The error in the logic was pointed out 40 years ago on the bottom of page 166: "These conclusions require a subtle but important extension of the meanč¶³ing assigned to a notation..." In other words, you have to assume there is a one-to-one correspondence between two or more measurements and two or more attributes of the thing being measured. That is a logical impossibility, when the thing being measured only possesses a single bit of information, redundantly encoded (AKA entangled) into two objects.
"would you tell me where that same logic fails in its application to the interrogation of two persons in adjacent rooms, please, and the inference that they must have been overhearing each other's interrogation when answering their own questions?"
Sure. The entire conversation consists of uttering a sequence of single bits of information, being unreliably measured by the interrogator (naive measurements produce correlated measurement errors of the only existing bit's value). The two always only utter a single bit value, which the interrogator mishears, because it never occurred to him or her, that the correct value can ONLY ever be measured, by "listening" from two very specific phase angles, which differ for each of the two persons. Any attempt to listen at a different phase, is doomed to produce a high probability of a bit-error in the measurement, particularly when the measured phase is nearly orthogonal to the unique, required phase angle (the only measurement angle guaranteed to produce no intersymbol interference and thus a bit-error). The problem is related to avoiding intersymbol interference, in Quadrature Amplitude Modulated (QAM) signaling.
In the link from my previous post, I used the term "cross talk", since few physicists are familiar with the more technically correct concept of "intersymbol interference": measurements of one attribute (symbol) being mistaken for measurements of another, that happens to be "local".
This may be easier to visualize, if you think of the conversation as using visual symbols, rather than sounds. Each person presents a sequence of anti-parallel coins, for the interrogator to see. But the coins are noisy and are not identical, except for the fact that if you observe one member of the pair along the correct viewing axis (which must always be perpendicular to the coins surface), then you will correctly decode the intended message - either heads or tails) And if you view the other person's coin from the exact opposite direction, you will always get the exact same bit value - the same, redundant message.
Now you might object, that the interrogator has no way of knowing, a priori, what the one and only correct viewing axis is, for each pair of coins, particularly since the people present each entangled pair at a different, random angle. Bingo! If the interrogator ignorantly assumes that he can simply measure the coins state from any old angle and subsequently deduce the correct message, he is sadly mistaken. But that is exactly what Bell's theorem has done. Bell has unwittingly assumed he can correctly make sense, out of a conversation encoded with an unknown one-time-pad. But that is known to be logically impossible. The correlation statistics of the resulting bit-errors, are exactly those observed in the quantum correlations produced by Bell experiments.
Rob McEachern