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Principles for Object-Linguistic Consequence: from Logical to Irreflexive.

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Summary
This summary is machine-generated.

This study addresses Curry's paradox by proposing strategies to weaken principles of object-linguistic consequence. The research introduces "grounded consequence" as a novel approach to achieve a self-applicable notion of logical consequence.

Keywords:
Irreflexive consequenceLogical consequenceObject-linguistic consequenceV-Curry paradox

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Area of Science:

  • Logic and Formal Philosophy
  • Foundations of Mathematics
  • Metaphysics

Background:

  • Examines the principles of a primitive, object-linguistic notion of consequence, as proposed by Beall and Murzi (2013).
  • Identifies that these principles lead to a version of Curry's paradox, a significant challenge in formal logic.
  • Highlights the need for strategies to manage or resolve paradoxes arising from self-referential or consequence-based principles.

Purpose of the Study:

  • To propose and investigate methods for weakening the problematic principles of object-linguistic consequence.
  • To develop and analyze new conceptions of consequence that overcome paradoxes.
  • To explore the implications of these strategies for theories of logical consequence and validity.

Main Methods:

  • Proposes several strategies based on the intuition that object-linguistic consequence internalizes meta-linguistic consequence.
  • Develops a theory of logical consequence by restricting the introduction rule for the consequence predicate.
  • Introduces a novel conception of 'grounded consequence' by restricting the structural rule of reflexivity, generalizing Kripke's inductive theory of truth (strong Kleene version).

Main Results:

  • Demonstrates that the proposed theory of logical consequence is conservative over a wide range of base theories via local reduction.
  • Establishes that 'grounded consequence' satisfies desirable principles for a naïve, self-applicable notion of consequence.
  • Presents distinct conceptions of consequence corresponding to the different proposed solutions.

Conclusions:

  • The study successfully develops strategies to overcome Curry's paradox within object-linguistic consequence frameworks.
  • Introduces 'grounded consequence' as a viable and principled alternative for self-applicable logical consequence.
  • The findings offer new insights into the nature of logical consequence, validity, and truth in formal systems.