Why do popular search engines not follow the usual AND, OR logic for queries?
Nice question!

An exact answer should be given by looking in the search engine source code but here is a possible explanation.

I run the queries at Google


burglar 33,800,000
burglar AND burglar  29,200,000
burglar OR burglar 26,500,000


The results indeed do not respect the expected Boolean relation burglar AND burglar &lt;= burglar OR burglar = burglar

However, that is since the search engine doesn't process the "and" and "or" as binary operator but just as search tokens.
Looking for them we get


And 25,270,000,000
Or  16,320,000,000


A term alone appears most times. "and" is more common than "or" so a term with "and" is more common than the term with "or"

Note that


burglar burglar 29,000,000  


Apparently looking for documents in which the term appears twice.

By the way, Google's Search operators documentation claim that "OR" should indeed act as a binary operator. 
You found a case in which they fail to do so.

Note that this behaviour is very specific to search engine.
In Bing you get the following results:


burglar 4,400,000 
burglar AND burglar  1,610,000
burglar OR burglar 1,610,000
And 10,400,000,000
Or  3,750,000,000
burglar burglar 1,610,000  


The number of results is similar for  "burglar AND burglar", "burglar OR burglar" and "burglar burglar" though we see that "And" is more popular than "OR". It seems that Bing treatments is the removal of "And" and "OR", possibly as stop words.

Bing documentation suggest the operators "&amp;&amp;" for "and" and "||" for "or".
 - burglar || burglar  4,400,000 = burglar
 - burglar &amp;&amp; burglar  1,610,000 = burglar burglar

These results fit the claim that when a term appears twice in the search query it should appear at least twice in the document too.