New Testament word study: agnōsia: “ignorance, ignorant, ignorantly”:

According to Vine, the Greek agnōsia denotes "ignorance" as directly opposed to gnosis (which signifies "knowledge" as a result of observation and experience); a (i.e. alpha) is used as a negative together with ginosko "to know”. [Compare the English: "agnostic"]

The word occurs only twice in the New Testament. According to Vine, in both passages reprehensible "ignorance" is suggested. But there appears, in our view, to be a deeper purpose behind this word.

1 Corinthians 15:34

“Be not led astray; evil communications corrupt good manners; awake up, as is right, and sin not; for certain have an ignorance[agnōsia] of God [KJV: “no knowledge”]; for shame to you I say it”. (1 Corinthians 15:33-34; Young’s Literal Translation; emphasis Rotherham)

1 Peter 2:15

“Dearly beloved, I beseech you “as strangers and pilgrims” [quote from Psalm 39:12] abstain from fleshly lusts, which war against the soul;

Having your conversation [Rotherham: “behaviour”] honest among the Gentiles: that, whereas they speak against you as evildoers, they may by your good works, which they shall behold, glorify God “in the day of visitation” [quote from Isaiah 10:1] Submit yourselves to every ordinance of man for the Lord's sake: whether it be to the king, as supreme; Or unto governors, as unto them that are sent by Him for the punishment of evildoers, and for the praise of them that do well.

For so is the will of God, that with well doing ye may put to silence the ignorance [agnōsia] of foolish men: As free, and not using your liberty for a cloke of maliciousness, but as the servants of God”. (1 Peter 2: 12-16, KJV; emphasis Rotherham)

The Septuagint (LXX) only uses agnōsia once throughout the whole of the Old Testament, namely when Elihu says in Job 35:16:

“Therefore doth Job open his mouth in vanity; He multiplieth words without knowledge”;

“without knowledge” (LXX: agnōsia) is the Hebrew bĕliy da`ath  with da`ath  here meaning “intelligence, understanding, wisdom”; hence “without da`ath” means “foolishness” (Gesenius, citing this very passage in Job). And note that Job did know and believed in God, but did not know Him sufficiently, wherefore the whole purpose of Elihu’s great discourse was to get Job to know the GREATNESS of God]

We may discern why the Holy Spirit has chosen agnosia in the two texts above, by comparing with a similar word also often translated ignorance”, namely the Greek agnoia.

The word agnoia means “lack of knowledge, ignorance,

• a) especially of divine things,
• b) of moral blindness” (Strong/Thayer);

It occurs 4 times in the New Testament:

1. “And now, brethren, I wot that through ignorance [agnoia] ye did it, as did also your rulers”. (Acts 3:17, KJV, emphasis Rotherham)

2. “And the times of this ignorance [agnoia] God winked at; but now commandeth all men every where to repent: Because He hath appointed a day, in the which He “will judge the world in righteousness” [quoting Psalms 9:8, 96:13; 98:9] by that man whom He hath ordained; whereof He hath given assurance unto all men, in that He hath raised Him from the dead” (Acts 17:30-31, KJV; emphasis Rotherham)

3. “This I say therefore, and testify in the Lord, that ye henceforth walk not as other Gentiles walk,
in the vanity of their mind,
Having the understanding darkened,
being alienated from the life of God through the ignorance [agnoia] that is in them,
because of the blindness of their heart:
Who being past feeling have given themselves over unto lasciviousness, to work all uncleanness with greediness.
But ye have not so learned Christ” (Ephesians 4:17-20, KJV; emphasis Rotherham)

4. “Wherefore gird up the loins of your mind, be sober, and hope to the end for the grace that is to be brought unto you at the revelation of Jesus Christ; As obedient children[Rotherham margin note: “children of obedience, i.e. persons who have so to speak derived their being from obedience”] not fashioning yourselves according to the former lusts in your ignorance [agnoia]:
But as He which hath called you is holy, so be ye holy in all manner of conversation [“i.e. behaviour”] Because it is written, “Be ye holy; for I am holy” [quoting Leviticus 11:44; 19:2; 20:7] (1 Peter 1:13-16, KJV; emphasis Rotherham)

Finally, we also have agnoēma which occurs only once in the New Testament:

“But into the second [i.e. the inner room or Most Holy Place] went the high priest alone once every year, not without blood, which he offered for himself, and for the errors [literally: “ignorances”: agnoēmatōn, i.e. plural of  agnoēma] of the people”.
(Hebrews 9:7, KJV; emphasis Rotherham)

The distinction between (1) agnōsia, and (2) agnoia and agnoēma, is as follows:

• agnōsia denotes "ignorance" as opposed to gnosis (which signifies "knowledge" as a result of observation and experience) and is a composite of a (i.e. alpha) used as a negative, together with ginosko "to know”;

• both agnoia and agnoēma derive from the verb agnoeō (“to be ignorant, not to know; not to understand, unknown; to err or sin through mistake, to be wrong” – Strong/Thayer);
  and agnoeō is a composite of a (i.e. alpha) used as a negative, together
  with noeō  ( “to perceive with the mind [nous], to understand, to have
  understanding; to think upon, heed, ponder, consider” – Strong/Thayer)

 
[Regarding nous (“the mind”) Vine has this important note:
“Mind (Noun and Verb):”mind,” denotes, speaking generally, the seat of reflective consciousness, comprising the faculties of perception and understanding, and those of feeling, judging and determining.
Its use in the NT may be analyzed as follows: it denotes

(a) the faculty of knowing, the seat of the understanding (Luk 24:45; Rom 1:28; 14:5; 1Cr 14:15, 19; Eph 4:17; Phl 4:7; Col 2:18; 1Ti 6:5; 2Ti 3:8; Tts 1:15; Rev 13:18; 17:9);

(b) counsels, purpose, Rom 11:34 (of the “mind” of God); 12:2; 1Cr 1:10; 2:16, twice
      (1) of the thoughts and counsels of God,
      (2) of Christ, a testimony to His Godhood; Eph 4:23;

(c) the new nature, which belongs to the believer by reason of the new birth, Rom 7:23, 25, where it is contrasted with “the flesh,” the principle of evil which dominates fallen man.
Under (b) may come 2Th 2:2, where it stands for the determination to be steadfast amidst afflictions, through the confident expectation of the day of rest and recompense mentioned in the first chapter”]

HENCE agnōsia in our two texts above in particular indicates the ignorance of believers arising from failure to obtain knowledge or further knowledge, especially that knowledge of GOD which is obtainable, including that such “knowledge” is obtained not by mere intellectual activity, but also by operation of the Holy Spirit consequent upon acceptance of Christ.

The ignorance referred to in the texts using agnoiais, by contrast, that particular ignorance which arises from willful blindness; hence the four  texts refer either to the “blindness” state of Gentiles, or of Israel.
This particular state of ignorance is often resulting from:

• man’s willful blindness and rejection of God (Romans 1:28 et seq.);
• the god of this world, even Satan, blinding men (2 Corinthians 4:4);
• in the case of Israel – blindness also from their own stubborn hearts and stiff-necked rebellions against God (2 Corinthians 3:14);

The word agnoēma, however, appears to be specially reserved by the Divine Author for ignorance in the sense of leading to error.
It also occurs only once in the Septuagint (LXX), in Genesis 43:12, when Joseph’s brethren intend to return to Egypt to buy more corn, and Jacob says regarding the money which was supposed to be payment for the corn they had already bought, and which they found hidden in their sacks:

“and take double money in your hand; and the money that was returned in the mouth of your sacks carry again in your hand; peradventure it was an oversight[mishgeh]” (ASV)

[mishgeh: “oversight”, from the root shagah: “to go astray, stray, err,  to commit sin of ignorance or inadvertence, err (ignorantly)”;  LXX translates mishgeh with agnoēma]

The interested Bible-student can follow through on the above by viewing the 25 New Testament occurrences of agnoeō (“to be ignorant, not to know; not to understand, unknown; to err or sin through mistake, to be wrong” – Strong/Thayer”) at www.blueletterbible.org

Similarly, the 223 New Testament occurrences of ginōskō can be viewed at www.blueletterbible.org