Emphatically means in a way that leaves no doubt about your feeling. Said firmly, and said once.
It attaches to denied, stated and rejected, and emphatically denied is close to a fixed phrase in reporting. There is a second use meaning decisively, so winning emphatically means winning by a wide margin, and sports writing uses it constantly. The word can also strengthen a whole statement, as in it is emphatically not the case. The adjective emphatic covers both, and the noun is emphasis.
1adverbemphaticallyIn a firm way that leaves no room for doubt.
“He emphatically denied ever having met her.”
2adverbdecisivelyDecisively and by a clear margin.
“They won emphatically in the second half.”
Too few real examples of this word to estimate which sense is commonest.
Proto-Indo-European
*leyǵ- — to bind, tie
Latin
alligāre
Old French
alier — to join together, unite; to alloy (metals); (cooking) to combine ingredients
Middle French
alier
Anglo-Norman
alier
Middle English
allien
English
ally
English, today
emphatically
Built from
From emphatic + -ally. ally comes from Middle English allien, alien (“to form an alliance, associate, join; to become an ally; to introduce (someone) as an ally; to marry; to become related (to someone); to attack, engage in combat; to combine; (cooking) to combine ingredients, especially to bind them together”)alie, alleye, allie, ally, allye, aly, alye, alyȝe, from Anglo-Norman alier, allier, Middle French alier, allierallyer, alyer, and Old French alier (“to join together, unite; to alloy (metals); (cooking) to combine ingredients”) (modern French allier), from Latin alligāre, the present active infinitive of alligō, adligō (“to bind around, to, or up (something), bandage, fasten, fetter, tie; to hold fast; to detain, hinder”), from al-, ad- + ligō (“to bind, tie; to bandage, wrap around; to unite”) (from Proto-Indo-European leyǵ- (“to bind, tie”)). a doublet of allay and alligate and alloy and ligament.
Etymology from Wiktionary, CC BY-SA