner today may yet make great strides in the long journey ahead if he does not give up. During her stay in Joseon, Dan-bi was forced to construct Jang Yeong-sil’s mathematically complicated water clock from scratch to dissuade ruthless ministers from executing her for supposed treason. She also unintentionally introduced his rain gauge and a modern-day banknote showing his design for a celestial globe to King Sejong, who subsequently ordered someone to build the globe. Curiously, too, just before she left  Joseon, she learnt that the man she took for Jang Yeong-sil, and whom King Sejong tasked with the construction of the globe, was actually Park Yeon. On the other hand, deeply impressed with her intellect and craftsmanship, King Sejong credited her for being his trusty aide and inventor in history but never seemed to know of her given name. Opening a scroll from him after her examination, she finds the first words he wrote with his alphabet: her surname and the given name he bestowed on her, “Yeong-sil.”

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