But what exactly is the social contract that they claim has power over individuals? What justification is used to apply this social contract in government legislation? Is there even a justification for a social contract?
Social contract arguments typically posit that individuals have consented, either explicitly or tacitly, to surrender some of their freedoms and submit to the authority of the ruler or magistrate (or to the decision of a majority), in exchange for protection of their remaining rights.1
To be short the social contract is a utilitarian justification. The greater good of all of society justifies its very use. If taking from the rich and giving to others increases the greater good of everyone else it's justified.
To even attempt to give a justification to the social contract we should first start with trying to define what society exactly is.
Here there will lie some disagreeing on what society is. But what you cannot argue is that society itself only contains the individuals that comprise it. Basically, society could not exist without individuals. Society therefor is an imaginary construct of the mind, there is no physical characteristics of "society". When we use the word society it is only to save time in describing any given group of individuals. For example when talking about a baseball team, we say baseball team because it is much easier to express than naming every individual that partakes in the actions of hitting a ball, throwing a ball, and catching a ball. There is no such physical entity as a baseball team, only a physical entity of the individuals that partake in the actions associated with the sport of baseball. In this regard it is then simpler to realize that society doesn't exist, it's only an expression used to talk about the multitude of individuals within some certain area.
Next it is as vitally important to discuss what contract is and how it is formed.
When two or more individuals express an interest in cooperating with one another for any given circumstance they are voluntarily (without coercion) accepting some type of agreement. If this agreement wasn't mutually beneficial for everyone involved it would simply not be agreed upon. When these agreements are made these individuals have formed a contract.
For example, Matt agrees to pay 100$ to Sam to renovate his yard. The payment can come before the labor is done or after, it doesn't matter. If Sam believes this 100$ is a greater want than the labor and time he will have to expend he'll agree to the terms and vice-versa if Matt believes the labor for the renovation is a greater want than the 100$ he too will agree to the terms. The two have entered a voluntary contract and neither are forced partake in the contract. So Sam will do the work and Matt will pay him. If Sam takes the money before the labor and doesn't renovate the yard he has stolen from Matt and if Matt doesn't pay Sam after the labor is done he has stolen from Sam. In both cases contracts have been violated.
The main point to be made here is that contract can only be made to protect physical properties. In our case it would be Sam's physical labor and Matt's money which comes from his physical labor.
Therefor, if society doesn't even exist outside of an imaginary construct then it cannot possibly have any physical properties to which a contract can be created. Society only comprises of individuals who can make contract with other individuals but society itself cannot hope to create a contract with society itself or over other individuals.
You disagree that society can't claim a contract over others? Fine. But there's also the other crucial entity that lies within any type of legal contract, the absence of coercion. You cannot possibly have a legal contract when someone is forced into it. Matt cannot point a gun at Sam's head and tell him if he doesn't renovate his yard he will shoot him. Sam would most likely choose to renovate the yard instead of getting shot, but since he was coerced into the situation Matt cannot claim he had a contract with Sam. Matt's threat of violence has infringed on the rights of Sam and therefore the contract is illegitimate.
So to those that say "social contract" is legitimate are completely wrong merely on the fact that if I disagree with this "contract" I must be forced to abide by it violating what contracts really are.
Furthermore, this "social contract" is a completely subjective term. Since it is a subjective term then my idea of a "social contract" or another person's idea of what it is has the same moral and legitimate claim to force it over you. It wouldn't matter if you thought my idea was right or wrong since there is no right or wrong in subjectivity.
The only legitimate contracts are those made by individuals voluntarily. If this is true then the most important law to uphold is the axiom of self-ownership. Since all individuals have self-ownership the greater good of all (which is utilitarian logic) can only be held if self-ownership is protected.