Part of the ongoing Verse Quick Reference project.

Joh 10:26 but you do not believe because [γάρ] you are not among my sheep.

Calvinists think this verse is about the mechanics of belief. People believe because they are predestined into Jesus. The people believe “because” they are among the sheep. They are not sheep “because” they believe. James White says as much:

Jesus says that the reason for unbelief is really rather simple: those who are not of His sheep do not believe. The standard human-centered idea is that we believe, and hence become Christ’s sheep. The Lord reverses this: those who are His sheep believe; those who are not His sheep do not believe. The decision as to who will constitute Christ’s sheep lies in the will of the Father, not the creature, man.
http://vintage.aomin.org/Believer.html

Michael Heiser offers a word of advice for people who want to take the Bible seriously: “Never base any doctrine on a preposition because they are notoriously elastic in meaning and translation”. Indeed, White’s take on John 10:26 relies on γάρ being a mechanical cause of what precedes it. But translating that word in that manner makes all sorts of weird renderings of other verses:

Mat 5:12 Rejoice and be exceedingly glad, for great is your reward in heaven, [γάρ] so they persecuted the prophets who were before you.

In Matthew 5:12, the word would be better rendered as “since”. The idea is that Jesus’ hearers can be safe in their beliefs because they know that the saints have great rewards as well. It is not that the persecution of the prophets causes Jesus’ hearers to get greater rewards.

In Mark 1:16, the word is used to illustrate:

Mar 1:16 And as He walked by the Sea of Galilee, He saw Simon and Andrew his brother casting a net into the sea; [γάρ] they were fishermen.

Being a fisherman does not “cause” them to caste their nets. It is illustrating that they were both casting their nets and are fishermen. They chose to be fishermen, and part of being a fisherman is the normal activity of fishing. The γάρ is using one sentence to reinforce the other, both meaning the same thing.

Thayer’s Greek Lexicon explains that γάρ is a very versatile word:

Now since by a new affirmation not infrequently the reason and nature of something previously mentioned are set forth, it comes to pass that, by the use of this particle, either the reason and cause of a foregoing statement is added, whence arises the causal or argumentative force of the particle, for (Latinnam,enim; German denn); or some previous declaration is explained, whence γάρ takes on an explicative force: for, the fact is, namely (Latinvidelicet, German nämlich). Thus the force of the particle is either conclusive, or demonstrative, or explicative and declaratory;

One use, as already discussed is illustrative or explaining:

III. It serves to explain, make clear, illustrate, a preceding thought or word: for equivalent to that is, namely;

Using “therefore” in the translation, a valid choice as confirmed by Strong’s Greek Dictionary as well as Thayer’s, renders the sentence:

Joh 10:26 but you do not believe [therefore] you are not among my sheep.
But, in the same manner of Mar 1:16 , the sentence can be rendered:
Joh 10:26 but you do not believe [therefore] for are not among my sheep. [compare to “for they were fishermen”]

This rendering actually fits the context much better. In context, Jesus is being approached by people pretending to be his disciples but who doubt Jesus:

Joh 10:23 and Jesus was walking in the temple, in the colonnade of Solomon.
Joh 10:24 So the Jews gathered around him and said to him, “How long will you keep us in suspense? If you are the Christ, tell us plainly.”

Instead of playing their game, Jesus explains that he has already answered, they did not believe him, and exposes them as not being his followers:

Joh 10:25 Jesus answered them, “I told you, and you do not believe. The works that I do in My Father’s name, they bear witness of Me.
Joh 10:26 But you do not believe, [therefore] you are not of My sheep, as I said to you.
Joh 10:27 My sheep hear My voice, and I know them, and they follow Me.

Jesus is not teaching total depravity, irresistible grace, or predestination. Why would he do that? What is he gaining by telling people who have no hope in the world that they are eternally without hope? Is that reading better than Jesus criticizing people who reject him?

The context makes clear that Jesus is exposing that the people have chosen not to believe in him.