This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

The well-known formal semantics of conditionals due to Stalnaker Studies in logical theory, Blackwell, Oxford, 1968), Lewis, and Gärdenfors The logic and 1140 epistemology of scientific change, North-Holland, Amsterdam, 1978, Knowledge in flux, MIT Press, Cambridge, 1988) all fail to distinguish between trivially and nontrivially true indicative conditionals. This problem has been addressed by Rott :345–370, 1986) in terms of a strengthened Ramsey Test. In this paper, we refine Rott’s strengthened Ramsey Test and the corresponding analysis of explanatory relations. We (...) show that our final analysis captures the presumed asymmetry between explanans and explanandum much better than Rott’s original analysis. (shrink)

Richard Bradley has initiated a new debate, with Brian Hill and Jake Chandler as further participants, about the implications of a number of so-called triviality results surrounding the Ramsey test for conditionals. I comment on this debate and argue that ‘Inclusion’ and ‘Preservation’, which were originally introduced as postulates for the rational revision of factual beliefs, have little to recommend them in the first place when extended to languages containing conditionals. I question the philosophical method of postulation that was applied (...) in the new debate, and instead base my arguments on explicit representations of belief states and canonical constructions of belief state revisions. (shrink)

This paper is about the situation in which an author (writer or speaker) presents a deductively invalid argument, but the addressee aims at a charitable interpretation and has reason to assume that the author intends to present a valid argument. How can he go about interpreting the author’s reasoning as enthymematically valid? We suggest replacing the usual find-the-missing-premise approaches by an approach based on systematic efforts to ascribe a belief state to the author against the background of which the argument (...) has to be evaluated. The suggested procedure includes rules for recording whether the author in fact accepts or denies the premises and the conclusion, as well as tests for enthymematic validity and strategies for revising belief state ascriptions. Different degrees of interpretive charity can be exercised. This is one reason why the interpretation or reconstruction of an enthymematic argument typically does not result in a unique outcome. (shrink)

According to the Ramsey Test, conditionals reflect changes of beliefs: α > β is accepted in a belief state iff β is accepted in the minimal revision of it that is necessary to accommodate α. Since Gärdenfors’s seminal paper of 1986, a series of impossibility theorems (“triviality theorems”) has seemed to show that the Ramsey test is not a viable analysis of conditionals if it is combined with AGM-type belief revision models. I argue that it is possible to endorse that (...) Ramsey test for conditionals while staying true to the spirit of AGM. A main focus lies on AGM’s condition of Preservation according to which the original belief set should be fully retained after a revision by information that is consistent with it. I use concrete representations of belief states and (iterated) revisions of belief states as semantic models for (nested) conditionals. Among the four most natural qualitative models for iterated belief change, two are identified that indeed allow us to combine the Ramsey test with Preservation in the language containing only flat conditionals of the form α > β. It is shown, however, that Preservation for this simple language enforces a violation of Preservation for nested conditionals of the form α > (β > γ). In such languages, no two belief sets are ordered by strict subset inclusion. I argue that it has been wrong right from the start to expect that Preservation holds in languages containing nested conditionals. (shrink)

I propose an account of indicative conditionals that combines features of minimal change semantics and information semantics. As in information semantics, conditionals are interpreted relative to an information state in accordance with the Ramsey test idea: “if p then q” is supported at a state s iff q is supported at the hypothetical state s[p] obtained by restricting s to the p-worlds. However, information states are not modeled as simple sets of worlds, but by means of a Lewisian system of (...) spheres. Worlds in the inner sphere are considered possible; worlds outside of it are ruled out, but to different degrees. In this way, even when a state supports “not p”, it is still possible to suppose p consistently. I argue that this account does better than its predecessors with respect to a set of desiderata concerning inferences with conditionals. In particular, it captures three important facts: that a conditional is logically independent from its antecedent; that a sequence of antecedents behaves like a single conjunctive antecedent ; and that conditionals restrict the quantification domain of epistemic modals. I also discuss two ways to construe the role of a premise, and propose a generalized notion of entailment that keeps the two apart. (shrink)

We present a semantic analysis of the Ramsey test, pointing out its deep underlying flaw: the tension between the “static” nature of AGM revision (which was originally tailored for revision of only purely ontic beliefs, and can be applied to higher-order beliefs only if given a “backwards-looking” interpretation) and the fact that, semantically speaking, any Ramsey conditional must be a modal operator (more precisely, a dynamic-epistemic one). Thus, a belief about a Ramsey conditional is in fact a higher-order belief, hence (...) the AGM revision postulates are not applicable to it, except in their “backwards-looking” interpretation. But that interpretation is consistent only with a restricted (weak) version of Ramsey’s test (in-applicable to already revised theories). The solution out of the conundrum is twofold: either accept only the weak Ramsey test; or replace the AGM revision operator ∗ by a truly “dynamic” revision operator ⊗, which will not satisfy the AGM axioms, but will do something better: it will “keep up with reality”, correctly describing revision with higher-order beliefs. (shrink)

According to Stalnaker’s Thesis, the probability of a conditional is the conditional probability. Under some mild conditions, the thesis trivialises probabilities and conditionals, as initially shown by David Lewis. This article asks the following question: does still lead to triviality, if the probability function in is replaced by a probability-like function? The article considers plausibility functions, in the sense of Friedman and Halpern, which additionally mimic probabilistic additivity and conditionalisation. These quasi probabilities comprise Friedman–Halpern’s conditional plausibility spaces, as well as (...) other known representations of conditional doxastic states. The paper proves Lewis’ triviality for quasi probabilities and discusses how this has implications for three other prominent strategies to avoid Lewis’ triviality: Adams’ thesis, where the probability function on the left in is replaced by a probability-like function, abandoning conditionalisation, where probability conditionalisation on the right in is replaced by another propositional update procedure and the approximation thesis, where equality in is replaced by approximation. The paper also shows that Lewis’ triviality result is really about ‘additiveness’ and ‘conditionality’. (shrink)

Dynamic epistemic logic, broadly conceived, is the study of logics of information change. This is the first paper in a two-part series introducing this research area. In this paper, I introduce the basic logical systems for reasoning about the knowledge and beliefs of a group of agents.