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 intermediate logic. Further, by a modification of a method of Fitting, we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics.
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.
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 …
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…
Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert's $\varepsilon$-calculus. The first and second $\varepsilon$-theorems for classical logic establish conservativity of the $\varepsilon$-calculus over its classical base logic. It is well known that the second $\varepsilon$-theorem fails for the intuitionistic $\varepsilon$-calculus, as prenexation is impossible. The paper investigates the effect of ad…
Several authors have recently defined intuitionistic logic based on team semantics (tIPC). In this paper we provide two alternative approaches to intermediate logics in the team semantics setting. We do this by modifying tIPC with axioms written with two different versions of disjunction in the logic, a local one and global one. We prove a characterization theorem in the first approach and we introduce a generalized team semantics in the second one.
Following a characterization [10] of locally tabular logics with finitary (or unitary) unification by their Kripke models we determine the unification types of some intermediate logics (extensions of {\sf INT}). There are exactly four maximal logics with nullary unification ${\mathsf L}(\mathfrak R_{2}+)$, ${\mathsf L}(\mathfrak R_{2})\cap{\mathsf L}(\mathfrak F_{2})$, ${\mathsf L}(\mathfrak G_{3})$ and ${\mathsf L}(\mathfrak G_{3}+)$ and they are tabular. There are only two minimal logics with hereditary finitary unification: ${\mathsf L}$($\mathbf F_{un}$), the least logic with hereditary un…
We report on the current status of our on-going project to develop well-behaved hypersequent calculi for intermediate predicate logics, such as the linearity axiom $\mathbf{LIN}\colon\left(φ\toψ\right)\lor\left(ψ\toφ\right)$ and the constant domain axiom $\mathbf{CD}\colon\forall x\left(φ\lorψ\left(x\right)\right)\toφ\lor\forall xψ\left(x\right)$.
This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We show they are elementary structures axiomatised by Horn sentences and we use our completeness result to state a version of algebraisability for inquisitive and dependence logics. We then foc…
For every univariate formula $χ$ we introduce a lattices of intermediate theories: the lattice of $χ$-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula $χ^2$, which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of $χ$-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We descr…
The thesis of this paper is that truth-relevant logic is a better foundation for mathematics than classical logic. It is a system proposed by Richard Diaz in 1981. In a certain sense t-relevant logic is based on Kleene strong tables. These define a system with three values: true, false, unknown. It turns out that there exist tautologies with the following property: there exists a proper subset of propositional variables (t-relevant variables) such that for all combinations true/false the tautology will be true, that is, the rest of the variables (redundant variables) occurring in the tautology…
We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral-completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov's notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally-complete logics. Using the Nerve Criterion w…
We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.
Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt Gödel (1932), and it is proved by Stanisław Jaśkowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). Gödel's proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Gödel or the Gödel-Dummet Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definablility of propositional connectives in this logic.
