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According to Marx, labor is what gives a commodity its value, specifically useful labor(das mudpie). My question here is what exactly is useful labor and why is labor itself what provides value and not the usefulness? Is it not a contradiction to say labor is what provides value when the usefulness is what is necessary for labor to have any value in the first place?

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I do not know what is the reference of "useful labor" ...i think use-value and value was used by Marx in the capital.

"The Two Factors of the Commodity: Use- Value and Value (Substance of Value, Magnitude of Value)"

Marx introduces us to his analysis of commodities.

A commodity is an external object that satisfies a human need either directly or indirectly. He says that useful things can be looked at from the point of view of quality and quantity. They have many attributes and can therefore be used in many ways. He uses the term use-value in relation to commodities' quality. "The usefulness of a thing makes it a use-value."

A commodity's use-value is a trait of the thing itself, and is independent of the amount of labor needed to make the commodity useful.

Exchange-value is the proportion by which use-values of one kind exchange for use-values of other kinds. It is a constantly changing relation, and is not inherent to the object.

For example, corn and iron have an exchange relation, which means that a certain amount of corn equals a certain amount of iron.

Each must therefore equal a third common element, and can be reduced to this thing. The common element cannot be a natural property of the commodity, but rather must be abstracted away from its use-value. Discarding use-values, only one property remains—the commodities are the products of abstract human labor. They are "congealed quantities of homogenous human labor."

This common factor in the exchange-value of the commodity is its value.

Thus, a use-value only has exchange-value when it consists of abstract human labor. This is measured by the amount of labor-time socially necessary to produce it. A commodity's value would stay constant if the labor-time also stayed constant.

With greater productivity, it takes less labor to produce a commodity, and thus, less labor is "crystallized" in the product, leading to a decrease in value.

"The value of a commodity, therefore, varies directly as the quantity, and inversely as the productivity, of the labor which finds its realization within the commodity."

Something can be a use-value without being a value. This occurs when something's usefulness is not produced through labor.

However, nothing can be a value without also being a use-value; if something is useless, so is the labor contained in it.

Marx presents several definitions that will be important throughout his work, so it is very important to be clear on their meanings.

A use-value corresponds to the usefulness of an object, and is internal to that object.

For example, a hammer is a use-value because of its contributions to building. Its use-value comes from its usefulness. In contrast, a hammer's exchange-value comes from its value relative to other objects. For example, a hammer might be worth two screwdrivers. An object doesn't have an exchange value in itself, but only in its relationship with other objects. However, the fact that the hammer and screwdriver can be exchanged at all suggests that there must be something common between them, some means of comparison.

Marx says that this is the object's value.

Value means the amount of labor it takes to make the commodities. It implies that the price of commodities comes from how much labor was put into them.

ref.-

http://www.sparknotes.com/philosophy/daskapital/section1/

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So far as I can make out, the use-value of commodities was discounted in Marx's account of capitalist as opposed to pre-capitalist economies.

[A commodity] has both a “use value” and an “exchange value”. These two sides, he emphasized, express an unresolved tension and conflict between satisfying human needs and wants, on the one hand, and making profits, on the other. This echoes, obviously, Aristotle’s contrast between the “natural” form of the acquisitive arts (oikonomiké), which focuses on use values, and the “unnatural” form, which serves the end of unlimited enrichment (chrematistiké) (see also Kurz, 2016b, chap. 1). Historically, in non-capitalist modes of production, men assessed commodities first and foremost in terms of their intrinsic use values —their objective properties to satisfy particular needs and wants. While there was exchange in ancient Greece, for example, it had not yet taken full possession of the economy, and exchange value therefore was accidental rather than reflecting some fundamental forces at work. For this reason, Marx argued, the focus of Aristotle’s analysis was on use value. In capitalist society, however, chrematistiké rules the roost and exchange is well established through interdependent markets: It is the dominant coordination mechanism of numerous processes of production and consumption and of the corresponding social division of labour. Exchange value is regulated by economic law and no longer reflects accident and even caprice. Exchange value in capitalist markets, Marx was convinced, expresses the “true” value of commodities.

He tried to establish this proposition in the following, well-known way. When two things are exchanged for one another in a given proportion, he argued, there must exist, “in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other. Each of them, so far as it is exchange-value, must therefore be reducible to this third” (Marx, 1954, p. 45; emphasis added). This is the famous doctrine of the existence of a tertium comparationis [the third part or element of the comparison : GT], which to Marx is a logical implication of his vivisection of commodities. As early as The German Ideology of 1845 he and Engels spoke of money as the tertium comparationis of all men and things. In Capital Marx explained: “As use-values, commodities are, above all, of different qualities, but as exchange-values they are merely different quantities, and consequently do not contain an atom of use-value” (Marx, 1954, p. 45; emphasis added). (Heinz D. Kurz, 'Marx and the Law of Value', Investigación Económica, Vol. 77, No. 304 (ABRIL-JUNIO DE 2018 2018), pp. 40-71 : 48-9.)

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References

Kurz, H.D. (2018) 'Marx and the Law of Value', Investigación Económica, Vol. 77, No. 304 (ABRIL-JUNIO DE 2018 2018), pp. 40-71

Kurz, H.D. (2016). Economic Thought: A Brief History. New York: Columbia University Press. Paperback 2017.

Marx, K. (1954). Capital, volume I. London: Lawrence and Wishart.

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