The Collaborative International Dictionary of English v.0.48:

High \High\, a. [Compar. Higher; superl. Highest.] [OE.
   high, hegh, hey, heh, AS. he['a]h, h?h; akin to OS. h?h,
   OFries. hag, hach, D. hoog, OHG. h?h, G. hoch, Icel. h?r, Sw.
   h["o]g, Dan. h["o]i, Goth. hauhs, and to Icel. haugr mound,
   G. h["u]gel hill, Lith. kaukaras.]
   1. Elevated above any starting point of measurement, as a
      line, or surface; having altitude; lifted up; raised or
      extended in the direction of the zenith; lofty; tall; as,
      a high mountain, tower, tree; the sun is high.
      [1913 Webster]

   2. Regarded as raised up or elevated; distinguished;
      remarkable; conspicuous; superior; -- used indefinitely or
      relatively, and often in figurative senses, which are
      understood from the connection; as
      (a) Elevated in character or quality, whether moral or
          intellectual; pre["e]minent; honorable; as, high aims,
          or motives. "The highest faculty of the soul."
          --Baxter.
      (b) Exalted in social standing or general estimation, or
          in rank, reputation, office, and the like; dignified;
          as, she was welcomed in the highest circles.
          [1913 Webster]

                He was a wight of high renown.    --Shak.
      (c) Of noble birth; illustrious; as, of high family.
      (d) Of great strength, force, importance, and the like;
          strong; mighty; powerful; violent; sometimes,
          triumphant; victorious; majestic, etc.; as, a high
          wind; high passions. "With rather a high manner."
          --Thackeray.
          [1913 Webster]

                Strong is thy hand, and high is thy right hand.
                                                  --Ps. lxxxix.
                                                  13.
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                Can heavenly minds such high resentment show?
                                                  --Dryden.
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      (e) Very abstract; difficult to comprehend or surmount;
          grand; noble.
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                Both meet to hear and answer such high things.
                                                  --Shak.
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                Plain living and high thinking are no more.
                                                  --Wordsworth.
      (f) Costly; dear in price; extravagant; as, to hold goods
          at a high price.
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                If they must be good at so high a rate, they
                know they may be safe at a cheaper. --South.
      (g) Arrogant; lofty; boastful; proud; ostentatious; --
          used in a bad sense.
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                An high look and a proud heart . . . is sin.
                                                  --Prov. xxi.
                                                  4.
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                His forces, after all the high discourses,
                amounted really but to eighteen hundred foot.
                                                  --Clarendon.
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   3. Possessing a characteristic quality in a supreme or
      superior degree; as, high (i. e., intense) heat; high (i.
      e., full or quite) noon; high (i. e., rich or spicy)
      seasoning; high (i. e., complete) pleasure; high (i. e.,
      deep or vivid) color; high (i. e., extensive, thorough)
      scholarship, etc.
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            High time it is this war now ended were. --Spenser.
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            High sauces and spices are fetched from the Indies.
                                                  --Baker.
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   4. (Cookery) Strong-scented; slightly tainted; as, epicures
      do not cook game before it is high.
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   5. (Mus.) Acute or sharp; -- opposed to grave or low; as,
      a high note.
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   6. (Phon.) Made with a high position of some part of the
      tongue in relation to the palate, as [=e] ([=e]ve), [=oo]
      (f[=oo]d). See Guide to Pronunciation, [sect][sect] 10,
      11.
      [1913 Webster]

   High admiral, the chief admiral.

   High altar, the principal altar in a church.

   High and dry, out of water; out of reach of the current or
      tide; -- said of a vessel, aground or beached.

   High and mighty arrogant; overbearing. [Colloq.]

   High art, art which deals with lofty and dignified subjects
      and is characterized by an elevated style avoiding all
      meretricious display.

   High bailiff, the chief bailiff.

   High Church, & Low Church, two ecclesiastical parties in
      the Church of England and the Protestant Episcopal Church.
      The high-churchmen emphasize the doctrine of the apostolic
      succession, and hold, in general, to a sacramental
      presence in the Eucharist, to baptismal regeneration, and
      to the sole validity of Episcopal ordination. They attach
      much importance to ceremonies and symbols in worship.
      Low-churchmen lay less stress on these points, and, in
      many instances, reject altogether the peculiar tenets of
      the high-church school. See Broad Church.

   High constable (Law), a chief of constabulary. See
      Constable, n., 2.

   High commission court, a court of ecclesiastical
      jurisdiction in England erected and united to the regal
      power by Queen Elizabeth in 1559. On account of the abuse
      of its powers it was abolished in 1641.

   High day (Script.), a holy or feast day. --John xix. 31.

   High festival (Eccl.), a festival to be observed with full
      ceremonial.

   High German, or High Dutch. See under German.

   High jinks, an old Scottish pastime; hence, noisy revelry;
      wild sport. [Colloq.] "All the high jinks of the county,
      when the lad comes of age." --F. Harrison.

   High latitude (Geog.), one designated by the higher
      figures; consequently, a latitude remote from the equator.


   High life, life among the aristocracy or the rich.

   High liver, one who indulges in a rich diet.

   High living, a feeding upon rich, pampering food.

   High Mass. (R. C. Ch.) See under Mass.

   High milling, a process of making flour from grain by
      several successive grindings and intermediate sorting,
      instead of by a single grinding.

   High noon, the time when the sun is in the meridian.

   High place (Script.), an eminence or mound on which
      sacrifices were offered.

   High priest. See in the Vocabulary.

   High relief. (Fine Arts) See Alto-rilievo.

   High school. See under School.

   High seas (Law), the open sea; the part of the ocean not in
      the territorial waters of any particular sovereignty,
      usually distant three miles or more from the coast line.
      --Wharton.

   High steam, steam having a high pressure.

   High steward, the chief steward.

   High tea, tea with meats and extra relishes.

   High tide, the greatest flow of the tide; high water.

   High time.
      (a) Quite time; full time for the occasion.
      (b) A time of great excitement or enjoyment; a carousal.
          [Slang]

   High treason, treason against the sovereign or the state,
      the highest civil offense. See Treason.
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   Note: It is now sufficient to speak of high treason as
         treason simply, seeing that petty treason, as a
         distinct offense, has been abolished. --Mozley & W.

   High water, the utmost flow or greatest elevation of the
      tide; also, the time of such elevation.

   High-water mark.
      (a) That line of the seashore to which the waters
          ordinarily reach at high water.
      (b) A mark showing the highest level reached by water in a
          river or other body of fresh water, as in time of
          freshet.

   High-water shrub (Bot.), a composite shrub (Iva
      frutescens), growing in salt marshes along the Atlantic
      coast of the United States.

   High wine, distilled spirits containing a high percentage
      of alcohol; -- usually in the plural.

   To be on a high horse, to be on one's dignity; to bear
      one's self loftily. [Colloq.]

   With a high hand.
      (a) With power; in force; triumphantly. "The children of
          Israel went out with a high hand." --Ex. xiv. 8.
      (b) In an overbearing manner, arbitrarily. "They governed
          the city with a high hand." --Jowett (Thucyd. ).

   Syn: Tall; lofty; elevated; noble; exalted; supercilious;
        proud; violent; full; dear. See Tall.
        [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

higher \higher\ adj. prenom.
   1. advanced in complexity or elaboration; as, higher
      mathematics.
      [WordNet 1.5]

   2. of or pertaining to education beyond the secondary level;
      as, higher education; higher learning.
      [WordNet 1.5]

WordNet (r) 3.0 (2006):

higher
    adj 1: advanced in complexity or elaboration; "higher finance";
           "higher mathematics"
    2: of education beyond the secondary level; "higher education";
       "higher learning"

Moby Thesaurus II by Grady Ward, 1.0:

38 Moby Thesaurus words for "higher":
   a cut above, above, ahead, ascendant, better, capping, chosen,
   distinguished, eclipsing, eminent, exceeding, excellent, excelling,
   finer, greater, highest, in ascendancy, in the ascendant, major,
   marked, of choice, one up on, outstanding, over, overlying, rare,
   rivaling, super, superincumbent, superior, surpassing, topping,
   transcendent, transcendental, transcending, upmost, upper,
   uppermost