Following up on the issues raised yesterday in "Feelings, beliefs, and thoughts", it might be helpful to explore the etymology of the various verbs that people commonly use to express the epistemic status of their assertions. From their entries in the Online Etymological Dictionary, we'll learn that several common propositional attitude verbs have roots in sensation, motion and emotion, just as feel does.
Old English belyfan "to believe," earlier geleafa (Mercian), gelefa (Northumbrian), gelyfan (West Saxon) "believe," from Proto-Germanic *ga-laubjan "to believe," perhaps literally "hold dear, love" (cognates: Old Saxon gilobian "believe," Dutch geloven, Old High German gilouben, German glauben), ultimately a compound based on PIE *leubh- "to care, desire, love"
Old English þencan "imagine, conceive in the mind; consider, meditate, remember; intend, wish, desire" (past tense þohte, past participle geþoht), probably originally "cause to appear to oneself," from Proto-Germanic *thankjan (cognates: Old Frisian thinka, Old Saxon thenkian, Old High German denchen, German denken, Old Norse þekkja, Gothic þagkjan). Old English þencan is the causative form of the distinct Old English verb þyncan "to seem, to appear" (past tense þuhte, past participle geþuht), from Proto-Germanic *thunkjan (cognates: German dünken, däuchte). Both are from PIE *tong- "to think, feel" which also is the root of thought and thank. The two Old English words converged in Middle English and þyncan "to seem" was absorbed, except for its preservation in archaic methinks "it seems to me."
c. 1200, recenen, from Old English gerecenian "to explain, relate, recount," from Proto-Germanic *(ga)rekenojan (cognates: Old Frisian rekenia, Middle Dutch and Dutch rekenen, Old High German rehhanon, German rechnen, Gothic rahnjan "to count, reckon"), from Proto-Germanic *rakina- "ready, straightforward," from PIE *reg- "to move in a straight line," with derivatives meaning "direct in a straight line, rule" (see regal).
Intransitive sense "make a computation" is from c. 1300. In I reckon, the sense is "hold an impression or opinion," and the expression, used parenthetically, dates from c. 1600 and formerly was in literary use (Richardson, etc.), but came to be associated with U.S. Southern dialect and was regarded as provincial or vulgar.
early 14c., "to assume as the basis of argument," from Old French suposer "to assume" (13c.), probably a replacement (influenced by Old French poser "put, place") of *suppondre, from Latin supponere "put or place under; to subordinate, make subject," from sub "under" (see sub-) + ponere "put, place" (past participle positus; see position (n.)). Meaning "to admit as possible, to believe to be true" is from 1520s.
early 14c., "end an argument," from Latin concludere "to shut up, enclose," from com- "together" (see com-) + -cludere, comb. form of claudere "to shut" (see close (v.)). Meaning "reach a mental conclusion, deduce" is from late 14c., a sense also in Latin.
mid-14c., "to form a mental image of," from Old French imaginer "sculpt, carve, paint; decorate, embellish" (13c.), from Latin imaginari "to form a mental picture, picture to oneself, imagine" (also, in Late Latin imaginare "to form an image of, represent"), from imago "an image, a likeness" (see image (n.)). Sense of "suppose, assume" is first recorded late 14c.
Old English felan "to touch or have a sensory experience of; perceive, sense (something)," in late Old English "have a mental perception," from Proto-Germanic *foljan (cognates: Old Saxon gifolian, Old Frisian fela, Dutch voelen, Old High German vuolen, German fühlen "to feel," Old Norse falma "to grope"), from PIE root *pal- "to touch, feel, shake, strike softly" (cognates: Greek psallein "to pluck (the harp)," Latin palpare "to touch softly, stroke," palpitare "to move quickly"), perhaps ultimately imitative.
In Germanic languages, the specific word for "perceive by sense of touch" has tended to evolve to apply to the emotions. The connecting notion might be "perceive through senses which are not referred to any special organ." Sense of "be conscious of a tactile sensation, sense pain, pleasure, illness, etc.; have an emotional experience or reaction," developed by c. 1200, also "have an opinion or conviction;" that of "to react with sympathy or compassion" is from mid-14c. Meaning "to try by touch" is from early 14c. From late 14c. as "know (something) beforehand, to have foreknowledge of." To feel like "want to" attested from 1829.