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I know that Street is abbreviated as St. But does the t in St represent the first t or the last t in Street?
Drive is abbreviated as Dr, which means it could be the first t, but Road is abbreviated as Rd, which means it could be the last t, if we were following the same pattern.
(Please note that this question is not opinion based. By looking at the patterns of other common abbreviations we can conjecture as to, or even deduce, the origin of St.)
The "t" in "st" should be taken as the first "t" of "street."
Consider that abbreviations of common nouns (especially in the domain of roadways) beginning with a consonant-vowel pair usually take the first consonant and the last consonant (or strategic consonants which appear throughout), as evinced by:
Abbreviations of common nouns beginning with a consonant-consonant pair usually take the first few consonants:
"Street"/"st" is of the latter category. As such, it seems like we should regard the "t" of the abbreviation "st" as the second consonant in "street", that is, the first occurrence of "t" in it. This agrees, I think, with the intuitions of most people.
EDIT: Note that the above reasoning also explains why "st" is also used as an abbreviation of "saint." Because "saint" starts consonant-vowel, the abbreviation uses the first and last consonant, "st".
As others have said, there's no real way to tell if the t in St represents the first or the second t in Street (or both).
What it means for a letter in a contraction or abbreviation to "represent" a letter in the corresponding full word is not clearly defined for all cases. For example, the abbreviation Rx is used to represent a medical "prescription." The abbreviation does historically come from a word that starts with r (recipe) but that word doesn't contain x either. Furthermore, most people don't know the history, in which case it's impossible for them to have a mental representation of these facts. I don't know how the abbreviation St originates, and there's no reason to suppose that my mental model of how it relates to Street is the same as the one used when it was first established.
Anyway, I have found one relevant non-opinion-based fact that points towards it being taken as representing the last t by at least some writers of British English.
Here it is: some British style guides make a distinction between shortened abbreviations missing letters at the end of the word (which generally are written with a full stop, such as Rev. "Reverend" and Col. "Colonel") and "contractions" that include the last letter, such as Dr "Doctor." (This is described by Barrie England's answer to this question: Does the abbreviation for Saint in a church name require a period?)
In all British punctuation guides that I have found so far that mention St, it is listed as a "contraction" that normatively does not take a final period, whether short for "Saint" or "Street."
This evidence is not definitive because it is not mandatory to use a period even after a shortened abbreviation (some people write Rev for "Reverend") and also, there's no reason to suppose speakers of other varieties of English analyse this contraction in the same way as the speakers who wrote these guides.
Examples:
I would go with the first t, because Street can be abbreviated down to Str as well, so perhaps St is just even shorter.
If it is written as St (which I prefer as, in addresses, there's often a comma written next which gives St., which looks awful) then following the "rules" of English abbreviations (as referenced above) then it is clearly a contraction meaning the "t" is the last one. If you write St. then you are indicating that the "T" written (in English abbreviations) is not the last letter. c.f. Dr for DoctoR, Mr for MisteR but Rev. for ReVerend.
i.e.
St is StreeT but St. would be STreet
(The dot serving its useful function of drawing the reader's attention to where letters are missing when the abbreviation is not a contraction.)
