You often hear, if you manage to stay awake during the lectures, a mathematician or physicist say, “The following is a necessary but not sufficient condition for my theory to be true.”
We say this so often that we tend to blend the words together: necessarybutnotsufficient, and we forget that it can be a confusing concept.
It means that there is an item or a list of items that must be the case in order for my theory to be true. But just because that item or those items on that list are true, it does not mean that my theory must be true. It could be the case that the item is true but my theory is false.
This is all important in the theory/model building that goes on in the sciences.
For example, it is necessary but not sufficient that my theory be able to explain already observed data that I have collected. If I cannot at least explain that data, then my theory cannot be true. This is necessarybutnotsufficient in the weak sense: all theories must be able to explain their already-observed data.
Again, it is a necessary but not sufficient condition that my theory be able to explain future data. This is necessarybutnotsufficient in the strong sense: if my theory is right, it must be able to explain data that is not yet seen.
Understand: it can still be the case that my theory can predict data that is not yet seen and my theory could be false. This is true in all cases where we cannot deduce (know with certainty) the true of a theory. Most theories (outside math) are not, of course, deduced.
How about an example? Let’s us the following game.
The game tests your inductive reasoning skills and asks you to infer the rule that accepts or rejects cards from a standard 52-card deck.
Let me be as clear as possible. The following conditions hold: (1) You believe that a rule that generates the card exists. (2) You will see a sequence of cards from which you will attempt to infer, through induction, the rule. (3) No matter how many cards are shown you will never know the rule with certainty; that is, you will never be able to deduce the rule from a set of premises.
Do not read further until you have played the game fully and discovered its secret.
Did you really play?
Tell the truth. Don’t cheat and read any more until you have played the game.
Their answer: “The rule was that cards which are not prime-numbers (i.e., any card apart from 2, 3, 5, 7, 11/Jack and 13/King), formed part of the sequence.” Notice that they do not define 1 as prime. In other words: “reject all prime-numbered cards.”
Use the shorthand that D = Diamond, H = Heart, C = Club, and S = Spade; J = Jack, Q = Queen, K = King, and A = Ace. So that KS = King of Spades, etc.
The sequence of cards that was presented was: QH, AH, 6H, AD, 7D, 3S, QD, 9C, 3C, 10D, 9D, 7S, 3D, 7H, QS, KC, 9C, 2D, 6H, JD, where the italicized cards were not accepted.
If, as is usual in many games, a face card (the Jack, Queen, or King) is worth 10 and and Ace worth 1 or 11 (choose 1 for ease; but either works), then there is at least one other rule that also generates that same sequence. (I’ll leave it to you guys to suggest more rules.)
A Different Answer
My rule is: Take the new card and add it to the last accepted card. If the total is odd, then accept the current card. If the total is even, then reject it. Also reject all 2s.
Start with QH, as provided. My first new card gave QH + AH = 11, which is odd, so accept the AH. The next is AH + 6H = 7, which is odd, so accept. The first reject is AD + 7D = 8, which is even, so reject.
The next offered card was 3S, and the last accepted was AD, so AD + 3S = 4, which is even, so reject. The third-from-last card was 2D, and 9C + 2D = 11, which I would normally accept because the total is odd, but since the new card is a 2, I reject it.
I have met the necessary but not sufficient condition for my theory/rule to be correct: my rule generates the observed series of data without error. I explain the already-observed data. I therefore have some confidence that mine is the correct rule/theory.
But suppose as you were playing the game, you surmised the rule: “Reject all prime-numbered cards.” You also have some confidence that yours is the correct rule/theory. You also explain the already-observed data.
The Old, Old Story
Now we have to go a step further, so pay attention. As we assumed, those cards, and the order in which we see them, represent a real physical or mathematical process. We now know that mechanism because we were told, but this example is highly artificial. In real life, unless we are in the rare cases where we can logically deduce a rule/theory, we have to guess. Anyway, let’s forget we were told the rule.
We have two proposed laws, yours and mine. Both could be right, and both explain the observed sequence so far. Neither of us knows with certainty we are right. You and I can each write a peer-reviewed article that says, “My theory produces the data exactly. Therefore, I am likely to be right.”
We could each submit grants asking for a staff to research the mysteries behind our theories. I would ask, “Why are 2s excluded? Could it have to do with quantum entanglement?” You would say, “Nature abhors prime numbered playing cards.” And off we would go.
To learn anything more, we both have to wait for new cards to show up.
Suppose that the cards only come to us slowly; perhaps we only see one new one per year. So we wait a year and the Ace of Spades shows up and is accepted. Both of our theories predicted this new data point: both have met the necessary but not sufficient condition of predicting new data. Both of us, and our followers, become more entrenched. We begin waiting for next year.
But year brings no comfort because last year was an Ace, so no matter what card shows up next, both of our theories will agree on whatever card comes up (except another Ace [thanks John]; prove this). Suppose that card is 8C. Another year must now go by.
Finally, one more year later, the 8D shows, which I did not predict and you did. I have finally been proven wrong. I have met the necessary and sufficient condition to prove my theory wrong. And it took a long, long time. However, even though I am wrong does not mean you are right with certainty. There still might be another theory that is better than yours.
On the positive side, you never had to modify your theory to account for new data. Your predictions stay the same regardless of what new data comes in. Your case is very strong.
What about me? Do I give up on my theory? Well, 8 is 2^3, and I’m already rejecting 2s. So maybe I would look at why powers of 2 should also be rejected. Not all the time, because I did accept that 8 of Clubs. Maybe it’s red 8s that are the trouble! Or I could posit that my theory is true up to measurement error (maybe I can’t see the cards clearly). I will modify my theory, keep my research staff, and prove, eventually, that I am right! Further, any modification means I have created a new theory!
However, I will certainly lose supporters, as I should. As more cards come in (any pair of even or odd numbered cards will be rejected; prove that), I will look more and more foolish if I cling to my original belief.
That’s a long story, and I’m sorry about that, but it’s not an atypical one. It shows once again the eternal wisdom that a theory is only good if it can predict data that was not, in any way, used to build the theory.