An Introduction and explanation of the origins
This school of literary theory is based on the political and philosophical writings principally of Karl Marx, but also Frederic Engels. In writings such as The German Ideology and The Communist Manifesto Marx described a model of history the keystone of which stated that economic and political conditions determine social conditions. Largely a reaction to and study of capitalism and the effects of its rise, Marx tried to connect a multitude of social and psycological effects that have arisen from the economic distinction of the classes in his work. These theories portrayed a society in perpetual upheaval because of the friction caused among citizens by unnatural social conditions.
Marx's philosophy as an approach to historical study is often associated with what is commonly know as the 'materialist', approach to political philosophy, as opposed to the 'idealist' position. As Marx said it, "it is not the consciousness of men that determines their being, but, on the contrary, their social being that determines consciousness." This is easily illustrated by the dillema of a "leisure class." That is, the theory that throughout history, those with the time and energy to write (create, idealize) were not usually the ones living in working class conditions, and the things they wrote about were defined by that circumstance. Likewise, the poorer authors were restricted by their participation in the capitalist work machine, which Marx believed eroded much of their humanity. This is also sometimes referred to as the base and the superstructure model, the majoity of philosophy and literature being the domain of the superstructure.
Marxist theorists tend to blieve that the ideas and meaning that inhabit literature are a direct result of this model. That is, literature is simply a vessel for propagating the dominant cultural ideologies among the people as a whole, preventing exploration of ideas as 'truth'. In many cases, it becomes a means for legitimizing the power of the superstructure class.
Georg Lukacs and the Social Realists school of thinkers take this model to the deterministic extreme, but it is likely that Marx actually believed that literature could affect social change, although not in and of itself. For such change to come to full fruition, action is required. Antonio Gramsci is more in line with this way of thinking, and came up with the idea of hegemony to resolve the apparent contradiction. Gramsci thought that behind the process of social determinism there was a sort of individual deception. That to say, that an individual is led to question and eventually disregard private convictions in favor for the dominant ideology. However, in hand with this is the implicit notion that an individual may reject ideologies and write out of true personal conviction, significantly magnifying literature's capacity for enacting real social change. For example, although a majority of women writers in the 19th century reinforced Victorian social sentiments, novelist Jane Austen rejected and parodied many of the dominant ideals of her time.
Getting to the actual literary theory
Given the Marxist affinity for dialectic movement in their thought, Marxist literary criticism aims to explose tensions and contradictions in writing. And obviously, Marxists see meaning as intrinsically linked to social status and convention, and thus they seek through literature for a greater understanding of the inner working of society as well as evidence of rejection/propoagation of the dominant ideology within a work of literature. A MArxist interpretation inevitably leads to larger social questions and often ends up as a critique in itself of current or historical material conditions.
In its methodolgy Marxist criticism is actually very similar to psychoanalytic criticism (it looks for conditions underneath the surface of society much in the way that psychologists look underneath hthe surface of consciousness). But Marxism has also had a significant influence on feminist criticism (see Austen exaple above), as well as Old and New Historicism.
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See also: The Economic and Philosophical Manuscripts, Read Hegel