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"Did Christ (being 'fully God and fully human') exist with a human soul, as a human person?"

First, two possible syllogisms demonstrating that the proposition "Christ existed, being fully human, with a human soul as a human person" is either true or false by the Law of the Excluded Middle, and must result in either Nestorianism or Apollinarianism, might be:

—1st Syllogism (Nestorianism)—
Premise 1: A human person has a human soul, a human body, a human will, and a human nature.

Premise 2: Christ was fully God and fully human.

Premise 3: The proposition "Christ existed, being fully human, with a human soul as a human person" is either true or false by the Law of the Excluded Middle.

1st Conclusion: If the proposition "Christ existed, being fully human, with a human soul as a human person" is true, then it follows that Christ had a human soul as a human person, and this would result in Nestorianism. 
According to Nestorianism, Christ is considered to have two persons, or natures, in one body, which follows from the first and second premises. 
Nestorians believe that the human and divine aspects of Christ are distinct, but united in the one person of Jesus Christ. 
This belief is different from the traditional view held by many Christians, who believe that Christ has only one person, or nature, that is both fully human and fully divine.

—2nd Syllogism (Apollinarianism)—
Premise 1: A human person has a human soul, a human body, a human will, and a human nature.

Premise 2: Christ was fully God and fully human.

Premise 3: The proposition "Christ existed, being fully human, with a human soul as a human person" is either true or false by the Law of the Excluded Middle.

Conclusion: If the proposition "Christ existed, being fully human, with a human soul as a human person" is false, then it follows that Christ did not have a human soul as a human person, and this would result in Apollinarianism. 
According to the doctrine of Apollinarianism, Christ does not have a human soul. 
The doctrine holds that the human and divine natures of Christ were united in such a way that the human nature was entirely subsumed by the divine nature, with only the divine Logos (the second person of the Trinity) remaining as the conscious subject of Christ's person. 
In other words, according to Apollinarianism, Christ's human nature was completely replaced by his divine nature, and he did not possess a fully human mind/consciousness or soul.

To demonstrate that affirmation of the first two premises, in the denial of the previous two conclusions (i.e. the orthodox position of the Trinity) is an incoherency in light of the Law of Identity and the Classical Laws of Logic, we can use the following syllogism:

—3rd Syllogism (Classical Incoherence)—
Premise 1: According to the Classical Laws of Logic, for any entity x, if x is identical to y, then any property or predicate that is true of x must also be true of y.

Premise 2: The orthodox position of the Trinity affirms that Christ has a human will, human soul, human nature, and human body.

Premise 3: A human person is identical to a human will, human soul, human nature, and human body.

3rd Conclusion: Therefore, according to the Law of Identity and the Classical Laws of Logic, Christ (in the orthodox Trinity) is a human person, and the orthodox position of the Trinity is an incoherency as it denies that Christ is a human person while being fully human.

This syllogism demonstrates that the orthodox position of the Trinity is an incoherency in light of the Law of Identity and the Classical Laws of Logic. 
The first premise states that according to the Classical Laws of Logic, for any entity x, if x is identical to y, then any property or predicate that is true of x must also be true of y. 
The second premise states that the orthodox position of the Trinity affirms that Christ has the properties of a human person. The third premise states that a human person is identical to a human will, human soul, human nature, and human body. 
Therefore, the conclusion of the syllogism is that according to the Law of Identity and the Classical Laws of Logic, Christ (in the orthodox Trinity) is a human person, and the orthodox position of the Trinity is an incoherency as it denies that Christ is a human person while being fully human.

Another possible syllogism demonstrating that a position which neither affirms nor denies the proposition "Christ existed, being fully human, with a human soul as a human person", but affirms both "A human person has a human soul, a human body, a human will, and a human nature" and "Christ was fully God and fully human" is an apparent incoherency and contradiction, might be:

—4th Syllogism (True/Nestorianism, False/Apollinarianism, Trinity/Incoherency)—
Premise 1: A human person has a human soul, a human body, a human will, and a human nature.

Premise 2: Christ was fully God and fully human.

Premise 3: The proposition "Christ existed, being fully human, with a human soul as a human person" is either true or false by the Law of the Excluded Middle.

Premise 4: It is incoherent and contradictory to affirm both "A human person has a human soul, a human body, a human will, and a human nature" and "Christ was fully God and fully human" without also affirming or denying the proposition "Christ existed, being fully human, with a human soul as a human person".

4th Conclusion: If a position affirms both "A human person has a human soul, a human body, a human will, and a human nature" and "Christ was fully God and fully human" without also affirming or denying the proposition "Christ existed, being fully human, with a human soul as a human person", then it is an apparent incoherency and contradiction.

This is because affirming both that Christ was fully God and fully human, and that a human person has a human soul, then by the Law of Identity, Christ had a human soul as a human person, if he was fully human. 
But if the proposition "Christ existed, being fully human, with a human soul as a human person" is neither affirmed nor denied, then it is unclear whether Christ had a human soul or not, which creates an apparent incoherency and contradiction in the position that "Christ was fully human and fully God".
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