In this talk we present a new truthmaker semantics for intuitionistic logic in which also Neil Tennant’s relevant intuitionistic logic, called Core Logic [2], can be captured. The semantics is based on Kit Fine’s truthmaker semantics for intuitionistic logic [1], but it interestingly differs from it on two levels: (1) no residuation or completeness requirement is needed for state spaces (mere join- semilattices suffice) and (2) it allows to distinguish between premise sets that are wholly relevant to their conclusion and those that are merely partially relevant to the conclusion. The latter enables us to define Core Logic as the exactified version of intuitionistic logic. The crucial technical difference with Fine’s semantics is that we add to Fine’s way of verifying sentences a dual way to verify them which is supposed to be the way premises of arguments are verified qua premises. Call the verification of sentences qua premises p-verification and that of sentences qua conclusions c-verification. Fine’s semantics is only concerned with c-verification, ours deals with both. The motivation behind this is the idea that, from a relevance- sensitive point of view, there is a fundamental difference between the inferential potential of connectives when they occur in premises and when they occur in conclusions. The case for disjunction is probably intuitively the clearest. In constructive mathematics, concluding a disjunction is always an act of weak- ening one’s previous results, i.e. in order to conclude a disjunction one must provide one of the disjuncts, and that is enough. In order to know that a dis- junction holds, one must only have information about a state of affairs that corresponds to one of the disjuncts. This is very different in case of reasoning contexts in which one assumes, hypothesizes, or postulates a disjunction (i.e. reasoning contexts where the disjunction is a premise). Because Disjunctive Syllogism (from A or B and nor A, conclude B) is valid in intuitionistic logic, a disjunction qua premise cannot merely state something about only one of the disjuncts. On the contrary: it needs to be about both and about the way they are related because it needs to afford the inference from the negation of any of the disjuncts to the other disjunct; it needs to guarantee the existence of a transformation of a proof for the absurdity of one of the disjuncts into a proof of the other disjunct. Disjunctions qua premises are therefore about different, richer, and more intensional subject-matter than disjunctions qua conclusions. In our semantics a conclusion is entailed by a set of premises iff each way in which the premises can be p-verified is a way for the conclusion to be c- verified. Our primitive notion of c-verification is exact: states that c-verify a sentence are wholly relevant to the verification of it qua conclusion. For instance a state that c-verifies A and B may not c-verify A, but a state that c-verifies A automatically c-verifies A or B. We may also say that a state inexactly c-verifies a sentence when it has a part that c-verifies it. Interestingly, our primitive notion of p-verification is what we may call dual-exact: p-verified sentences are wholly relevant to the states that p-verify them. For example, a state that p-verifies A or B may not p-verify A, but a state that p-verifies A and B automatically p-verifies A. When p-verification is dual-exact and c-verification is exact in the general characterization of entailment above, we obtain a relevant intuitionistic consequence relation which coincides with Core Logic as defined by Tennant [2]. When p-verification is dual-exact and c-verification is inexact, we obtain full (irrelevant) intuitionistic consequence. In case one sees intuitionistic logic along traditional lines without an exact truthmaker approach (relying on Kripke Frames, for example), these premise- conclusion meaning-distinctions are obfuscated and the same reading for premises and conclusion works everywhere. In a traditional non-exact semantics, verifiers are not local and a verifier for a disjunction immediately fixes the verification status of both disjuncts. Then Disjunctive Syllogism succeeds without problems. But this is not the case in the exact approach to semantics, which localizes the states of affairs and the subject-matter that correspond to sentences. This is the reason why it makes a lot of sense for truthmaker semanticists to make a fundamental distinction between the meaning of sentences (and the connectives occurring in them) when they play the inferential role of a premise and when they play the role of a conclusion. It is an interesting fact that, when setting things up as described above, the relevance aspect implicitly present in the motivation behind exact truthmaker semantics (Fine explains exactness as whole relevance) can be reflected in the relevance of the consequence relation by making the most direct consequence relation falling out of this (what has been argued to be) a relevant consequence relation, namely Core Logic. In the talk we define the new truthmaker semantics, show that it is adequate for Core Logic (and therefore, indirectly, also for intuitionistic logic) via an orig- inal multiset-multiset sequent calculus for the latter, and discuss and motivate the difference between p-verification and c-verification. References [1] Fine, Kit (2014). Truth-Maker Semantics for Intuitionistic Logic. Journal of Philosophical Logic 43 (2-3):549-577. [2] Tennant, Neil (2017). Core Logic. Oxford, England: Oxford University Press.