Consider this dialogue:

John: If I bet you a dollar this quarter would land heads, would you accept?

Jane: Yes.

John: If I bet you a dollar this nickel would land heads, would you accept?

Jane: No.

John: Do you believe that the probabilities of the coins landing heads are different?

Jane: No.

I seek the name of the form of the following (implicit) argument:

John: You would bet on a quarter, so why wouldn't you bet on a nickel? That is inconsistent!

What is this called, who are some philosophers that discussed it, and in what works?

Google results for "argument forms" concentrate on strictly logical arguments, e.g. modus ponens, modus tollens, etc. However this form seems more akin to

  If A, Then B.
  A' Is Like A.
  Not B.
  Therefore, Contradiction (or something Like one).

I guess it resembles some variant of modus tollens, but something seems different.

This form of argument always struck me as a powerful one, transcendent even, in a way. Whereas other forms of logical arguments (aside from being misunderstood and misused) seem to require a certain level of learned "sense" to evoke meaning, this form seems to strike some intrinsic chord in us, that even those who know nothing of "logic" inherently understand and abide by it...

I'm also interested in hearing some more clever, entertaining, and/or memorable versions of my dialogue, which I only hastily skimmed off the top of my head. (This is, in part, why I ask about prior works: I'd like to collect better examples and metaphors.)

Thanks, all.


I'm aware that humans possess subjective preferences, e.g. as in Rex Kerr's I-like-watching-quarters-flip-but-not-nickels example. Many studies have shown that human beings can be "primed" or conditioned to prefer one thing over another even against their own rationale. If it helps to redirect the focus, consider a second quarter instead of a nickel, one identical to the first as far as Jane can tell. One can poke holes in my example ad nauseam but hopefully someone gets what I mean to mean.

Said again in different words, I would like to learn more about the history and the basis of an argument such as, "This situation, and that situation, are similar. (Similar enough to justify these words.) However, you are acting one way in one situation, and a different way in another. You must stop!" (Especially in a political context, I feel this argument is most potent.)

  • 1
    I'd say it really is a contradiction if one recognises the analogy. If A and A' share relevant properties, the conclusion is likely to be true for both. If you fail to see that you're ignoring the modus tollens. The thing is, analogies aren't just logical, but you certainly see that point yourself.
    – iphigenie
    Commented Sep 7, 2012 at 19:51
  • 2
    Jane: The expected value of the bet is zero either way. I like watching quarters flip. Nickels are too small. So put your nickel and your logic away, and flip that quarter!
    – Rex Kerr
    Commented Sep 8, 2012 at 1:19
  • @RexKerr - Yes, I considered that there may be "hidden" preferences such as the one you highlight; one can imagine that we must have hidden preferences we're not even aware of, if our neural networks are primed or conditioned a certain way. In the question however (I thought it was clear enough) that I'm asking about the contradiction (or whatever it is) that appears when Jane decides exclusively on one choice while admittedly having no base on which to distinguish it from the other. Commented Sep 12, 2012 at 3:15
  • @RexKerr - (If there is still an objection, perhaps we can replace "nickel" with another quarter that is identical to the first as far as Jane can tell.) Commented Sep 12, 2012 at 3:21
  • @acheong87 - My point is that there is likely no basis to decide one way or the other on either choice. So if Jane doesn't agree with herself, a hidden reason is likely. "I pick randomly" is fine also, and may result in "inconsistent" preferences.
    – Rex Kerr
    Commented Sep 12, 2012 at 5:39

2 Answers 2


Formally, it probably looks like this:

N ⇔ C   // a nickle is a two-sided coin with a 50-50 chance of flipping heads/tails
Q ⇔ C   // a quarter is a two-sided coin with a 50-50 chance of flipping heads/tails
N ∧ ¬Q  // I'm willing to flip a nickle and not flip a quarter
C ∧ ¬C  // A coin is not a coin -> contradiction (nonsense)

No primes necessary and no analogies necessary. A nickle is not like a quarter, it's the exact same thing (a two-sided coin with a 50-50 chance of flipping heads/tails) - a fact which Jane seems to accept. The inconsistency is a simple contradiction akin to saying it's the year 2012 and the year 1012 or up is down. As some of you may have noticed, my sketch is slightly incorrect (I sort of mix up prepositions with predicates). A more correct implementation would be:

N ⇔ C   // a nickle is a two-sided coin with a 50-50 chance of flipping heads/tails
Q ⇔ C   // a quarter is a two-sided coin with a 50-50 chance of flipping heads/tails
N → F   // a nickle has a flipping preference
Q → ¬F  // a quarter does not have a flipping preference
F ∧ ¬F  // a flipping preference is not a flipping preference -> contradiction
C ∧ ¬C  // a coin not being a coin can also be derived -> contradiction

The above conclusions can be derived in a number of ways (material implication pops out as an obvious solution). Of course, things would be more complicated if Jane would argue that the coins are not, in fact, "the same thing." Here is where you may need an analogy. In short, if Jane would argue that:

  • (N ⇔ ¬C) ∨ (Q ⇔ ¬C),

there would be some problems. But that is outside of the scope of the question.


There seems to be a misunderstanding on the subject of the bet, whether it is about the odds only or the whole situation. To highlight it even more: ‘I’m going to play Russian roulette with this man. If I win will you give me a dollar?’ ‘Yes,’ ‘I’m going to play Russian roulette with your relative. If I win will you give me a dollar?’ ‘No,’

Not sure how to classify it.

  • Thanks for the response, but please see my response to Rex in the question comments. I intentionally made my example mild (boring) to minimize plausible alternative factors, such as the death of a relative resulting from the toss. I do not mean to ask about odds or psychology. Commented Sep 12, 2012 at 3:22

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