In Chapter 5 of Second Treatise of Government, Locke argues that man has a right to appropriate common property by the act of mixing his labor with it. Although he provides many justifications for and defenses of this claim, ultimately it is logically invalid, and his own defenses of this claim serve to demonstrate its falsehood.
	Locke believes that the right of property is a natural right, and his core argument in Chapter 5 is focused on how a man can justly appropriate common property for his own. Locke believes that in the beginning, all property and all the bounties of the earth are held in common amongst all people (5.26). Yet every man has the right to secure what is necessary for his own survival (5.25). Locke assumes that there are enough resources for everyone to sustain themselves without harming the self-preservation of others (5.27), so long as they take only what they are able to use and enjoy, and not what will spoil and go to waste (5.31). Upon establishing these initial conditions, he goes on to describe the method by which man makes common property his own. His argument can be summarized as thus:
Premise 1: If X is constituted by something I own, then I own X.
Premise 2: If I have an exclusive use right to Y, I own Y.
According to Locke, every man owns his own body (5.27). If my labor is a product of my body, then I also own my labor (Premise 1). Uncultivated land has almost no value of its own (5.36). By mixing my labor with uncultivated land, it becomes many times more valuable (5.37). The value in this mixture is almost entirely constituted by my labor (5.40). Thus if someone else were to make a claim to this mixture, they would be making a claim to my labor. Since I have exclusive ownership over my labor, nobody else has any claim to it. Since nobody else therefore has a claim to the mixture, I have an exclusive use right to that mixture. Since I have an exclusive use right to the mixture, I own the mixture (Premise 2). Since nearly the entire value of this land mixture is constituted by my labor, and I own my labor, I own the land mixture. (Premise 1).
	Essentially, because I have forced together my labor with something which was previously held in common, no one else has any claim to it (leaving me the sole possessor), and because I have added to the value by combining it with what was already mine (my labor), I own the entire thing. This two pronged argument suffers from a clear logical flaw. If we focus in on the logical structure of Locke's argument, we can isolate this claim: "Since I own my labor, and my labor has produced most of the value in this mixture, I own this mixture" and reduce it to "Since I own what I produce, and I produced nearly the entire value of Y, I own Y." The consequent claim does not follow from the antecedent. A part of something is not the whole. It is true that no one else has a claim to the mixture, because it contains my labor, but there is no proof that I myself have a claim to the mixture. I only own as much of the mixture as is equivalent to the value of the labor I placed into it. If the value of my labor is X, the value of the land is Y, and the value of the mixture is Z, and assuming that Y is greater than 0 (5.40), then X+Y = Z. If I own X, but not Y, X ≠ Z and therefore I do not own Z. What I own [X] is equivalent to Z - Y. Therefore I only own as much of the mixture [Z] as is equivalent to the labor I put into it [X].
	Additionally, the implicit notion that my labor inherently has worth is not necessarily true. It is quite possible that my labor is fruitless. If I cultivate a plot of land, but it yields no crops, does that still mean that I own it? Locke does not clarify whether the criteria for private ownership, that I merely mix my labor with something owned in common, is based upon my labor actually improving the land. The logical structure of his argument rests upon the notion that my labor gives the land additional value, but his language suggests that so long as I mix my labor with common land, no matter the productivity of that labor, I become the private owner of that land.
	In order to address the concerns raised in the previous section, ignoring the possibility that my labor may not add value to the land, and still working under the assumption that mixing my labor with common property makes it my property, Locke would argue in this manner for the practicality of the appropriation of public property in order to defend his position:
My labor belongs to myself (established). No one has any right to my labor (established). To cultivate land is to use my labor. If I cultivate land for common use, others will be using my labor (contradicts 2.6). I will therefore not cultivate land for the common use. I will thus only cultivate land if that land belongs to me. The cultivation of land increases its efficiency (5.37). I can easily obtain as many resources from 10 acres of cultivated land as I can from 100 acres of uncultivated land (5.37). Therefore, by privatizing and cultivating those 10 acres, I save at least 90 acres of land for the rest of society to use (5.37). Therefore, the privatization of land increases the common stock.
	Although this seems to solve the issue of the 'missing Y', whereby I appropriate all of the value of the land [Y] which I privatize (by effectively giving back to the community rather than taking from it), it is further complicated by Locke's own premises. Locke sets limits upon the amount of common property which an individual may appropriate; that is that nobody can make a claim to more than he is able to use and enjoy, but from his writings on money we can see this:
Man has no right to take for himself anything which will go to waste in his possession (5.31). Property can be exchanged for money (5.46). Money does not spoil (5.46). Therefore a man can appropriate as much as he wishes from the common property, so long as he exchanges perishable goods for money (5.50). According to Locke, this harms nobody (5.50).
	Locke assumes that resources are non-scarce, to the point where all men should be fully satisfied in their desires (5.31). This is demonstrably false, even within the confines of the conditions he has outlined, such that no man may take more than he can use or exchange for non-perishables (5.31). Money must be scarce, otherwise it holds no value. If resources are non-scarce, no rational human being will exchange money for those resources. If money is being exchanged for resources, then resources must also be scarce. The very existence of money in Locke's system proves that his views on scarcity are incorrect.
	Locke's system of privatization prides itself on efficiency, but it is not Pareto efficient (that is, an alternate distribution of resources could benefit the collective without causing any individual harm). Under Locke's system, an extremely inefficient farmer could take hold of an incredible amount of land, slightly increase its productivity, sell any excess product, and Locke would happily accept this. In order to produce the most societal good, the allocation of land should be based upon comparative advantage; that is, it should be distributed to those most capable of organizing its cultivation in order to yield the most product. Those who are not given land through this process can contract their labor to the landowners in order to earn money.
A proof of this, based on Locke's principles: Property can be exchanged for money (5.46). My labor is my property (previously established). Therefore I can exchange my labor for money.
	These wage earners can then exchange money for resources. In this way, yield would be maximized as well as total societal good (based on satisfaction of desires).
	Locke's views on property rights are firmly cemented in his concept of natural rights, which is consistent with his views on the state of nature, wherein we are subject to the laws of nature (2.4). Other thinkers, such as Hume, base their concept of property rights not upon a natural entitlement, but as a product of social convention. This may be largely influenced by their views on underlying premises about the fundamentals of mankind and nature. Whereas Locke believes that God has provided richly (5.31), Hume believes that resources are scarce, and man is unable to provide for himself all of the things he needs and desires without increasing efficiency through cooperation and social order (312). Locke believes that society emerges to protect property rights (7.85), but Hume believes that property does not exist until society does (312), and society emerges from man's need to overcome his inability to secure what he requires (312). Locke's view on possession is much more clear when it comes to property. A man comes to own property by mixing his labor with it (5.27), but for Hume, ownership is subject to a different standard. Rather than add the provision that a man must mix his labor with property for it to become his, Hume allows the first possessor to lay claim to it (323). Although he provides alternative means (324), as over time the claim to first ownership can become unclear (326).
	The preeminent logical flaw in Locke's argument is that ownership of a part does not constitute ownership of the whole. Although his argument is developed in a system of non-scarce resources, his discussion of money contradicts this initial condition. He prides his system of appropriation on its supposed efficiency, but it is clearly not a Pareto efficient system, and therefore not the optimal distribution of resources. Other authors such as Hume who acknowledge scarce resources, reject natural rights to property, and take a more practical approach to the topic are able to produce real life practicable solutions to the property dilemmas mankind faces, instead of becoming stuck in the mire of proving the existence of natural rights.