In the context of impeachment of a president, the US constitution refers to "high crimes and misdemeanors".
The context is
... or other high crimes and misdemeanors.
Does that mean
... or other high crimes or other misdemeanors.
or
... or other high crimes or misdemeanors.
Is it a matter of dialect of English? (I would expect it is not because it is about logic, which is independent of language in a fundamental way)
Historically, "high crimes and misdemeanors" is a set phrase by itself; you cannot really decompose it into linguistic parts and it has a historical (though somewhat vague) meaning in English law.
Importantly, the word "high" is referring to the privileged nature of the type of offense described: specifically, these are offenses committed by a "high" authority: an abuse of power. In that sense, I suppose if one had to choose between your options it would be to have "high" modifying both crimes and misdemeanors.
McDowell writes:
In all of the English cases the political nature of the offenses charged in impeachments was revealed by the use of the word "high" to modify both "crimes" and "misdemeanors." The use of "high" in "high crimes and misdemeanors" did not refer to the substantive nature of the offense, that it was a particularly serious offense, but that it was a "crime or misdemeanor" carried out against the commonwealth itself. This use of "high" to distinguish crimes and misdemeanors against the society as a whole derived from its use in distinguishing "high" treason from "petit" treason.
McDowell continues, quoting Alexander Hamilton (parenthetical added by me):
The objects of impeachment, he (Hamilton) noted, "are those offenses which proceed from the misconduct of public men, or, in other words, from the abuse or violation of some public trust. They are of a nature which may with peculiar propriety be denominated POLITICAL, as they relate chiefly to injuries done immediately to the society itself."
There is also an informative Q&A on Law.SE:
https://law.stackexchange.com/questions/30736/what-is-the-meaning-of-high-crimes-and-misdemeanors (via @Bobson's link in a comment to the OP)
McDowell, G. L. (1998). High Crimes and Misdemeanors: Recovering the Intentions of the Founders. Geo. Wash. L. Rev., 67, 626.
We can find evidence, more than 150 years old, by looking at the Journal of the Senate from the impeachment of Andrew Johnson:
In his articles of impeachment, article XI ends with (emphasis and any typos/transcription errors are likely mine):
... whereby the said Andrew Johnson, President of the United States, did then, to wit, on the twenty-first day of February, A. D. eighteen hundred and sixty-eight, at the city of Washington, commit, and was guilty of, a high misdemeanor in office.
The Chief Justice of the Supreme Court also phrased his question for the vote:
Mr Senator --------, how say you? Is the respondent, Andrew Johnson, President of the United States, guilty, or not guilty, of a high misdemeanor, as charged in this article of impeachment?
So, at least in 1868, it was understood by the Chief Justice of the Supreme Court and the Senate that high applies to both crimes and misdemeanors.
The text of the Constitution says, "high crimes and misdemeanors". While the expression may have been developed from English common law, which law may distinguish between "high crimes" and "high misdemeanors", it has no other application to the U.S. Constitution.
Significantly, wrongs punishable by impeachment are discussed distinctly in Chapter 9, entitled “Of Misprisions and Contempts, affecting the King and Government.” There, Blackstone employed the term “high misdemeanors” in a specialized sense. The “first and principal” illustration of “high misdemeanors,” Blackstone wrote, is “the mal-administration of such high officers, as are in public trust and employment. This is usually punished by the method of parliamentary impeachment.”
Keep Blackstone’s identification of “high misdemeanors” with “mal-administration” of high officers in mind. It comes back around in the Constitutional Convention’s discussion of the language that ultimately became the Constitution’s impeachment standard.1
But Mason’s first choice of a catch-all term – “mal-administration” – drew Madison’s objection as to vagueness. This might reduce the executive to service at the pleasure of two-thirds of the Senate, Madison feared. Whereupon, Mason proposed the substitute “other high crimes and misdemeanors against the state” – the more familiar term of art drawn from longstanding English practice. The substituted language was a reasonably close substitute for mal-administration. (Recall that Blackstone had classed as a “high misdemeanor” the mal-administration of high officers of government.) But “high Crimes and Misdemeanors” arguably had a more precise and more familiar meaning drawn from longstanding English practice and usage, of which the framers evidently were aware. The specific historical term-of-art clearly retained the idea of abuse of authority or official misconduct as impeachable offences. But it gently shaded more in the direction of requiring some form of culpable, wrongful conduct and away from any intimation that ordinary political disagreements over administration were sufficient grounds for removal – which had been Madison’s concern.
It is thus a slight overstatement to say, as some commentators have said, that “mal-administration” was rejected as a ground for impeachment. It is more accurate to say that concerns over the possible vagueness of the word were redressed by repairing to the familiar, general term-of-art “high Crimes and Misdemeanors,” a term probably nearly as capacious but at least somewhat more precise, and arguably importing a sense that the officer must be found to have behaved in a way that can be judged wrongful (in some sense) and not properly permitting removal over ordinary disagreements “at the pleasure” of the legislature.2
A president may be impeached for "high crimes and misdemeanors" because that is the term developed for the U.S. Constitution and only its use is consistent with the obligation "to support this Constitution" (Art. 6, Cl. 3). To impeach a president for "high misdemeanors" is tantamount to impeaching the president under English common law. The president of the United States cannot be impeached under English common law.
High crimes and misdemeanors is an undefined class of offenses, as such, the term is considered indivisible and there should be no consideration of or or and/or. That is, one should not attempt to define any offense as one or the other.
There have been attempts to provide a definition. This is a resolution sponsered by Rep. Ted Yoho (R-FL-3) (Introduced 04/13/2015)
H.Res.198 — 114th Congress (2015-2016)
SECTION 1. DEFINITION OF PRESIDENTIAL IMPEACHABLE OFFENSES.
The House of Representatives declares the following Presidential actions shall constitute impeachable ``high crimes and misdemeanors'' within the meaning of Article II, section 4, which will cause the House to vote an article or articles of impeachment to send to the Senate for trial--
(1) initiating war without express congressional authorization;
(2) killing American citizens in the United States or abroad who are not then engaged in active hostilities against the United States without due process (unless the killing was necessary to prevent imminent serious physical danger to third parties);
(3) failing to superintend subordinates guilty of chronic constitutional abuses;
(4) spending appropriated funds in violation of conditions imposed for expenditure;
(5) intentionally lying to Congress to obtain an authorization for war;
(6) failing to take care that the laws be faithfully executed through signing statements or systematic policies of nonenforcement;
(7) substituting executive agreements for treaties;
(8) intentionally lying under oath to a Federal judge or grand jury;
(9) misusing Federal agencies to advance a partisan political agenda;
(10) refusing to comply with a congressional subpoena for documents or testimony issued for a legitimate legislative purpose; and
(11) issuing Executive orders or Presidential memoranda that infringe upon or circumvent the constitutional powers of Congress.
1 The Original Meaning of “High Crimes and Misdemeanors,” Part I, AUGUST 8, 2018.
2 The Original Meaning of “High Crimes and Misdemeanors,” Part II, AUGUST 9, 2018.
