Necessary but not sufficient


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

Go to The Philosopher’s Mag and play this game called “Dealing with Induction”. It asks the question: “how easy is it to draw a wrong conclusion about the future from the evidence of the past?”

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.

The Answer

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.


  1. Damn, never got to the rationale behind it. I ridiculously thought that it had a sequence of some sorts, like “this card + a geometric progression,” etc. But then a funny thing happened, a spurious coincidence, that illuded me, and I got several cards right, the “aha” moment, though the rule was kinda awkward.

    And then, puff, wrong, wrong.

    It was fun, but it’s like going to the wrestling arena knowing a priori you’re going to get beaten. As much fun you have giving punches, there’s always the depression with the thought that you’re going KO no matter what.

  2. Nice one! The classical case is Newtonian Mechanics versus General Relativity. Computation in GR greatly exceeds what’s needed for NM. However it’s only in cases at extreme velocities and masses that the extra work pays off. NM remains in use even though it has been falsified.

    And we should induce that GR in its turn will succumb to a preferable theory…

  3. I think the “Dealing with Induction” game is an interesting way to do experiments on the web. Psychologist Richard Wiseman has conducted some experiments this way.

    The answer given at ‘The Philosopher’s Mag’ is simply wrong.
    “The game you have just played assumed a realist position, in other words there was actually a law (a rule) that dictated whether the cards were accepted or not, and this law was written in terms that humans are familiar with.”

    The Ace is a court card (and not a 1), and although a Queen is higher than a Jack, giving it a number is wrong. I cannot argue against “the law”, but IMO “the law” will have to be reformulated.

  4. Cool game. When the 6H appeared for a second time I went bonkers. Wasup w/dat? [Assuming each game played was the same – or was that part of the test?]

    Was feeling pretty cocky with my score until saw the average was 0.01. How deflating.

  5. I made 5 errors, and it said that was “pretty good”, and then it said the average was 0.01…I’m not buying it. I wonder how many errors Mr. Briggs or any other noncheater here made.

    Prime values didn’t occur to me either, but of the 20 cards given, I got the last 11 right in a row. I went through rule iterations “accept only all hearts”, “accept only all reds”, and “accept only some pairs” before arriving at my fourth and final rule, “accept only some values”; specifically:

    “Accept only all sixes, nines, tens, queens, and aces.”

    Not immediately recognizing a pattern to the values I theorized they were a random selection, in which case probability dictated trying to assign each new unique value to the set with fewer unique values. Which just happened to work out. Theory validated!

  6. All,

    I think the “average error” calculation has a bug on their site. I can’t imagine most people who go through that game for the first time do not make any errors.

  7. For me the rule went from hearts, to red cards, to aaargh.

    I didn’t bother with converting J, Q, K to numbers since there is Harold’s point of whether the Ace is a one or an 11 or even a 14 and I assumed they’d play fair on that.

    Since the subject was induction, I was half suspecting a very simple rule with a twist/shift like the grue or bleen concepts. There was also the famous experiment on predicting rules from card sequences.

    You highlight many points that reveal potential for ‘social construction’ or at least confusion – to include 1 or not as prime, (it works as an 11 but not a 14), conversion of court cards etc.

    I suspect that many simple what’s next in this sequence tests fail to recognise more complex but still valid sequences as a right answer. When you’re only given four or five numbers in a sequence there are probably many more rules, especially when you include additional knowledge spheres like primes, number of vowels, begins with a vowel/non-vowel when converted to german etc etc.

    Great post.

  8. My theory is that there is no “error calculation”, it is an automated response. The interesting thing was that I did my best to get it “right”, I did not try to falsify my own theory (not that it makes an iota of a difference).

  9. “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. ”

    this rule is wrong right off the bat because the first card is even (QH = 12) and when it was supplied, the total was null +12 = 12. Even though the program mandates the QH is accepted, it would have to still follow the rule when it is introduced.

  10. “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 (prove this).”

    I’d rather disprove it

    if another Ace comes up, you get 1+1 = 2. Your rule rejects this card because the total would be even, the “exclude primes” rule accepts it as 1 is not prime.

  11. The first card is only a given because they give it to you. For it to legitimately be accepted it had to follow the rule when introduced.

    12+ null = 12.

  12. John,

    Oh, ouch. The Ace thing is right, the NULL is not.

    The rules give you the first card. It’s not that interesting.

  13. “Further, any modification means I have created a new theory!”

    What’s more, you get to write a new paper and add a line to your CV. Indeed, you get a CV line for every paper resulting from a modification of your theory. That poor sap who got it right the first time only gets 1 publication!

    Further, every paper that claims to refute your evolving theory gives you citations!


  14. I was lazy and just clicked “reject” to all of them. I decided at the outset that there were so many possible answers that I need not bother trying (without some real reward for just trying).

    In global warming the weather has so many possible rules for its patterns that I’m not at all surprised that people who set out to discover patterns end up “seeing” them. And the smarter they are the more sophisticated a pattern they can detect. They know for example that to see the pattern you need to ignore the near future if it cools.

    I kinda got this from when I studied design. You start with a brief of requirements/rules, but there is a huge variety of possible solutions that would meet the requirements. Later one tries to justify the design solution to the client by saying it “led from” the requirements, “form follows function”. But in your heart you know that the design was the product of many little subjective choices. That one insight is probably the one reason why I’m so skeptical of AGW theory.

    Thank you for getting us do do that game/test. I had no idea stuff like that existed.

  15. Enjoyed this post.

    I accepted only honor cards. I made no errors in carrying out my rule and was told that I made 6 errors.

    This game made me feel hopeless. You know, the purpose of any experiments is to gain a better understanding about the question under investigation. After three tries, I still had no idea what the rule could be.

    With the exception of 2, all prime numbers are odd, but not every odd number is prime. So your rule doesn’t work on odd, non-prime numbers, say, 1 & 9? No?

    The simple rule of accepting odds numbers is not perfect but good. Here is another rule: accept all numbers that are greater than 3 and divisible by 2 or 3. Obviously, this rule cannot be applied to all prime number. Here is another one: accept 1, 4, 6, 8, 9, 10 & 12(Q). ^_^

  16. I’ll wager Pascal was a Spaniard!
    What you describe is not faith, it’s some sort of lie or denial of one’s “better judgement” to oneself, a delusional state, exactly like the Bond example, (phew, but he would never submit.) At this point the individual does not have faith, they just repeat a mantra.
    The children example is no different. Their knowledge base and ability to think sophisticatedly may be lacking but that’s a different issue from the sequence I mentioned.
    This is utterly different from the concept of faith or belief, and you can substitute any other belief in there instead of God and the example doesn’t change.
    What Pascal spoke of can’t be taken seriously; probably Catholicism or Christianity inspired? It sounds like that sort of language. It’s not great on even cursory inspection.
    As for 1984, you can fool someone into anything, but you can’t scare someone into thinking anything, because you’re asking them to forget what they know. One’s mistaken thoughts might, often do, cause fear but that’s the other way round. So you would first have to convince someone that 2+2 = 5 or whatever, which could not happen to start with, but in a work of fiction, dare I say fairy tale?
    Deterministic in a probabilistic way! Too much, stop!
    H2O and the weather does not have free will, it’s just complicated.
    Just as neurology and its interfaces are complex. This does not explain what put me in me, and you in you.
    As for 1 + 1 =2, The numbers were created by us, with our free will, we made a stencil that compares very well with what we discover or create as we bumble along. I see now that you must deny free will in order to continue with your disbelief; but then you must submit that you are a droid? I mean this in the nicest way that such a statement could ever be made.

    3 wrong, and I didn’t cheat. I nearly did though.
    No idea what my theory was, it never formed itself even until the end; I think that was the point.

    Steve Brookline,
    So true!

  17. Harold: Thank you for the links. I also enjoy reading about all kinds of games.

    Joy: I forgive you for frequently filling up my screen with no-valued-added content. So many words, so little meaning. Do start your own blog and attract your type of readers. Yes, you may have the last word.

  18. William

    “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. ”
    I am not sure that that is wise .
    Actually physical theories always predict parameters that have been also used to build the theory .
    If your theory is that the gravitationnal force is inversely proportionnal to the squared distance between 2 bodies , then you don’t really need the value of the proportionality constant .
    But it is better to have it .
    So you will measure several forces at several distances (= data about forces) and discover that :
    a) you were right
    b) the proportionality constant is G.M.M’
    Then you proceed to predict the data about the same forces for other distances and it still works .
    Your theory will be good .
    At least as long as nobody finds a case where it doesn’t work .
    Then you must drop it but you wouldn’t really know why – is it because the proportionality doesn’t hold ?
    Is the squared bit wrong ?
    Is there sometimes something else than only distance that plays ? If yes when and why ?
    All of the above in unknown proportions ?
    Did you already look at the PNP conjecture (one of the 1 M$ Clay problems) ?
    I find that it has the essence of similar musings like in this thread when one asks if some particular algorithm ever stops .
    Countable infinities and such .

Leave a Comment

Your email address will not be published. Required fields are marked *