transitive property in statistical comparisons I want to know if the exposure to a given dose of a toxic chemical alters the concentration of a given substance in blood in my animals. I make this super general, bear with me, because I think this is a general question.
I have 100 individuals at the beginning of the trial. I then randomly select 10 individuals and measure them as the initial, pre-exposure value. Then I introduce the toxic compound and 10 minutes later I measure another 10 individuals (this is my "during exposure" value because my organisms are reacting). I wait another 10 minutes (the effect is no supposedly gone) and measure another set of 10 individuals. 
So at the end of the experiment I have measured 30 different animals. One factor ANOVA yields significant differences and I use tukey to know which is different from which. To my surprise I get something I can not interpret in a straightforward manner.
The measured variable gets lower during exposure and then goes back to values similar to the pre-exposure state. 
See the plot here: 
https://www.dropbox.com/s/euzrqdccyyxd3xw/is%20this%20recovery.jpg?dl=0
But when I see the posthoc letters, in my mind there are two competing explanations:
1) The variable recovers: because the post-exposure values are statistically equivalent to those at the beginning.
2) The variable does not recover: because the post-exposure values are statistically equivalent to those during the exposure (which in turn are statistically different from the pre-exposure values!!).
So, in math A=B and B=C therefore A=C. But does this work also in statistical comparisons??? I can't figure out if this is just a paradox and which is the appropriated interpretation (if any).
 A: There is a 3rd option that you don't state: 3) there is partial recovery and the final value is higher than during the exposure but not as high as the initial values.
Remember that statistical significance is a "rule-out" test which means that we can sometimes rule out certain hypotheses, but any of those that we don't rule out could be true, we just don't know which one.  The summary that you give suggests that we can rule out A=B, we know that the value during exposure is lower than that at the beginning.  You just don't have enough data to rule out your statement 1, 2 or my 3.  While logically we know that 1 and 2 (and 3) cannot both be true at the same time, the Tukey test does not say that they are both true, just that we can't rule either out and that therefore at least one of them could be true (outside logic tells us that they cannot both be true at the same time, but the test does not worry about that).
Think of it this way:  there is a box with 5 small balls inside (small enough that all 5 can be hidden in a closed hand).  I reach my empty hand into the box and bring it out closed and ask you how many of the balls are in my hand (you don't get to look in the box).  You can rule out negative numbers as impossible, also you can rule out numbers greater than 5 since my hand was empty going in (we are assuming no slight of hand or multiplying balls), but while it is impossible for me to be holding both 0 balls and 5 balls at the same time (or any number in between), you cannot rule out 0 or 5 with the information that you have.  The Tukey test is similar, it is not saying that A=B has to be true and that B=C has to be true (and if A does not equal C then they cannot both be true) it is just saying that it can not rule either out and they could be true.
