Definition:

A system of punctuation determined by the grammatical structure of a sentence and its meaning. Also called grammatical punctuation.

In modern English, logical punctuation has, in part, superseded rhetorical (or elocutionary) punctuation, a system determined by the breaths that mark the act of reading a text aloud.

The expression logical punctuation is sometimes applied specifically to the British style of punctuating quotations as opposed to the American style.

(See the observations below.)

See also:

• Basic Rules of Punctuation
• A Brief History of Punctuation
• End Punctuation: Periods, Question Marks, and Exclamation Points
• Grammatical Meaning
• Guidelines for Using Quotation Marks Effectively

Examples and Observations:

• Logically Punctuating Quotations
  "I'm not sure what is meant by 'fail-safe'. This is logical punctuation, since fail-safe is just an isolated term, not a statement or question; only the complete sentence deserves a period. Nevertheless it is wrong [according to the conventions of American English]; it should be I'm not sure what is meant by 'fail-safe.' It isn't logical, it's just the way it is. Commas and periods always go within closing quotation marks."
  (Edward D. Johnson, The Handbook of Good English. Simon & Schuster, 1991)
  
  "For at least two centuries, it has been standard practice in the United States to place commas and periods inside of quotation marks. This rule still holds for professionally edited prose: what you'll find in Slate, the New York Times, the Washington Post--almost any place adhering to Modern Language Association (MLA) or AP guidelines. But in copy-editor-free zones--the Web and emails, student papers, business memos--with increasing frequency, commas and periods find themselves on the outside of quotation marks, looking in. A punctuation paradigm is shifting. . . .
  
  "[T]he British way simply makes more sense. Indeed, since at least the 1960s a common designation for that style has been 'logical punctuation.' . . .
  
  "[L]ogical punctuation isn't likely to break through to the rule-keepers any time soon. The old way is just too established."
  (Ben Yagoda, "The Rise of 'Logical Punctuation'." Slate, May 12, 2011)

• From Rhetorical to Logical Punctuation
  "With the advent of printing, readers grew more proficient at reading silently. Logical punctuation had to be introduced for silent readers to serve as intonation did for oral readers."
  (John H. Fisher, The Emergence of Standard English. Univ. Press of Kentucky, 1996)
  
  "The virgule, colon and full stop or period (/:.) are the only marks found in early books printed in England, but from 1520-40 the comma replaced the virgule. The structuring reflected by these marks was in the main rhetorical, until the use of logic-based punctuation increased in the seventeenth century (though it never achieved the logical consistency of, say, German punctuation). The phases leading from rhetorical to logical punctuation and the reasons for this development are not quite clear. The transition is possibly related to the changing communicative functions of written English as it developed further and further away from being merely a representation of spoken English (or of texts meant for oral delivery) towards greater autonomy as a system. Most striking is the reinterpretation of the colon (:), which originally was a mark indicating a pause of medium length, but which developed into an indicator of text coherence expressing logically consecutive or adversative relations ('therefore'; 'on the other hand'). This change is clearly related to the introduction of the semicolon (or comma-colon) from 1580-90, which more or less assumed the syntactical function of the earlier colon, that of separating two closely connected main clauses.
  
  "The question and exclamation marks did not become common until the seventeenth century."
  (Manfred Görlach, Introduction to Early Modern English. Cambridge Univ. Press, 1991)