Solution: Truth

SOLUTION:

In each line, the speakers use a phrase which is the motto of a university, translated from Latin. This is clued in multiple ways:

The title of the puzzle is Truth, and "Veritas" is the motto of Harvard
The two debaters, Mitter and Timothy, are actually Mens and Manus, of the "mens et manus" motto of MIT. This also comes across in the substance of the debate, sort of.
The flavor text clues Latin specifically, since the mottos are from Latin.

The first letters of the names of the universities that Mitter uses spell "OBS OED." Those for Timothy spell ESTAMIN. The solution is ETAMINE, an obsolete form of which is ESTAMIN, which you can verify by looking it up in the OED. MIT has a site license for the OED, so this is not a problem. The lists are given below -- the original language is given, then a translation.

OBS OED:

Oxford (Dominus Illuminatio Mea) -- the Lord is my light
Brown University (In Deo speramus) -- In God We Trust
South, University of the (Ecce quam bonum) -- Behold How Good
Ohio Wesleyan (In tuo lumine videbimus lumen) -- in thy light shall we see light
Emmanuel College (Deus nobiscum) -- God with us
Dartmouth (Vox clamantis in deserto) -- a lone voice crying out in the wilderness

ESTAMIN:

Edinburgh, University of (Nec temere, nec timide) -- Neither rashly nor timidly
St. John's College (Facio liberos ex liberis libris -- Free adults from books and a balance
Trinity College (Pro ecclesia et patria) -- for church and country
Amherst (Terras irradient) -- Let them enlighten the lands
Michigan, University of (Artes, scientia, veritas) -- arts, knowledge, truth
Immaculata College (Scientia floret virtute) -- knowlege flourishes in virtue
New York University (Perstare et praestare) -- To Persevere and Surpass