Why did Jose give Cole the gun?

The scientists use Jose to force Cole into trying to kill the virus-spreader because they needed to learn the virus-spreader's identity.

Remember, the scientists knew Cole was at the airport because Cole had left a voicemail from there. But in the voicemail, while Cole said the Army of the 12 Monkeys did not spread the virus, he didn't say who the real culprit was.

So Jose's task at airport is to give Cole the gun and then watch who Cole goes after. But while Jose and Cole are talking, Catherine runs up and says she's figured out it is the redhead assistant to Dr Goynes who spreads the virus. At this point, Jose fades out of the scene, having got what he needed. (for confirmation see scene as written in production draft of script)

Of course, airport security shoots Cole when Cole points the gun at the virus-spreader. The scientists may have provided Cole a non-working gun to encourage this outcome.

I'm pretty sure the scientists from the future did not know Dr. Peters was carrying the virus. As explained on the movie FAQ at IMDB.com:

It is strongly implied that the scientists from the future planned Cole's death because he refused to return to his own time, and they could not allow him to disturb the past any further. They send Jose (Jon Seda) through time to give Cole an ancient gun and instructions to complete his mission. They might have intended that Cole try to use the gun in a(n) (futile) attempt to kill the carrier of the virus.The scientists probably knew it would not work and would cause security personnel to neutralize him, while the female scientist herself made sure the mission was completed.

The scientists exploited Cole. It was essential that Cole die at the airport as it provided the memory of his older self being shot, which the Scientists manipulated for his mission, as the memory was recurrent as a dream that continued to develop providing him new investigative paths or possibilities as well as drawing him closer to Dr. Raily, a psychologist whose entire job is understanding her patient's pasts and psychology.

Dr. Raily's sympathies for Cole helped their relationship develop, which was key into pushing them toward the final discoveries at the airport. With Dr. Raily identifying Dr. Peters as the trigger man from her past interaction with him, Dr. Jones from the Future stepped in. The Scientists had no interest in stopping the viral outbreak but lied to Cole about that, evident by his comment "I can't save you, nobody can." They could have.

Instead the Scientists chose to allow the virus to spread and to study the outbreak patterns and develop an anti-virus that they would provide in the future, allowing them to be seen as heroes, allowing them to continue and possibly expand their power and governance over the underground society as they returned to the surface of the Earth. This technique was used for the plot of V for Vendetta ("Imagine you have a virus and only you have the cure.")

Meanwhile, Cole is truly an essential messiah. James Cole has the same initials as Jesus Christ. However, he would probably never be known as a messiah.

Jose gave Cole the gun so that he would be killed and preserve the timeline. The scientist did not want to prevent the virus outbreak because if the virus is not spread they would have never had the chance to send anyone back to stop it, which creates a paradox.

The survivors had an immunity to the virus. Also in the end the woman that said "Jones, I'm in insurance," is one of the scientists. I think she means, sarcastically, I'm here to ensure the future. They needed the virus to find a cure, not to stop the epidemic. She is probably there to grab a vile of the virus and take it to the future.

The gun was given to James Cole so that the airport guards would shoot him. As a child, James saw himself die, so that element of the past (which was unchangeable) HAD to take place. However, I do wonder why the gun was an antique....

Cole was pardoned. The scientist stopped the virus spreader on the plane at the end. By identifying the person responsible, Cole stopped the virus from spreading and his childbood continues as normal after the airport incident. He is now free, he is a hero. That is why the Dr (his new girlfriend) smiles when she sees the young Cole standing there, she understands the circle.

This is one of the few movies that actually identifies a time travel reality. You cannot change events in your own timeline (baring parallel universes). One should look up the John Titor hoax for a good explanation of that type of time travel. Please stay with me on this, I'm veering from the movie so I can explain why I think Gilliam did what he did, then I'll get back to the movie: Personally I think Titor's explanation of time travel to alternate universes leaves too many holes. Sure, if you travel to a parallel universe you can make all the changes that you want, but if there is one of you time traveling and you make a paradoxical error (i.e. kill your own father before you're conceived) then surely there are others of you making this same error in other dimensions. In fact, the number of alternate you's could be as countless as the grains of sand on a beach. Just because there are parallel dimensions does not negate the shear truth of paradox because if you make this mistake they are also very likely making this same mistake, too much of a chance for a crossover. My thought is that if you did go back in time and you tried to kill your grandfather, well you obviously didn't succeed so your gun must have jammed. Intend all you want, you will not succeed because you cannot change events that have passed. Now, back to the movie- (hopefully you can see how that connects, because no matter what James does he cannot change anything). One thing that doesn't make sense with the explanations given in this forum is in the very end where the astrophysicist states "I'm in insurance." One person stated (as the reason why they wanted James dead) "and they could not allow [James] to disturb the past any further." I don't think you can change anything in the timeline and leaving him in the past wouldn't affect anything, so that explanation has a hole in my opinion (unless the scientists do actually believe you can make changes). It seems to me that she was there to be insurance in case James failed, therefore I don't think they planned on exploiting James into doing something to kill himself. Consider this, perhaps the movie acknowledges that you cannot change time, but the scientists in the movie may be under different assumptions, therefore they give James the gun to try to change history. At the same time other people have made a good point, this was an ancient gun so why would they give him that particular gun if they wanted him to succeed? Also, the mission is to get the virus, not to stop the virus, so when she states she is in insurance my gut tells me she had no plans on killing him or stopping the spread.

Then again, I come back to the gun, why the gun? So, ultimately here is my answer to the OP's question: This whole issue seems to be paradoxical in nature and maybe paradox in the plot is the point. This is from the same guy who brought you Brazil and The Zero Theorem, and he likes to be vague, as well as make the point you cannot really make changes (haven't seen Zero, but consider in Brazil the whole escape is a dream and the system remains intact).

Obviously this is a Terry Gilliam movie, so it is more clever and humorous to show Jose (who in the end scene of the movie is recovered from his World War I injuries) who now shows up and hands Cole a World War I gun...

Well, here goes... 1. You can't change the timeline or the outcomes, no matter what you do ( there are an infinite alternate dimensions and timelines, but the one your in cannot be altered) 2. If you're a scientist from the future looking for the original virus to extrapolate a universal cure what do you do? Only 1% of the population is left. You somehow find a person who was there the day the virus was released at Philadelphia airport. Whose memory of the entire event ( even though he doesn't understand what he saw) is indelibly burned into his mind by the trauma of a shooting that has taken place in front of his eyes as a child. You start from there and slowly piece together the who, what, where, and when. Cole is used by the scientists to recover the info as others have stated. 3. As soon as they know who the female scientist Dr. Jones steps in to get the virus. How? She shakes his hand! If you look at the scene it's very deliberate. As soon as they shake hands she recoils back stating she's in insurance. She now is a carrier of the original virus ( which he handled at the airport) She can go back and a cure extrapolated from her blood. She's not there to stop him or the spread of the virus, it's done you can't change the past. 4. Final scene of Cole's eyes; the traumatic vision burned into his psyche for future use. 5. it would be nice if the Scientists could have gotten it right from the start, but that's not the way science works. You have a theory, you go down a path, it's wrong, reassess, learn from your mistakes and try over and over till you get the answer, hopefully.