That's a very good question! But also a complicated one. It's not so easy to answer with a simple 'yes' or 'no'. This is one of those gray areas of the Bible: it doesn't say clearly, one way or the other. However, there are many hints in the Bible that allow us to arrive at a solid conclusion.
First, we do know that animals have the "breath of life", just like we do (
Genesis 6:17,
Genesis 7:15,
Genesis 7:22). However, they are not made "in the image of God" like we are (
Genesis 1:26-27 ), so there is already a big difference. Thus, if they do have a soul, it is a very different kind of "soul" than we have, just as God's "soul" is very different from our souls (yes, God has a soul:
Leviticus 26:11,
Jeremiah 32:41). However, the "breath of life" corresponds to the life-giving "spirit", not the "soul". Some theologians say that "spirit" and "soul" are the same thing, but upon careful analysis, it is clear that they are actually two different things, Biblically.
Also, in the Bible, there is no mention at all of animals in Hades, Sheol, Hell, or Heaven, which is very interesting. Of course, that doesn't mean they won't be there ("absence of evidence is not evidence of absence"). But it's a strong indicator that they won't. They are mentioned many times in the Bible on Earth, from the beginning (they were even created before man), and up to the end of the period of time known as "the millennium" (just prior to the end of everything and the beginning of eternity), but then there is no further mention of them.
However, the biggest clue to the answer, is that Jesus did not die for animals: only for people. Animals have no morals or ethics, they have no sense of right and wrong, so it would not be "justice" to condemn them to hell, and they cannot go to heaven, since only those who consciously recognize Jesus as their personal savior and submit to him, can go to heaven. So it becomes clear that there will be no animals in heaven or hell, which are the only two possibilities, according to the Bible.
That, in turn, implies that any "soul" an animal might have, can die... unlike our souls, or the souls of angels, which are immortal. Our souls cannot die: they go to either heaven or to hell, but they do not die. The same is true for the souls of angels: they stay in heaven, or they go to hell. (Yes, hell is a real place!)
But it seems that the "souls" of animals
can die: they go neither to heaven nor to hell.
Therefore, if animals have a soul, it would be a type of mortal soul, very different from what we human beings have, since we are "made in the image of God" and are therefore immortal.
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