On Tuesday, in a previous post titled “Presuppositions,” I shared this picture:

and wrote:

Four cards are shown, each card showing a character on the side shown, namely A, B, 2, and 3, respectively.

Consider the following proposition:

Proposition P: If one side of a card has a vowel, then on the other it has an even number.

To establish that none of these four cards falsifies Proposition P, which cards must one turn over?


At the original post, several people graciously shared their thoughts in the comment field. Thanks to you all for your valuable contributions!

I encountered the picture at the home of Per Skedinger, who presented us with the puzzle. Different answers were suggested by members of our dinner group.

Proposition P would be falsified by a card that on one side had a vowel and on the other side did not have an even number. The question is, which of the four cards may falsify Proposition P?

At the dinner, one member of our group included the “2” card in his answer, I think because he took “If one side of a card has a vowel” to mean “If and only if one side of a card has a vowel.” But Proposition P does not say if and only if. So that reason for including the “2” card is faulty.

At the dinner, I—like some of the commenters at the first post—suggested, the “A” card and the “3” card. But discussion made me realize that I was presupposing a condition that I didn’t have great grounds for, namely, that all cards have a letter on one side and a number on the other. Once we drop that condition, we see that we need to turn over the “B” card, for it may have a vowel on the other side. (Kudos to commenters Joel, Capt. J, robc, and Francisco.)

Our formulations always involve inarticulate assumptions or presuppositions.

Among the commenters at the first post, almost all operate on the assumption that we might read the card-side “data” as showing that one card has only an A on it, one only a B, one only a 2, and one only a 3.

But are we certain that the “2” card-side has only a 2 on it? We do not see the entire side. Maybe there is a vowel or an odd number in the unshown portion of the “2” card-side.

And you could go further, and wonder about teeny-weeny characters (“the fine print”). And Capt. J came up with some other creative responses to the problem.

If you keep digging you can always argue that “the facts” are theory-laden.

Suppose we turn over all four cards and find that none falsifies Proposition P. Does that mean that Proposition P has been verified?

If Proposition P refers, not just to the four cards, but to some larger set of cards of which the four are but elements, then Proposition P has been confirmed for those four cards but has not been fully verified.

I take this opportunity to excurse a bit.

My thinking in philosophy of science tends toward Thomas Kuhn, Michael Polanyi, and Deirdre McCloskey, and I read David Hume along such lines. The tendency is skeptical about claims of a strict logic of science, a principle for demarcating between science and non-science, or a definite scientific method. (If I am not mistaken, the Kuhn work most relevant to these matters is his 1970 piece quoted here.)


Elsewhere, I have written:

A sect might tailor its language so as to ensure the analyticity of the any of the following sentences: “The triangle has three sides.” “The child was born of a mother.” “The sum of the assets equals liabilities plus equity.” “Y = C + I + G + NX.” “The person maximized his utility.” “If transaction costs were negligible and parties were aware of the relevant opportunities, those parties achieved an efficient outcome.” “The moral sentiment relates to a sympathy.”

For such a language community, the analytic statement in question would be non-falsifiable. Does that mean it’s not scientific? Here I think of something Thomas Schelling wrote: “It is sometimes said, in textbooks and in learned volumes, that these accounting statements, being unfalsifiable, do not count as science. I don’t care.”

Here, someone might respond: Well, the falsifiability criterion is about demarcating between science and non-science for empirical statements, and those analytic statements are not empirical. So, by themselves, they don’t speak to the merit of the falsifiability criterion.

I then say: A language community maintaining one of those analytic sentences may be taken to be saying, tacitly: Maintaining the analytic sentence is good. (Didn’t Gary Becker say that about one of them? Didn’t Adam Smith say it about another?) That statement, though vague, would seem to be empirical, and falsifiable, in principle. The big challenge then is in providing falsifying evidence. The challenging researcher must show that maintaining the analytic sentence is not good. Here we see why the philosophy of science merges with the history and sociology of science: Did the way in which the sociology of judgment actually unfolded conduce to the good? Was it perverted by certain forces, as in Soviet science or the governmentalization of science generally? Michael Polanyi wrote about the conditions under which the sociology of judgment conduced to the good.

Like grammar, strict logic and the hunt for falsifying evidence have their place, but they leave important things underdetermined. It’s fine to call foul on errors in strict logic, and it’s fine to present purportedly falsifying evidence, but it’s misguided to think that strict logic and the hunt for falsifying evidence alone can establish important claims. Science is a moral activity. It is purposeful, interpretive, even aesthetic, and the rules of aesthetics are rather different from the rules of grammar (the differentness is presented in Figure 6.3 here). A blank page contains neither violations of grammar nor errors in logic, but it does not satisfy the standards of good science.

Saying that science is interpretive is not saying that science is arbitrary. It is not saying that no interpretation is better than another.

I have often wondered about the following proposition:

Proposition R: No half-way serious, reputable thinker has ever said that no interpretation is better than another.

I don’t know whether Proposition R is true. (Of course, “half-way serious, reputable thinker” is vague, but we have to draw a line somewhere.)

Comments are open. In particular, if you can falsify Proposition R, please provide the falsifying evidence in the comment section.