First-Order Bi-Intuitionistic Logic: A Proof-Theoretic Investigation via Polytree Sequents

Core Concepts

The authors present the first sound and complete proof system for first-order bi-intuitionistic logic with increasing domains, which avoids the collapse of domains that occurs in previous approaches.

Abstract

The paper investigates first-order bi-intuitionistic logic, which extends first-order intuitionistic logic with an exclusion operator dual to implication. The authors note that previous attempts to axiomatize first-order bi-intuitionistic logic with non-constant domains have failed, as the resulting logic collapses to one with constant domains.
To overcome this issue, the authors introduce a proof system based on labeled polytree sequents, which internalize an existence predicate into the syntax. This allows them to control the scope of free variables and avoid the collapse of domains. The key insights are:

Explicitly considering the variable context in formulae, which reveals a dependency between the residuation principle and the existence predicate in the first-order setting.
Introducing an explicit eigenvariable context in the sequent calculus, which encodes the existence predicate without introducing it explicitly in the language.

The authors prove that their calculus enjoys cut-elimination and is conservative over first-order intuitionistic logic. They also discuss the relationship between the existence predicate and the exclusion operator, which was not obvious and required careful treatment.

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Quotes

"It is well-known that extending the Hilbert axiomatic system for first-order intuitionistic logic with an exclusion operator, that is dual to implication, collapses the domains in the model into a constant domain."
"A key insight in avoiding the collapse of domains in BIQ(ID) is to consider the universal quantifier as implicitly carrying an assumption about the existence of the quantified variable."
"What this example highlights is that a typical proof-theoretical argument used to show the provability of the quantifier shift axiom (and hence the collapse of domains), like the one we have seen earlier, implicitly depends on an existence assumption on objects in the domains in the underlying Kripke model."

Key Insights Distilled From

Taking Bi-Intuitionistic Logic First-Order: A Proof-Theoretic Investigation via Polytree Sequents

by Tim S. Lyon,... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15855.pdf

Deeper Inquiries

How can the insights from this work on first-order bi-intuitionistic logic be applied to other non-classical logics with non-constant domains

The insights gained from the study of first-order bi-intuitionistic logic with increasing domains can be applied to other non-classical logics that involve non-constant domains. By introducing the concept of an existence predicate and carefully managing the use of this predicate in proofs, it may be possible to develop sound and complete proof systems for other logics with similar domain constraints. The explicit consideration of variable contexts and the interplay between quantifiers and the existence predicate can provide a framework for tackling the challenges posed by non-constant domains in various non-classical logics. Additionally, the use of labeled polytree sequents and reachability rules can be extended to other logics to ensure the structural integrity of proofs and to check for the existence of data along certain paths.

What are the potential applications of first-order bi-intuitionistic logic with increasing domains in computer science and other fields

The applications of first-order bi-intuitionistic logic with increasing domains in computer science and other fields are diverse and impactful. In computer science, this logic can be utilized in the development of programming languages and formal verification systems. The ability to handle non-constant domains can be particularly useful in modeling complex systems where the domain may change dynamically. For example, in artificial intelligence and machine learning, where uncertainty and changing environments are common, first-order bi-intuitionistic logic with increasing domains can provide a more flexible and expressive framework for reasoning about knowledge and beliefs. In fields like robotics, natural language processing, and automated reasoning, this logic can enhance the representation and manipulation of information in a more nuanced way.

Is there a deeper connection between the existence predicate and the exclusion operator that could lead to a better understanding of the semantics of first-order bi-intuitionistic logic

There is indeed a deeper connection between the existence predicate and the exclusion operator in first-order bi-intuitionistic logic that can shed light on the semantics of the logic. The existence predicate, which postulates the existence of objects in the domain, plays a crucial role in controlling the scope of free variables and avoiding the collapse of domains. By explicitly considering the interaction between the existence predicate and the exclusion operator, we can gain a better understanding of how these two concepts influence the semantics of the logic. This deeper connection may lead to insights into the nature of existence and exclusion in logical systems, potentially paving the way for new developments in the understanding of first-order bi-intuitionistic logic and its applications.