Psyche: Animals and Souls

1 Corinthians 15:45 (NASB). So also it is written, “The first man, Adam, became a living soul ['psychē'].” The last Adam became a life-giving spirit ['pneuma'].

I was doing a study on another subject and ran across the definition of the Greek word for soul.

I cannot see around the realization that animals have to, by definition, have a soul.

The word, "psyche," pre-exists the word, "soul," so it must have precedence.

Matthew 10:28 (NASB).  Do not fear those who kill the body but are unable to kill the soul; but rather fear Him who is able to destroy both soul and body in hell.

In addition, I not see how that animals do not have immortal souls, either.

I must remember to treat my dog better. Maybe strip steaks for Saturday dinner.

psychē
Strong: G5590
GK: G6034
breath; the principle of animal life [emphasis added]; the life, Mt. 2:20; 6:25; Mk. 3:4; Lk. 21:19; Jn. 10:11; an inanimate being, 1 Cor. 15:45; a human individual, soul, Acts 2:41; 3:23; 7:14; 27:37; Rom. 13:1; 1 Pet. 3:20; the immaterial soul, Mt. 10:28; 1 Pet. 1:9; 2:11, 25; 4:19; the soul as the seat of religious and moral sentiment, Mt. 11:29; Acts 14:2, 22; 15:24; Eph. 6:6; the soul, as a seat of feeling, Mt. 12:18; 26:38; the soul, the inner self, Lk. 12:19

pneuma
Strong: G4151
GK: G4460
wind, air in motion, Jn. 3:8; breath, 2 Thess. 2:8; the substance spirit, Jn. 3:6; a spirit, spiritual being, Jn. 4:24; Acts 23:8, 9; Heb. 1:14; a bodiless spirit, specter, Lk. 24:37; a foul spirit, δαιμόνιον, Mt. 8:16; Lk. 10:20; spirit, as a vital principle, Jn. 6:63; 1 Cor. 15:45; the human spirit, the soul, Mt. 26:41; 27:50; Acts 7:59; 1 Cor. 7:34; Jas. 2:26; the spirit as the seat of thought and feeling, the mind, Mk. 8:12; Acts 19:21; spirit, mental frame, 1 Cor. 4:21; 1 Pet. 3:4; a characteristic spirit, an influential principle, Lk. 9:55; 1 Cor. 2:12; 2 Tim. 1:7; a pervading influence, Rom. 11:8; spirit, frame of mind, as distinguished from outward circumstances and action, Mt. 5:3; spirit as distinguished from outward show and form. Jn. 4:23; spirit, a divinely bestowed spiritual frame, characteristic of true believers, Rom. 8:4; Jude 19; spirit, latent spiritual import, spiritual significance, as distinguished from the mere letter, Rom. 2:29; 7:6; 2 Cor. 3:6, 17; spirit, as a term for a process superior to a merely natural or carnal course of things, by the operation of the Divine Spirit, Rom. 8:4; Gal. 4:29; a spiritual dispensation, or a sealing energy of the Holy Spirit, Heb. 9:14; the Holy Spirit, Mt. 3:16; 12:31; Jn. 1:32, 33; a gift of the Holy Spirit, Jn. 7:39; Acts 19:2; 1 Cor. 14:12; an operation or influence of the Holy Spirit, 1 Cor. 12:3; a spiritual influence, an inspiration, Mt. 22:43; Lk. 2:27; Eph. 1:17; a professedly divine communication, or, a professed possessor of a spiritual communication, 1 Cor. 12:10; 2 Thess. 2:2; 1 Jn. 4:1, 2, 3

Word study from Biblegate.com; Mounce Reverse-interlinear New Testament.