In this paper, Graham Priest's understanding of dialetheism, the view that there exist true contradictions, is discussed, and various kinds of metaphysical dialetheism are distinguished between. An alternative to dialetheism is presented, namely a thesis called 'dimathematism'. It is pointed out that dimathematism enables one to escape a slippery slope argument for dialetheism that has been put forward by Priest. Moreover, dimathematism is presented as a thesis that is helpful in rejecting the claim that logic is a normative discipline. 3 , a distinction will be drawn between truth (falsity) at a state in a model and support of truth (falsity) at a state from a model. With that contrast explicitly made, truth at a state from a model may be seen as a metaphysical concept. If a statement then is true at a state, what the statement says is the case at that state . If a statement is false at a state, what the negation of the statement says is the case at that state . Support of truth (falsity), however, does not imply truth (falsity) in a metaphysical sense, that is, does not imply real truth (real falsity). For our presentation of dialetheism we have to consider some other, more specific setups as well. They point out that in a modal expansions of Priest’s (or Asenjo and Priest’s) Logic of Paradox, LP , with necessity, □ , and possibility, ◊ , being interdefinable and the necessitation rule being valid, ∼ ◊ ( A ∧ ∼ A ) is provable for any formula A , so that by rejecting certain features of LP , a coincidence between the third and the fourth level of paraconsistency can be avoided. However, there is more than one way of representing the claim that some inconsistent but non-trivial theories may be true as a statement in the object language. Although Priest’s favourite paraconsistent logic, LP , has a trivial model in which every formula receives the designated value “both true and false”, why should one assume that if there is an actual structure, this structure is represented by the trivial LP model? Also, axiomatic extensions of KD LP , the normal modal logic KD based on LP , have trivial models in which every formula has the value “both true and false” at every world, but why should one assume that if there is an actual world, this world is represented by a trivial world from a trivial model? If semantic consequence is of type ( ⊧ a ), logically provable dialetheias should be true at any state, for entailment of type ( ⊧ b ) , they should be true at every designated state d M , and for semantic consequence of type ( ⊧ c ) logically provable dialetheias should be true at every normal world. For a strongly paraconsistent logic the dialetheists are in need of, this means that this logic should be a non-trivial but inconsistent logic. The Logic of Paradox, LP , is, however, a non-trivial but non-inconsistent logic. Dimathematism, however, conceives of logic as the science of the most general laws of information flow. In the semantics of the basic paraconsistent logic FDE , first-degree entailment logic, it is possible to replace the classical truth values, true and false, and their metaphysical understanding by four semantical values, T, F, N, and B, which have an informational reading. Therefore, even trivialists with respect to natural languages would not be justified in maintaining that dimathematism is a trivial theory. What is the relationship between dialetheism and dimathematism? Is the latter a variant or a generalization of dialetheism? Is every dialetheia a dimathema? These questions are problematic. A dialetheia is, by definition, a contradiction that is both true and false at the actual world of some model. Dimathematism is, however, not committed to assuming actual worlds. Logic as a normative discipline would then produce norms, i.e., obligations, prohibitions, or permissions, of correct reasoning as an activity or of rational belief formation, supposing that the latter is in an at least indirect sense agentive. Although there is the distinction between ought-to-be and ought-to-do, I assume that norms in the first place apply to agents and not to states of affairs. If it ought to be the case that A , for example, this may be understood as there being a norm that obligates some addressees of that norm to see to it that A . This is, perhaps, surprising because Frege himself believed that logic is a prescriptive normative science. Frege ( 2013 , p. XV) draws the distinction between descriptive and prescriptive laws: It is commonly granted that the logical laws are guidelines which thought should follow to arrive at the truth; but it is too easily forgotten. The ambiguity of the word “law” here is fatal. In one sense it says what is, in the other it prescribes what ought to be. Only in the latter sense can the logical laws be called laws of thought, in so far as they The four cases distinguished between by the four values N, F, T, B are on a par, and the statement that a sentence A receives one of these values at a state is merely descriptive and if true this does not amount to instantiating or having any positive or negative value. Dimathematism is a view that is helpful in rejecting the claim that logic is a normative discipline in both of the senses distinguished above. Summary In this paper, I have discussed dialetheism as presented by its co-founder and most prominent protagonist, Graham Priest. A distinction has been drawn between various versions of dialetheism as a metaphysical view. It is derived from the ancient Greek word μ α ´ θ η μ α , which means “that which is learned” or “what one gets to know.” Here it is to be understood as “what one has been told,” and the prefix ‘di’ is meant to indicate that an atomic proposition can be both told to be true and told to be false (by different information sources or maybe even one and the same source). 12 In Priest et al. ( 2018 ), it is added that there are at least these four grades of paraconsistent involvement.