Plausibility vs Probability http://whatho.in/2013/plausibility-versus-probability/ refers to pp 155-156 of 533 of Thinking, Fast and Slow by Daniel Kahneman. I'll use one of Kahneman's other  questions from p 156:

  
*
  
*A massive flood somewhere in North America next year, in which
  more than 1,000 people drown
  
*An earthquake in California sometime next year, causing a flood
  in which more than 1,000 people drown
The California earthquake scenario is more plausible than the North
  America scenario, although its probability is certainly smaller. As
  expected, probability judgments were higher for the richer and more
  entdetailed scenario, contrary to logic. This is a trap for forecasters and
  their clients: adding detail to scenarios makes them $\color{darkred}{more \, persuasive}$, but
  less likely to come true.

Since Pr('adding detail') means adding more events to the intersection in $\Pr(\cap A_j)$,
 and $Pr(A \wedge B) \le Pr(X)$ where X is either A or B, thus:
$\Pr(\cap_{j \le 1} A_j) \le \Pr(\cap_{j \le 1} A_{j+1})$, or in words, 'adding detail to scenarios makes them...less likely to come true' (♦). 
Yet how does this make them  $\color{darkred}{more \, persuasive}$ or more plausible? Doesn't (♦) prove 'less probable'? What did I miss in the definition of plausible which contains the word probable?
 A: Like Michael, I disagree with the basic premise that adding details makes something more plausible. Certainly, if the extra details outline a specific scenario that people recognize, then such familiarity lends itself to agreement (this is called "anchoring", where you based assessments on what you can remember vs. the actual statistics).
Also, a plausibility can be thought of as a measure of belief $B(\cdot)$ regarding statement $S$:
$Pl(S)=B[P(S)>0]$ So plausibility is meta-probabilistic concept.
Applying this to the earthquake example: "An earthquake in California that causes a flood where 1,000+ drown" (event $C$) is a subset of the set of events constituting "A flood in North America that kills 1,000+ people." ($N$)
Thus, if our beliefs are to have any consistency:
$C\subset N \implies P(C)\leq P(N) \implies Pl(C)=B[P(C)>0] \leq Pl(N)=B[P(N)>0]$ 
In fact, since we know that both earthquakes and floods do happen in California, presumably we know that $P(C),P(N)>0$ hence both should have the same plausibility value. If one denies this, then one is equating plausibility with something like probability. 
A: Perhaps people are subconsciously doing a conditional probability, and they are actually comparing Pr(Flood with 1000 dead in U.S.) with Pr(Flood with 1000 dead in CA | Earthquake in CA).  Then the second (conditional) probability could well be greater than the first (unconditional) probability.
[The people doing this would not necessarily need to know any probability theory to make a qualitative comparison of this type.]
A: Well, i guess this is not really a mathematical question but rather aims at psychology and intuition (as you tagged it yourself).
Of course, your small proof says exactly the opposite, and with that statement in mind it seems absolutely plausible, that the second scenario is less probable. I assume, this example is adressed towards beginners in this field (as the text seems to be intended to give an introduction to probability theory) or at least not meant to confuse people who already have some profound knwoledge/experience with probability theory. As user 14982305 points out, those people usually do not (yet) benefit from the mathematical education to, as he puts it "see through this".
I also agree with paw88789 that (some) people might actually mistake an intersection of two events as a conditional event however I presonally don't agree with the authors conjecture that more details make an event seem more plausible. 
