We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$. To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$. We see several examples which give us separations among omniscience principles.

This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL${}_{\scriptsize\mbox{e}}$-algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we provide some philosophical upshots like, e.g., a discussion on the relevance of the above operation in relationship with G. Polya’s logic of plausible inference, and some characterization results on weak and strong connexivity.

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects.

Chapter 1. Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.

Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined.

Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.

Abstract prepared by Will Stafford extracted partially from the dissertation.

E-mail: [email protected]

URL: https://escholarship.org/uc/item/33c6h00c

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 we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.

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 adding critical $\varepsilon $ - and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.

The Gödel translation provides an embedding of the intuitionistic logic $\mathsf {IPC}$ into the modal logic $\mathsf {Grz}$ , which then embeds into the modal logic $\mathsf {GL}$ via the splitting translation. Combined with Solovay’s theorem that $\mathsf {GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf {PA}$ , both $\mathsf {IPC}$ and $\mathsf {Grz}$ admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions $\mathsf {MIPC}$ , $\mathsf {MGrz}$ , and $\mathsf {MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds $\mathsf {M^{+}IPC}$ into $\mathsf {M^{+}Grz}$ and the splitting translation embeds $\mathsf {M^{+}Grz}$ into $\mathsf {MGL}$ . As proven by Japaridze, Solovay’s result extends to the monadic system $\mathsf {MGL}$ , which leads us to a provability interpretation of both $\mathsf {M^{+}IPC}$ and $\mathsf {M^{+}Grz}$ .

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$-varieties. We prove that the lattice of $\mathtt {DNA}$-logics is dually isomorphic to the lattice of $\mathtt {DNA}$-varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$-varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$-formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

We show that monadic intuitionistic quantifiers admit the following temporal interpretation: “always in the future” (for $\forall $ ) and “sometime in the past” (for $\exists $ ). It is well known that Prior’s intuitionistic modal logic ${\sf MIPC}$ axiomatizes the monadic fragment of the intuitionistic predicate logic, and that ${\sf MIPC}$ is translated fully and faithfully into the monadic fragment ${\sf MS4}$ of the predicate ${\sf S4}$ via the Gödel translation. To realize the temporal interpretation mentioned above, we introduce a new tense extension ${\sf TS4}$ of ${\sf S4}$ and provide a full and faithful translation of ${\sf MIPC}$ into ${\sf TS4}$ . We compare this new translation of ${\sf MIPC}$ with the Gödel translation by showing that both ${\sf TS4}$ and ${\sf MS4}$ can be translated fully and faithfully into a tense extension of ${\sf MS4}$ , which we denote by ${\sf MS4.t}$ . This is done by utilizing the relational semantics for these logics. As a result, we arrive at the diagram of full and faithful translations shown in Figure 1 which is commutative up to logical equivalence. We prove the finite model property (fmp) for ${\sf MS4.t}$ using algebraic semantics, and show that the fmp for the other logics involved can be derived as a consequence of the fullness and faithfulness of the translations considered.

A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to be decidable both for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$ . This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

Rybakov (1984a) proved that the admissible rules of $\mathsf {IPC}$ are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.

Positive logics are $\{ \wedge , \vee , \to \}$-fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

We introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.

Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

Stable logics are modal logics characterized by a class of frames closed under relation preserving images. These logics admit all filtrations. Since many basic modal systems such as K4 and S4 are not stable, we introduce the more general concept of an M-stable logic, where M is an arbitrary normal modal logic that admits some filtration. Of course, M can be chosen to be K4 or S4. We give several characterizations of M-stable logics. We prove that there are continuum many S4-stable logics and continuum many K4-stable logics between K4 and S4. We axiomatize K4-stable and S4-stable logics by means of stable formulas and discuss the connection between S4-stable logics and stable superintuitionistic logics. We conclude the article with many examples (and nonexamples) of stable, K4-stable, and S4-stable logics and provide their axiomatization in terms of stable rules and formulas.

This paper contains a proof–theoretic account of unification in intermediate logics. It is shown that many existing results can be extended to fragments that at least contain implication and conjunction. For such fragments, the connection between valuations and most general unifiers is clarified, and it is shown how from the closure of a formula under the Visser rules a proof of the formula under a projective unifier can be obtained. This implies that in the logics considered, for the n-unification type to be finitary it suffices that the m-th Visser rule is admissible for a sufficiently large m. At the end of the paper it is shown how these results imply several well-known results from the literature.

It is a celebrated result of McKinsey and Tarski [28] that S4 is the logic of the closure algebra Χ+ over any dense-in-itself separable metrizable space. In particular, S4 is the logic of the closure algebra over the reals R, the rationals Q, or the Cantor space C. By [5], each logic above S4 that has the finite model property is the logic of a subalgebra of Q+, as well as the logic of a subalgebra of C+. This is no longer true for R, and the main result of [5] states that each connected logic above S4 with the finite model property is the logic of a subalgebra of the closure algebra R+.

In this paper we extend these results to all logics above S4. Namely, for a normal modal logic L, we prove that the following conditions are equivalent: (i) L is above S4, (ii) L is the logic of a subalgebra of Q+, (iii) L is the logic of a subalgebra of C+. We introduce the concept of a well-connected logic above S4 and prove that the following conditions are equivalent: (i) L is a well-connected logic, (ii) L is the logic of a subalgebra of the closure algebra $\xi _2^ + $ over the infinite binary tree, (iii) L is the logic of a subalgebra of the closure algebra ${\bf{L}}_2^ + $ over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logic L above S4 is connected iff L is the logic of a subalgebra of R+, and transfer our results to the setting of intermediate logics.

Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics for S4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames for S4 and descriptive spaces are isomorphic. It follows that every logic above S4 is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of both C and R. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients of L2 is an interior image of R.