Groucho Marxism

In classical logic, statements or propositions are always either true or false. One of the justifications given for this rule is that contradictions – statements that are both true and false – entail everything. Which is to say that if you allow a single contradiction, you can prove any proposition you like. This feature is known as the principle of explosion and can be easily proved, as follows. Suppose the proposition P is both true and false; then P is true, so (P or Q) is true for any proposition Q; but P is also false, so Q must be true; and since Q was arbitrary, the result is proved. However, there are alternatives to classical logic that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. These logics are referred to as ‘paraconsistent’.

It is clear from the above that in order to avoid the principle of explosion, we must abandon either the principle of ‘disjunction introduction’ – P implies (P or Q) – or the principle of ‘disjunction syllogism’ – if (P or Q) then (not P) implies Q. In practice, we can choose to abandon either or both of these principles if we want to. Some might object to tampering with the laws of logic in this way. Logic is the foundation upon which the whole of mathematics rests, and messing around with this foundation may seem rather foolhardy. But it is important to point out that, much as we are free to choose the axioms of mathematics however we like, we are free to choose the rules of logic however we like as well. Just like mathematics, logic is, at the end of the day, a human invention.

The view that (some) statements can be both true or false is known as ‘dialetheism’. Dialetheism is not a system of formal logic; it is a thesis about truth that influences the construction of a formal logic. We have already seen that there exist systems of formal logic that allow for contradictions without leading to the principle of explosion. Whether we choose to use one of these systems rather than classical logic depends on our views on dialetheism. One argument in favour of dialetheism is that it resolves well-known paradoxes that involve contradictory statements. The most famous of these is the liar paradox, which is encapsulated by the statement ‘I am lying’. The paradox arises when trying to determine whether this statement is true or not.

The classical way to solve this problem is to revise the axioms of the logic so that self-contradictory statements are not allowed. Dialetheists, on the other hand, respond to this problem by simply accepting the statement ‘I am lying’ as both true and false. To a dialetheist, therefore, there is no paradox at all. Another argument in favour of dialetheism is that it is more closely aligned with human reasoning and human language. Ambiguous situations may cause humans to affirm both a proposition and its negation. For example, if someone stands in the doorway to a room, it may seem reasonable both to affirm that the person is in the room and to affirm that they are not in the room. Another example arises in statements such as ‘that car is red’, which one person may evaluate as true and another as false.

One prominent advocate for dialetheism is the British philosopher Graham Priest. According to Priest, there is a close connection between dialetheism and the dialectics of Hegel and Marx. In fact Priest goes so far as to argue that both Hegel and Marx were dialetheists. Priest gives two examples of Marx’s apparent dialetheism. The first concerns the notion of a commodity, which, according to the argument set out in Das Kapital, is both a use-value and an exchange-value. This entails a contradiction as use-values and exchange-values are incommensurate. The second example concerns wage labour. Under capitalism, wage labourers are free to sell their labour power as they choose; yet they are hardly free in any meaningful sense as the alternative is starvation and death.

The close connection between Marxism and dialetheism is not surprising as one of the things Marx is most famous for is pointing out the contradictions of capitalism. In formal logic, a contradiction is simply a statement which is both true and false. Whether Marx meant the term in this technical sense, or in a more colloquial sense, is open to debate. Regardless, in a recent (2024) paper, Priest shows that paraconsistent logic, which allows for true contradictions, can provide a basis for formalizing the somewhat nebulous concept of the dialectic. Specifically, Priest provides a formal logical model of a dialectical progression, a dynamic concept found in the writings of both Hegel and Marx. I shall return to this model in a future blog post.