An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

We examine some combinatorial properties of parallel cut elimination in multiplicative linear logic (MLL) proof nets. We show that, provided we impose a constraint on some paths, we can bound the size of all the nets satisfying this constraint and reducing to a fixed resultant net. This result gives a sufficient condition for an infinite weighted sum of nets to reduce into another sum of nets, while keeping coefficients finite. We moreover show that our constraints are stable under reduction. Our approach is motivated by the quantitative semantics of linear logic: many models have been proposed, whose structure reflect the Taylor expansion of multiplicative exponential linear logic (MELL) proof nets into infinite sums of differential nets. In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion. It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative. We moreover show that the set of differential nets that occur in the Taylor expansion of an MELL net automatically satisfies our constraints. Interestingly, our nets are untyped: we only rely on the sequentiality of linear logic nets and the dynamics of cut elimination. The paths on which we impose bounds are the switching paths involved in the Danos--Regnier criterion for sequentiality. In order to accommodate multiplicative units and weakenings, our nets come equipped with jumps: each weakening node is connected to some other node. Our constraint can then be summed up as a bound on both the length of switching paths, and the number of weakenings that jump to a common node.

S5-Style Non-Standard Modalities in a Hypersequent Framework

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem.

A note on cut-elimination for classical propositional logic

In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

Sequent Calculi for the Propositional Logic of HYPE

In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

A focused linear logical framework and its application to metatheory of object logics

Linear logic (LL) has been used as a foundation (and inspiration) for the development of programming languages, logical frameworks, and models for concurrency. LL’s cut-elimination and the completeness of focusing are two of its fundamental properties that have been exploited in such applications. This paper formalizes the proof of cut-elimination for focused LL. For that, we propose a set of five cut-rules that allows us to prove cut-elimination directly on the focused system. We also encode the inference rules of other logics as LL theories and formalize the necessary conditions for those logics to have cut-elimination. We then obtain, for free, cut-elimination for first-order classical, intuitionistic, and variants of LL. We also use the LL metatheory to formalize the relative completeness of natural deduction and sequent calculus in first-order minimal logic. Hence, we propose a framework that can be used to formalize fundamental properties of logical systems specified as LL theories.

Schema Complexity in Propositional-Based Logics

The essential structure of derivations is used as a tool for measuring the complexity of schema consequences in propositional-based logics. Our schema derivations allow the use of schema lemmas and this is reflected on the schema complexity. In particular, the number of times a schema lemma is used in a derivation is not relevant. We also address the application of metatheorems and compare the complexity of a schema derivation after eliminating the metatheorem and before doing so. As illustrations, we consider a propositional modal logic presented by a Hilbert calculus and an intuitionist propositional logic presented by a Gentzen calculus. For the former, we discuss the use of the metatheorem of deduction and its elimination, and for the latter, we analyze the cut and its elimination. Furthermore, we capitalize on the result for the cut elimination for intuitionistic logic, to obtain a similar result for Nelson’s logic via a language translation.

A Binary Quantifier for Definite Descriptions for Cut Free Free Logics

This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.

The consistency of arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

Ordinal notations and induction

In order to prove that the simplification process for arithmetic eventually reaches a simple proof, it is necessary to measure the complexity of proofs in a more sophisticated way than for the cut-elimination theorem. There, a pair of numbers suffices, and the proof proceeds by double induction on this measure. This chapter develops the system of ordinal notations up to ε0 which serve as this more sophisticated measure for proofs in arithmetic. Ordinal notations are presented as purely combinatorial system of symbols, so that from the outset there is no doubt about the constructive legitimacy of the associated principles of reasoning. The main properties of this notation system are presented, and it is shown that ordinal notations are well-ordered according to its associated less-than relation. The basics of the theory of set-theoretic ordinals is developed in the second half of the chapter, so that the reader can compare the infinitary, set-theoretic development of ordinals up to ε0 to the system of finitary ordinal notations. Finally, Paris-Kirby Hydra game and Goodstein sequences are presented as applications of induction up to ε0.