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🔗 Source: consequently.org
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Summary

((A ! B) ! B)
c7: (A ! (B ! C)) ! (B ! (A ! C))

Restall's proposed semantics overgenerates, leading to a problem known as validating the disjunctive syllogism and allowing for inconsistent base points in models of relevant logic R. This issue extends beyond Restall's work, affecting Graham Priest's textbook "An Introduction to Non-Classical Logic" which utilizes similar techniques.

Problem:

The error arises from the conditions used to define the accessibility relation R in the semantics. Specifically:

d5, d6, and d20: These conditions, intended to model relevant implications, lead to inconsistent base points when combined with the definition of R.
d7: This axiom also fails due to its dependence on d6 and d5.

Diagnosis:

The problem stems from a flawed completeness proof that constructs canonical models for refuting invalid arguments. While the initial construction seems sound, Restall's proposed modification to handle permutation in R is insufficient. Even with the modified definition of R*, not all frame conditions (like d5 and d20) are satisfied, leading to inconsistent base points.

Solution:

The authors conclude that modifying at least one of the frame conditions (d5-d20) is necessary. They do not specify which condition to modify, leaving this as an open question for future research.

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  • File Format: PDF
  • File Size: 230 KB
  • Pages: 8
  • Language: EN
  • Total Downloads: 35
  • Last Updated: 3 months ago

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