I don't think that the combination of pro- and portare had anything to do with the emergence of a negative connotation in the word purport. I say this because dictionary coverage of the term shows no sign that such a connotation was widely understood in contemporaneous use for at least the 255 years between between Elisha Coles's dictionary of 1676 and Webster's Fourth Collegiate Dictionary of 1931.
The dictionary understanding of 'purport'
Elisha Coles, An English Dictionary: Explaining the Difficult Terms That Are Used in Divinity, Husbandry, Physick, Phylosophy, Law, Navigation, Mathematicks, and Other Arts and Sciences (1676) has a single entry for purport, as a noun:
Purport, the true meaning.
John Kersey, The New World of Words: or, Universal English Dictionary, sixth edition (1706) treats the word similarly:
Purport, Meaning, the Tenour or Substance of a Writing.
Fifty years later, Samuel Johnson, A Dictionary of the English Language (1756) has entries for purport as a noun and as a verb:
PURPORT, s. {pourporte, French.} Design ; tendency of a writing or discourse. Norris.
To PURPORT, v. a. {from the noun.} To intend ; to tend to show. Bac[on]. Rowe.
Similarly, Noah Webster, A Compendious Dictionary of the English Language (1806) has these entries:
Purport, n. a design, meaning, tendency
Purport, v. t. to intend, design, mean, tend, show
And Webster's full-size An American Dictionary of the English Language (1828) has these:
PURPORT, n. 1. Design or tendency; as the purport of Plato's dialogue. Norris. 2. Meaning; import; as the purport of a word or phrase.
PURPORT, v. t. To intend; to intend to show. Bacon. 2. To mean; to signify.
The 1864 edition of Webster's American Dictionary of the English Language hinted that the word purport might at times convey a sense of something potentially deceptive, but only with regard to a newly identified definition drawn from Spenser and labeled obsolete:
Purport, n. 1. Design or tendency; meaning; import. [Examples:] "The whole scope and purport of that dialogue." Norris. "With a look so piteous in purport/As if he had been loosed out of hell/To peak of horrors, he comes before me." Shak. 2. Disguise; covering. {Obs.} [Example:] For she her sex under that strange purport/Did use to hide. Spenser.
Purport, v. t. To intend to show; to intend; to mean; to signify. [Example:] They in most grave and solemn wise unfolded/Matter, which little purported, but words/Ranked in right learned phrase. Rowe.
But as late as Webster's Fourth Collegiate Dictionary (1931), the entries for purport give no strong sense that the word carries a negative connotation:
purport v. t. To have as its purport or the like ; to mean or seem to mean or intend.
purport n. Meaning.
The detailed comparison of purport (as a noun) to similar words (meaning, sense, signification, significance, and import) under the entry for meaning in the same dictionary doesn't indicate any sort of cloud hovering over purport:
Purport is meaning esp. in the sense of general tenor.
But the Fifth Collegiate Dictionary (1936) tells a very different story:
purport v. t. To convey or profess outwardly, as one's or its meaning or intention; to have the appearance, often specious, of being, intending, etc.
purport n. Meaning; import; tenor; often, substance; gist.
So what happened between 1931 and 1936 to the verb form of purport? My guess is that the people at Merriam-Webster noticed the suggestive way that lawmakers and courts were using the verb purport in the 1930s—and as it turns out, long before.
The legal meaning of 'purport'
The Federal Food, Drug, and Cosmetic Act of 1938 includes the following instance of purport (cited in the evocatively named case of United States v. 306 Cases Containing Sandford Tomato Catsup with Preservative, tried in a federal district court in New York in 1944):
The libel [that is, the government's plea for judicial approval of its seizure and condemnation of the cases of catsup] charges that the seized article is misbranded within the meaning of the Federal Food, Drug, and Cosmetic Act, 21 U.S.C.A. § 343(g) (1). If it is misbranded, it is subject to seizure and condemnation by authority of 21 U.S.C.A. § 334. Under the applicable section of the Act, a food shall be deemed to be misbranded "if it purports to be or is represented as a food for which a definition and standard of identity has been prescribed * * * unless (1) it conforms to such definition and standard * * *."
...
The evidence clearly shows that the seized product is a food which purports to be tomato catsup, a food for which a definition and standard of identity has been promulgated, and that it does not conform to such standard in that it contains benzoate of soda, an ingredient not approved by the standard. It is therefore misbranded within the meaning of 21 U.S.C.A. § 343(g) (1) and is herewith condemned.
Here, purports means something along the lines of "claims on its face," "professes," "represents," "alleges on its own behalf," or "gives the appearance"—the meaning that the Fifth Collegiate attributes to the word. Black's Law Dictionary, revised fourth edition (1957) gives a distinctly legal cast to both the noun and the verb forms of purport:
PURPORT, n. Meaning; import; substantial meaning; substance; legal effect. The "purport" of an instrument means the substance of it as it appears on the face of the instrument, and is distinguished from "tenor," which means an exact copy. [Citations omitted.]
PURPORT, v. To convey, imply, or profess, outwardly; to have the appearance of being, intending, claiming, etc. [Citation (to U.S. v. 306 Cases of Catsup) omitted.]
The earliest law dictionary entry for purport that I've been able to find is from Alexander Burrill, A New Law Dictionary and Glossary (1851), which seems to be the source of the noun entry for purport in Black's Law Dictionary (the first edition of the latter was published in 1891):
PURPORT. {L. Lat. proportum, from Fr. pour, for, and porter, to carry.} Meaning; import; substantial meaning; substance. The purport of an instrument means the substance of it as it appears on the face of the instrument, and is distinguished from tenor, which means an exact copy. MS. Buller, J., 2 East's P. C. 983. Wharton's Am. Crim. Law, 83.
And the wording of entry of the verb form of purport from Black's Law Dictionary seems to have come from Webster's New International Dictionary, second edition (1934):
purport v. t. ... To convey, imply, or press outwardly, as one's (esp. a thing's) meaning, intention, or true character ; to have the appearance, often specious appearance, of being, intending, claiming etc. (that which is implied or inferred) ; ...
Of course, in law when someone purports (that is, represents) something to be true, the court pays considerable attention to the question of whether the representation is true or false. And as points of contention at trial, purported facts necessarily become matters of considerable controversy and close investigation.
But in the context of law, purport seems to have been used in the sense of "claim on its face" long before Merriam-Webster caught up with the usage. For example, Samuel Phillipps, A Treatise on the Law of Evidence, second edition (1815) uses the words purports and purporting on multiple occasions, including this one:
Upon what principle, it may be asked, is a party, by whom an instrument purports to be made, incompetent to prove it forged? In Watt's case, on an information for the forgery of a deed purporting to be the revocation of a will, it was adjudged by the barons of the exchequer after a conference with the judges of the King's Bench, that no legatee named in the will, nor any other person who is a loser by the deed, or who may receive any advantage from the verdict, can be a witness for the prosecution: ...
Here, the sense of purports and purporting is "claims/claiming on its face." And similarly, Henry Roscoe, Thomas Granger & George Sharswood, A Digest of the Law of Evidence in Criminal Cases, third edition (1852) has this language:
And by the 5 & 6 Vict. c. 122, s. 25, in the event of the death of any witness deposing to the petitioning creditor's debt, trading, or act of bankruptcy, under any fiat in bankruptcy, his deposition purporting to be sealed with the seal of the court of bankruptcy, or a copy thereof purporting to be so sealed, shall be receivable in evidence of the matters therein contained.
Again, purporting means something like "seeming by all appearances." It is not a very great step to go from purport in the sense of "have the semblance" to purport in the sense of "have the appearance, often specious," especially after decades of legal usage in settings where the trial result might depend on the question of whether the appearance matched the reality.
Conclusions
In my view, the negative connotation of purport in present-day usage owes far more to the word's use over many decades as a term of art, in law, to describe an asserted but not yet proved claim of fact than to the English term's Latin roots.
Also of considerable interest is the fact that for more than a hundred years, dictionaries seemed oblivious to legal usage of purport, focusing instead on the ways that Norris, Bacon, Shakespeare, and Rowe (and belatedly, Spenser) had used the term in more-literary settings. Perhaps the legal sense did not penetrate into the vocabulary of society at large, but it seems odd nevertheless that dictionaries—and in particular, Merriam-Webster—paid so little attention to the way the word was being used in the legal system.
