Analyzing Proof-theoretic Validity and Base-extension Semantics for Intuitionistic Propositional Logic

Core Concepts

The authors explore the relationship between proof-theoretic validity (P-tV) and base-extension semantics (B-eS) in intuitionistic propositional logic (IPL), demonstrating how B-eS encapsulates the declarative content of a basic version of P-tV. They diverge from previous approaches by considering the constructive, hypothetical setup of P-tV.

Abstract

The paper delves into proof-theoretic semantics based on proof rather than truth, focusing on two major branches: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). It explains how B-eS for IPL captures constructiveness while addressing incompleteness results in other approaches. The authors provide a detailed analysis of atomic systems, canonical proofs, and the relationship between P-tV and B-eS. They establish a comprehensive framework for understanding the semantic foundations of IPL through natural deduction arguments and logical constants.

Stats

A foundational idea in P-tS is the use of atomic systems to supply the meaning of atomic propositions.
Arguments without assumptions and detours are said to be canonical proofs; they are inherently valid.
The study of such reductions is the technical background to P-tV provided by Prawitz [39].
Sandqvist [44] has given support judgments inductively according to the structure of formulas with base cases given by proof in an atomic system.

Quotes

"In eliminating a symbol, we may use the formula with whose terminal symbol we are dealing only ‘in the sense afforded it by the introduction of that symbol’." - Gentzen [52]
"An argument is valid if it represents a canonical proof, which is defined by substituting open assumptions for valid arguments of those assumptions." - Paper Analysis

Key Insights Distilled From

From Basic Proof-theoretic Validity to Base-extension Semantics for Intuitionistic Propositional Logic

by Alexander V.... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2210.05344.pdf

Deeper Inquiries

How does inferentialism differ from model-theoretic semantics?

Inferentialism and model-theoretic semantics are two different approaches to understanding the meaning of logical statements.
Inferentialism:

Focus: Inferentialism focuses on the rules of inference and how they determine the validity of arguments.
Meaning: According to inferentialism, meaning arises from rules of inference rather than properties of structures or truth values.
Validity: Validity is determined by whether an argument follows the correct rules of inference, making it a process-oriented approach.
Example: In natural deduction systems, such as Gentzen's NJ, validity is established through the application of specific rules for introducing and eliminating logical connectives.
Model-Theoretic Semantics:

Focus: Model-theoretic semantics focuses on interpreting logical statements based on their truth values in mathematical structures or models.
Meaning: Meaning is derived from how propositions correspond to elements in these models and whether they hold true under certain interpretations.
Validity: Validity is determined by whether a statement holds true in all possible interpretations or models, emphasizing truth conditions over proof procedures.
Example: In classical logic, a statement like "A ∧ B" is considered valid if both A and B are true in a given interpretation/model.
Overall, while inferentialism emphasizes the process of reasoning and rule-following for determining validity, model-theoretic semantics looks at how statements relate to truth values within specific models or interpretations.

What implications do incompleteness results have on intuitionistic logics?

Incompleteness results have significant implications for intuitionistic logics:

Limitations: Incompleteness results show that there are limits to what can be proven within intuitionistic logics. Gödel's incompleteness theorem demonstrated that no consistent formal system can prove all truths about natural numbers. This applies not only to classical but also to intuitionistic systems.

Constructive Nature: Intuitionistic logic rejects the law of excluded middle (LEM) which states that every proposition must either be true or false. As a result, incompleteness highlights that there may be propositions where neither their truth nor falsity can be established constructively within an intuitionistic framework.

Proof Theory vs. Truth Values: The focus on proof theory in intuitionistic logic means that incompleteness results impact what can be proven through valid arguments rather than establishing absolute truth values for all statements.

Semantic Interpretation: Incompleteness challenges traditional semantic interpretations based solely on mathematical structures by highlighting gaps in what can be captured purely through model-based reasoning.

How does reductive logic bridge constructiveness and validity conditions?

Reductive logic serves as a bridge between constructiveness and validity conditions by providing a framework where proofs themselves act as realizers within an argumentative context:

Constructive Reasoning: Reductive logic emphasizes constructing proofs step-by-step using explicit evidence rather than relying solely on abstract principles like LEM (Law of Excluded Middle). This aligns with the constructive nature inherent in many non-classical logics including intuitionistic logic.

2.Realizers: Realizers refer to objects within reductive logic that demonstrate how conclusions follow logically from premises through concrete steps akin to building blocks forming an argument structure.
3.Bridging Constructiveness & Validity: By focusing on realizers acting as arrows connecting premises with conclusions via intermediate steps (constructive reasoning), reductive logic bridges constructiveness with notions of validity - ensuring each step taken towards proving something contributes directly towards its overall validation
4.Operationalizing Logic: Reductive Logic operationalizes logical concepts into tangible processes allowing us not just understand why something holds but also see explicitly how we arrive at those conclusions - thereby bridging theoretical constructs with practical applications