Old English hælþ "wholeness, a being whole, sound or well," from Proto-Germanic *hailitho, from PIE *kailo- "whole, uninjured, of good omen" (source also of Old English hal "hale, whole;" Old Norse heill "healthy;" Old English halig, Old Norse helge "holy, sacred;" Old English hælan "to heal"). With Proto-Germanic abstract noun suffix *-itho (see -th (2)).
Of physical health in Middle English, but also "prosperity, happiness, welfare; preservation, safety." An abstract noun to whole, not to heal. Meaning "a salutation" (in a toast, etc.) wishing one welfare or prosperity is from 1590s. Health food is from 1848.
late 14c., probleme, "a difficult question proposed for discussion or solution; a riddle; a scientific topic for investigation," from Old French problème (14c.) and directly from Latin problema, from Greek problēma "a task, that which is proposed, a question;" also "anything projecting, headland, promontory; fence, barrier;" also "a problem in geometry," literally "thing put forward," from proballein "propose," from pro "forward" (from PIE root *per- (1) "forward") + ballein "to throw" (from PIE root *gwele- "to throw, reach").
The meaning "a difficulty" is mid-15c. Mathematical sense of "proposition requiring some operation to be performed" is from 1560s in English. Problem child, one in which problems of a personal or social character are manifested, is recorded by 1916. Phrase _______ problem in reference to a persistent and seemingly insoluble difficulty is attested from at least 1882, in Jewish problem. Response no problem "that is acceptable; that can be done without difficulty" is recorded from 1968.
"favorable to health, wholesome," 1540s, from Latin salubris "promoting health, healthful," from salus (genitive salutis) "welfare, health" (from PIE root *sol- "whole, well-kept"). Originally of foods, medicine; in reference to air, climate, etc., by 1610s. Related: Salubriously; salubriousness.
early 15c., sanite, "healthy condition, health," a sense now obsolete, from Old French sanité "health," from Latin sanitatem (nominative sanitas) "health, soundness of body; sanity, soundness of mind; reason; correctness of speech;" from sanus "healthy; sane" (see sane). Meaning "soundness of mind" is attested from c. 1600.
"recovering strength and health after sickness," 1650s, from French convalescent, from Latin convalescentem (nominative convalescens), present participle of convalescere "thrive, regain health, begin to grow strong or well," from assimilated form of com-, here perhaps an intensive prefix (see com-), + valescere "to begin to grow strong," inchoative of valere "to be strong" (from PIE root *wal- "to be strong").
As a noun, "one who is recovering strength and health after sickness or debility," attested from 1758.
c. 1600, "doubtful, questionable, uncertain, unsettled," from French problematique (15c.), from Late Latin problematicus, from Greek problēmatikos "pertaining to a problem," from problēmatos, genitive of problēma (see problem).
Specific sense in logic, differentiating what is possible from what is necessarily true, is from 1610s. The sense of "constituting, containing, or causing a difficulty" is modern, probably from a noun use in sociology (1957). Related: Problematical (1560s); problematically.