A common charge one finds against the Book of Mormon is that the Book of Mormon teaches Modalism (basically, the view that the Father and the Son are the same person; it has different manifestations [pun intended] historically and in modern times). The main text critics (e.g. Dan Vogel) cite to support this claim is Mosiah 15:1-4:
(1)And now Abinadi said unto them: I would that ye should understand that God himself shall come down among the children of men, and shall redeem his people. (2)And because he dwelleth in flesh he shall be called the Son of God, and having subjected the flesh to the will of the Father, being the Father and the Son—(3)The Father because he was conceived by the power of God and the Son, because of the flesh; thus becoming the Father and the Son—(4)And they are one God, yea, the very Eternal Father of heaven and of earth.
There have been many excellent rebuttals to the claim that this, and similar texts (e.g. Ether 3:14) teach that the Father and the Son are the same person (I will list some of them at the end of this post). For this post, I will focus on investigating this issue applying the “Indiscernibility of Identicals.” Simply put, the Indiscernibility of Identicals states that for any x and any y, x=y if x and y (1) never have differed, (2) don't differ, (3) will not ever differ, and (4) could not differ.
In Mosiah 15:2, we read of how Jesus will “be called the Son of God, and having subjected the flesh to the will of the father,” and in 15:7 Abinadi states that “[Jesus] shall be crucified, and slain, the flesh becoming subject even unto death, the will of the Son being swallowed up in the will of the Father.” In these two verses, there are two wills; the Father’s and the Son’s, the latter submitting to the former. Each person has their own will, showing that the person of the Son is distinguished from the person of the Father in that Jesus submits his will to God the Father, but the latter does not submit his will to himself and/or the Son.
Furthermore, in 15:8, we read another text that differentiates the Father and the Son:
And thus God [the Father] breaketh the bands of death, having gained the victory over death; giving the Son power to make intercession for the children of men.
In this verse, the Father gives to the Son the power to make intercession for the children of men (cf. Heb 7:24-25; Rom 8:33-34; 2 Nephi 2:9-10). It is the Son, not the Father, who offers intercession, and it is offered by the Son to the Father.
Applying the Indiscernibility of Identicals, we can see that, even within the context of Mosiah 15 itself, ignoring the plethora of texts in the Book of Mormon that differentiates between the person of the Father and the Son, Jesus and His Father differ from one another and cannot be identified as being one and the same person.
Online Resources on Modalism and the Book of Mormon
Blake T. Ostler, "Re-vision-ing the Mormon Concept of Deity"
Ari D. Bruening and David L. Paulsen, "The Development of the Mormon Understanding of God: Early Mormon Modalism and Early Myths"
Martin S. Tanner, "Review of 'Book of Mormon Christology' by Melodie Moench Charles"