We introduce a new semantics for justification logic based on subset relations. Instead of using the established and more symbolic interpretation of justifications, we model justifications as sets of possible worlds. We introduce a new justification logic that is sound and complete with respect to our semantics. Moreover, we present another variant of our semantics that corresponds to traditional justification logic.
These types of models offer us a versatile tool to work with justifications, e.g.~by extending them with a probability measure to capture uncertain justifications. Following this strategy we will show that they subsume Artemov's approach to aggregating probabilistic evidence.
We study abstract intermediate justification logics, that is arbitrary intermediate propositional logics extended with a subset of specific axioms of (classical) justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev's, Fitting's or Lehmann's and Studer's models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying…
We show that the standard Gödel modal logics, as initially introduced by Caicedo and Rodriguez in \cite{CR2009,CR2010}, are not realized by the basic Gödel justification logics although being related by the forgetful projection.
Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators "t:", indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with Gödel logi…
The Logic of Proofs, LP, and other justification logics can have self-referential justifications of the form t:A. Such self-referential justifications are necessary for the realization of S4 in LP. Yu discovered prehistoric cycles in a particular Gentzen system as a necessary condition for S4 theorems that can only be realized using self-referentiality. It was an open problem whether prehistoric cycles also are a sufficient condition. The main results of this paper are: First, with the standard definition of self-referential theorems, prehistoric cycles are not a sufficient condition. Second, …
A Note on Fixed Points in Justification Logics and the Surprise Test Paradox
本文研究了在调整逻辑中加入不动点的效果。我们介绍了两种扩展的理由逻辑：扩展的不动点（或对角）运营商，和扩展的最少不动点。前者是Smorynski的对角化算子逻辑的证明版本，后者是Kozen的模态$μ$ -演算的证明版本。我们还介绍了Fitting的量化证明逻辑的不动点扩展，并在这些扩展中形式化了Knower Partial和Surprise Test Partial。通过将惊奇陈述解释为没有正当性的陈述，我们给出了数量化证明逻辑中惊奇测试paragraph的自指版本的一个解决方案。我们也给出了形式化的惊喜测试Parket在时间模态认知逻辑，并在哥德尔-勒布证明逻辑。
I show that propositional intuitionistic logic is complete with respect to an adaptation of Dummett's pragmatist justification procedure. In particular, given a pragmatist justification of an argument, I show how to obtain a natural deduction derivation of the conclusion of the argument from, at most, the same assumptions.
Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学院数学研究所）；School of Computer Science, University of Birmingham（伯明翰大学计算机学院）
Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be "contained" in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied in a set of known premises. To address this problem, we offer an alternative account of the episte…
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the logic of partitions, that is at the same mathmatical level as Boolean subset logic since subsets and quotient sets (partitions or equivalence relations) are dual to one another in the category-theoretic sense. Our purpose here is to explore the notions of negation and implication in that other mathematical logic of partitions.
It is known that the satisfiability problems of the product logics K4xS5 and S4xS5 and of the logic SSL of subset spaces are in N2EXPTIME. We improve this upper bound for the complexity of these problems by presenting ESPACE-algorithms for these problems. In another paper we show that these problems are EXPSPACE-hard. This shows that all three problems are EXPSPACE-complete.
It is known that the satisfiability problems of the product logics K4xS5 and S4xS5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. We improve these lower bounds for the complexity of these problems by showing that all three problems are EXPSPACE-hard under logspace reduction. In another paper we show that these problems are in ESPACE. This shows that all three problems are EXPSPACE-complete.
In this paper we adhere to the definition of infra-topological space as it was introduced by Al-Odhari. Namely, we speak about families of subsets which contain empty set and the whole universe X, being at the same time closed under finite intersections (but not necessarily under arbitrary or even finite unions). This slight modification allows us to distinguish between new classes of subsets (infra-open, ps-infra-open and i-genuine). Analogous notions are discussed in the language of closures. The class of minimal infra-open sets is studied too, as well as the idea of generalized infra-spaces…
We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $κ>λ\geq \aleph_0$, a continuous theory $T$ has a model with density character $κ$ which has a definable subset of density character $λ$, then $T$ has a model with density character $\aleph_1$ which has a separable definable subset. We also show that if we assume that $T$ is $ω$-stable, then if $T$ has a model of density character $\aleph_1$ with a separable definable set, then for any uncountable $κ$ we can find a model of $T$ with density character $κ$ which has a separable definable subset. In order to …
