A terminating sequent calculus for intuitionistic propositional logic is obtained by modifying the R$\supset $ rule of the labelled sequent calculus $\mathbf {G3I}$. This is done by adding a variant of the principle of a fortiori in the left-hand side of the premiss of the rule. In the resulting calculus, called ${\mathbf {G3I}}_{\mathbf {t}}$, derivability of any given sequent is directly decidable by root-first proof search, without any extra device such as loop-checking. In the negative case, the failed proof search gives a finite countermodel to the sequent on a reflexive, transitive, and Noetherian Kripke frame. As a byproduct, a direct proof of faithfulness of the embedding of intuitionistic logic into Grzegorcyk logic is obtained.

Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic $\mathbf {GL}$ is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.

We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$-sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.

We study various properties of formalised relativised interpretability. In the central part of this thesis we study for different interpretability logics the following aspects: completeness for modal semantics, decidability and algorithmic complexity.

In particular, we study two basic types of relational semantics for interpretability logics. One is the Veltman semantics, which we shall refer to as the regular or ordinary semantics; the other is called generalised Veltman semantics. In the recent years and especially during the writing of this thesis, generalised Veltman semantics was shown to be particularly well-suited as a relational semantics for interpretability logics. In particular, modal completeness results are easier to obtain in some cases; and decidability can be proven via filtration in all known cases. We prove various new and reprove some old completeness results with respect to the generalised semantics. We use the method of filtration to obtain the finite model property for various logics.

Apart from results concerning semantics in its own right, we also apply methods from semantics to determine decidability (implied by the finite model property) and complexity of provability (and consistency) problems for certain interpretability logics.

From the arithmetical standpoint, we explore three different series of interpretability principles. For two of them, for which arithmetical and modal soundness was already known, we give a new proof of arithmetical soundness. The third series results from our modal considerations. We prove it arithmetically sound and also characterise frame conditions w.r.t. ordinary Veltman semantics. We also prove results concerning the new series and generalised Veltman semantics.

Abstract prepared by Luka Mikec.

E-mail: [email protected]

URL: http://hdl.handle.net/2445/177373

Vardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$—the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$.

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 $\mathsf {M^{+}Grz}$ .

By Solovay’s celebrated completeness result [31] on formal provability we know that the provability logic ${\textbf {GL}}$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result in [22] by considering a polymodal version ${\mathsf {GLP}}$ of ${\textbf {GL}}$ with modalities $[n]$ for each natural number n referring to ever increasing notions of provability. Modern treatments of ${\mathsf {GLP}}$ tend to interpret the $[n]$ provability notion as “provable in a base theory T together with all true $\Pi ^0_n$ formulas as oracles.” In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity $\Pi ^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $\Sigma ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $\alpha $ will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper. The paper proves soundness and completeness for the proposed interpretation for a wide class of theories, namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic.

We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.

We obtain modal completeness of the interpretability logics IL $\!\!\textsf {P}_{\textsf {0}}$ and ILR w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL $\textsf {P}_{\textsf {0}}$ and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions of ILW w.r.t. the generalised semantics in the future, and demonstrate its usage with $\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$ .

The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.

The logic IL (All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this article we raise the previously known lower bound of IL (All) by exhibiting two series of principles which are shown to be provable in any such theory. Moreover, we compute the collection of frame conditions for both series.

For the Heyting Arithmetic HA, $HA^{\text{*}} $ is defined [14, 15] as the theory $\left\{ {A|HA \vdash A^\square } \right\}$, where $A^\square $ is called the box translation of A (Definition 2.4). We characterize the ${\text{\Sigma }}_1 $-provability logic of $HA^{\text{*}} $ as a modal theory $iH_\sigma ^{\text{*}} $ (Definition 3.17).

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$ (Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

In this article we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.

A formula φ is called n-provable in a formal arithmetical theory S if φ is provable in S together with all true arithmetical ${{\rm{\Pi }}_n}$-sentences taken as additional axioms. While in general the set of all n-provable formulas, for a fixed $n > 0$, is not recursively enumerable, the set of formulas φ whose n-provability is provable in a given r.e. metatheory T is r.e. This set is deductively closed and will be, in general, an extension of S. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic $PA$ can be axiomatized by ${\varepsilon _0}$ times iterated local reflection schema over $PA$. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of n-provability of a sentence can be much shorter than its proof from iterated reflection principles.

Let T and U be any consistent theories of arithmetic. If T is computably enumerable, then the provability predicate $P{r_\tau }\left( x \right)$ of T is naturally obtained from each ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of T. The provability logic $P{L_\tau }\left( U \right)$ of τ relative to U is the set of all modal formulas which are provable in U under all arithmetical interpretations where □ is interpreted by $P{r_\tau }\left( x \right)$. It was proved by Beklemishev based on the previous studies by Artemov, Visser, and Japaridze that every $P{L_\tau }\left( U \right)$ coincides with one of the logics $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α and β are subsets of ω and β is cofinite.

We prove that if U is a computably enumerable consistent extension of Peano Arithmetic and L is one of $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α is computably enumerable and β is cofinite, then there exists a ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of some extension of $I{{\rm{\Sigma }}_1}$ such that $P{L_\tau }\left( U \right)$ is exactly L.

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.

For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then

$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$

In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).

Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,

$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$

We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.

Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.