An Odd Use of the Second Supine

The Maverick Philosopher, 'I Don't Mind Losing':

In the middle paragraph I was quoting an actual person, whereas on the two other occasions I was not quoting, strictu dictu, but mentioning a sentence.

Unlike many people who use Latin phrases, the Maverick Philosopher actually understands Latin. But I'm puzzled by strictu dictu meaning "strictly speaking." He is not alone in using this phrase. It is possible to find other examples, even in printed books. The alternative stricte dictu seems slightly more common than strictu dictu, at least judging from a Google search. But I have trouble construing either strictu dictu or stricte dictu.

Dictu is a supine. Allen and Greenough, New Latin Grammar § 508, define a supine as "a verbal abstract of the fourth declension...having no distinction of tense or person, and limited to two uses." The two uses are:

1. "The Supine in -um is used after verbs of motion to express purpose." (§ 509)
   
2. "The Supine in -ū is used with a few adjectives and with the nouns fās, nefās, and opus, to denote an action in reference to which the quality is asserted." (§ 510)
   

Clearly dictu by its form falls into the second category. It is a supine from the verb dicere ("say, tell, relate"). Adjectives used with dictu are often neuter adjectives of the third declension, e.g. horribile dictu ("horrible to relate"), mirabile dictu ("wonderful to relate"), difficile dictu ("difficult to relate"), etc.

How are we to understand strictu? I can think of the following types of Latin words that end in -u:

• Supines, as above. In order to qualify as such, strictu would have to be a supine from stringere ("draw tight"). I am not aware of any examples of a supine strictu.
  
• Nominative or accusative singular of neuter fourth declension nouns. Such nouns are rare (cornu, genu, pecu, veru, plus some odd examples quoted by ancient grammarians). So far as I know, there is no neuter fourth declension noun strictu.
  
• Dative or ablative singular of fourth declension nouns of any gender. To my knowledge, there is no fourth declension noun strictus or strictu.
  

So strictu doesn't appear to be a proper Latin word at all.

I'm baffled by strictu dictu, but what about stricte dictu? The perfect passive participle strictus, -a, -um (from stringere) can be used as an adjective meaning "strict," and the corresponding adverb stricte can mean "strictly." But can an adverb like stricte modify a verbal abstract, a supine, like dictu? I can't find such a usage discussed in books on Latin grammar, and so even stricte dictu seems odd to me, although not quite as odd as strictu dictu.

Strictu (if it existed) and stricte are not adjectives suitable for use with dictu, in contrast to difficile, horribile, and mirabile.

Maybe I'm just obtuse, but both strictu dictu and stricte dictu puzzle me. I can't find any classical or even medieval examples or analogues. If you correct me, please cite chapter and verse.

I do not have access to:

• Nils Sjöstrand, De vi et usu supini secundi Latinorum (Lund: Hjalmar Möller, 1891)
  
• C.H.M. Kroon, "Rarum dictu: the Latin second supine construction," Glotta 67 (1989) 198-228