Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

Maffezioli, Paolo; Orlandelli, Eugenio

doi:10.18778/0138-0680.48.2.04

Cited by 11 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, [14] demonstrates the usefulness of the single-premise approach to quantification in obtaining the Craig's interpolation theorem. We also plan to extend those results to the systems developed here.…”

Section: Discussionmentioning

confidence: 88%

See 1 more Smart Citation

A More Unified Approach to Free Logics

Pavlović

Gratzl

2020

J Philos Logic

View full text Add to dashboard Cite

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (namely self-identity) are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (i), are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics (free and classical). The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. Keywords Positive free logic • Negative free logic • Sequent calculus • G3 • Modal logic An anonymous reviewer has offered numerous suggestions which have, combined, amounted to a significant improvement of the paper, and for that we extend our gratitude. Special thanks goes to O. Foisch.

show abstract

Section: Discussionmentioning

confidence: 88%

“…by adding the condition that E!t. This approach from [16,20] was explicitly used in [14] to simplify the quantifier rules of intuitionistic logic with existence predicate. However, in the more general framework of this paper, E!…”

Section: Positive Free Logic G3 Pfmentioning

confidence: 99%

A More Unified Approach to Free Logics

Pavlović

Gratzl

2020

J Philos Logic

View full text Add to dashboard Cite

show abstract

“…Then, from the left-to-right (right-to-left) direction of this semantic clause we easily obtain the rules LD i (RD). As shown in [14], for intuitionistic logic with existence predicate it is possible to obtain a simpler calculus by replacing the two premisses rule…”

Section: Definition 41 (Weight Of Terms and Formulas)mentioning

confidence: 99%

Labelled calculi for quantified modal logics with definite descriptions

Orlandelli¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce labelled sequent calculi for quantified modal logics with definite descriptions. We prove that these calculi have the good structural properties of G3-style calculi. In particular, all rules are height-preserving invertible, weakening and contraction are height-preserving admissible and cut is admissible. Finally, we show that each calculus gives a proof-theoretic characterization of validity in the corresponding class of models.

show abstract

“…The definitional hallmarks of a free logic are: we take a logic to be free iff (1) it is free of existential presuppositions with respect to its singular terms, (2) it is free of existential presuppositions with respect to its general terms and finally (3) its quantifiers have existential import, or are more broadly limited to the predicate E! (most commonly read as existence [3,30,31,38] and definedness [2,12]). Its main variants are positive FL, which allows for some atoms with terms outside of E!…”

Section: Introductionmentioning

confidence: 99%

Neutral Free Logic: Motivation, Proof Theory and Models

Pavlović

Gratzl

2022

J Philos Logic

View full text Add to dashboard Cite

Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka The Journal of Philosophy, 56, 125–137 1959; Lambert Notre Dame Journal of Formal Logic, 8, 133–144 1967, 1997, 2001). What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics reject the claim that names need to denote in (ii). Positive free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, negative free logic treats them as uniformly false, and neutral free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli (Bulletin of the Section of Logic, 48(2), 137–158 2019) as well as classical logic in Pavlović and Gratzl (Journal of Philosophical Logic, 50, 117–148 2021), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad (Studia Logica, 105(1), 93–119 2017, Journal of Applied Non-Classical Logics, 30(3), 272–289 2020), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties.

show abstract

Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

Cited by 11 publications

References 8 publications

A More Unified Approach to Free Logics

A More Unified Approach to Free Logics

Labelled calculi for quantified modal logics with definite descriptions

Neutral Free Logic: Motivation, Proof Theory and Models

Contact Info

Product

Resources

About