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According to classical logic, a single contradiction is sufficient to completely undermine any formal system, or even an informal argument. As the philosopher William of Soissons demonstrated in the 12th century, a contradiction implies that any statement is true, resulting in a “deductive explosion.” William’s argument proceeds as follows:
Consider any pair of statements P and Q. If P and ¬P (not P) are true, i.e. if there is a contradiction, then P ∨ Q (P or Q) is also true, since P is true. But because P is also false, Q must then be true in order for P ∨ Q to remain true. Q can be any statement, no matter how absurd it is.
This argument may be hard to follow without a background in symbolic logic. Here’s an example, inspired by Wikipedia’s entry on the principle of explosion:
Consider the statements “I have ten fingers” and “toads breathe fire.” If “I have ten fingers” and “I do not have ten fingers” are both true, then it is also true that “I have ten fingers” or “toads breathe fire”; in order for one statement or another to be true, it is sufficient that only one of them is true. But, due to the contradiction, it is also false that “I have ten fingers.” Because we have already shown that “I have ten fingers” or “toads breathe fire” is true, then it must be true that “toads breathe fire.”
Alternative systems of logic
A paraconsistent logic is one in which a contradiction does not entail that every statement is true. As philosopher Graham Priest explains, many natural phenomena obey paraconsistent logic. For instance, Bohr’s theory of the atom predicts that an electron orbits an atom without radiating energy. However, Maxwell’s theory states that an electron will radiate energy when it is accelerating and orbiting an atom. But this contradiction does not imply that all properties of an electron, however absurd, must be true; it would be incorrect to infer the statement “an electron is purple” from the contradiction. Some logicians have also argued that quantum mechanics follows paraconsistent logic, in the sense that the theory suggests that quantum phenomena display self-contradictory properties like being located in two places at the same time. However, quantum mechanics does not collapse as a result of these “contradictions”; in fact, the contradictions are at the heart of the theory.
Priest describes several systems of paraconsistent logic, but for the sake of brevity, I will discuss just one of them here: discursive logic. This system, developed by Stanisław Jaśkowski, is concerned with situations in which the truth of a statement is contested, like in a discourse where one person argues that a certain proposition is true whereas the other claims that it is false. Discursive logic considers each participant in the discourse as belonging to a different “world” within the system. Hence, in one world, an assertion A may be true, and in another world, ¬A (not A) may be true. According to Priest, what is true in the overall discourse is the sum of all the assertions made within the individual worlds (i.e. by the participants), so both A and ¬A would be true in the discourse. However, the fact that A and ¬A both hold does not imply that any statement B is true. Indeed, Priest states, “Consider a [discourse] M, such that A holds at [the world] w1, ¬A holds at a different world w2, but B does not hold at any world… Then both A and ¬A hold, yet B does not hold in M.” Thus, discursive logic is paraconsistent.
The crucial idea of discursive logic is that it addresses the question of where something is true, rather than whether something is true, in an absolute sense. (In this way, discursive logic is similar to the topos, a topic in math that I recently blogged about, which embeds a notion of “space” into truth. Thus, certain statements are true in one region of the topos, which pertains to a particular kind of mathematical structure, while they are false in another.) Therefore, a statement A and its negation ¬A can both be true, in separate worlds, while the statement A ∧ ¬A (A and ¬A) is never true in one particular world. 
Paraconsistency in the brain and in consciousness
A system of discursive logic appears to undergird the functions of the brain. In a process known as selective signal enhancement (SSE), populations of neurons compete with each other about their interpretations of stimuli. For instance, if someone sees a vague face in the distance, then the neural populations in her brain will have varying interpretations of the person’s identity; one population may label the face as “Jennifer” and another as “Mary.” As the face becomes clearer, the signal from one population will dominate and the person will be able to identify the face. The neurons are essentially participating in a cognitive discourse; in a recent blogpost, I quoted neuroscientist Michael Graziano’s metaphor for SSE: “Neurons act like candidates in an election, each one shouting and trying to suppress its fellows. At any moment only a few neurons win that intense competition, their signals rising up above the noise and impacting the animal’s behavior.”
So, we can associate different information-encoding networks in the brain with different worlds in a logical system.  In one world, the statement “this is Jennifer’s face” (A) holds, whereas in another world, the statement “this is not Jennifer’s face” (¬A) is true. The coexistence of these two statements, albeit in two separate worlds, does not entail that all statements about the perceptual stimulus at hand are true; there isn’t a single neuronal population that would categorize the stimulus as a tiger.
Moreover, there is typically no single network that encodes both A and ¬A. But this seemingly inherent property of the brain does not always hold true. In altered states of consciousness, such as those induced by psychedelics, one can genuinely believe that he is experiencing multiple parallel realities at the same time, or that he is reliving more than one memory simultaneously. In formless consciousness, where the boundary between self and other dissolves, consciousness is no longer localized at one point in space or time; one has the utterly sublime feeling of simultaneously being everywhere and everybody, of becoming one with the universe, of returning to the “eternal, infinite sky” in which all the impermanent forms of our reality appear as “passing clouds.” The brain embraces contradiction in formless consciousness; it simultaneously encodes, for instance, the pair of statements “I am this body” and “I am everything that has ever lived,” or “I am here” and “I am here, there, and the whole of the cosmos.” Ordinarily, these two statements would be produced by two different worlds in the discursive logic of the brain, but in formless consciousness, these worlds merge together.
Thus, the paraconsistency of the brain collapses in formless consciousness; a single contradiction implies that all the information that could be generated by the brain is true. Indeed, this idea is consistent with my recent definition of formless consciousness as a state that maximizes the repertoire of experiences that the brain can project into our awareness. In other words, we are conscious of everything – we perceive color, recall every memory, imagine our sense of self merging with everyone else’s, and so on.
The physical instantiation of the coalescing of logical worlds likely lies in the literal unification of neuronal networks. That is, the networks that produce our conscious experiences become more interconnected with each other. This claim is substantiated by recent empirical findings about psychedelics, which tend to bring about formless consciousness. According to neuroscientist Robin Carhart-Harris, psychedelics “reduce the degree of separateness or segregation” between well-established brain networks, especially those that normally do not communicate with one another. Furthermore, the degree of connectivity between these networks corresponds to the extent of “ego dissolution,” or loss of sense of self, during the psychedelic trip. In particular, LSD strengthens the connections between the regions of the brain that are responsible for self-awareness and the networks that process sensory information about the external world.
 Incidentally, this property of discursive logic is known as non-adjunction.
 This statement presupposes that the brain is a formal system, i.e. one that strictly obeys axiomatic rules of inference. However, this is a contested idea in neuroscience and philosophy of mind. For example, philosopher Tim Bayne states that “axiomatic methods are most closely associated with mathematics and logic, and one will not find any mention of them in accounts of explanation in the mechanical or life sciences.” Thus, according to Bayne, one ought to be skeptical of the axiomatic approach to neuroscience.