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Full Idea
It is a virtue of the Friedman-Sheard axiomatisation that it is thoroughly classical in its logic. Its drawback is that it is ω-inconsistent. That is, it proves &exists;x¬φ(x), but proves also φ(0), φ(1), φ(2), …
Gist of Idea
The FS axioms use classical logical, but are not fully consistent
Source
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 4.3)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.10
A Reaction
It seems the theory is complete (and presumably sound), yet not fully consistent. FS also proves the finite levels of Tarski's hierarchy, but not the transfinite levels.
| 19120 | Semantic theories need a powerful metalanguage, typically including set theory [Halbach/Leigh] |
| 19121 | We can reduce properties to true formulas [Halbach/Leigh] |
| 19122 | Nominalists can reduce theories of properties or sets to harmless axiomatic truth theories [Halbach/Leigh] |
| 19124 | A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh] |
| 19126 | If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new [Halbach/Leigh] |
| 19125 | If we define truth, we can eliminate it [Halbach/Leigh] |
| 19127 | The T-sentences are deductively weak, and also not deductively conservative [Halbach/Leigh] |
| 19128 | If a language cannot name all objects, then satisfaction must be used, instead of unary truth [Halbach/Leigh] |
| 19129 | The FS axioms use classical logical, but are not fully consistent [Halbach/Leigh] |
| 19130 | KF is formulated in classical logic, but describes non-classical truth, which allows truth-value gluts [Halbach/Leigh] |