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Full Idea
If axioms are formulated for a language (such as set theory) that lacks names for all objects, then they require the use of a satisfaction relation rather than a unary truth predicate.
Gist of Idea
If a language cannot name all objects, then satisfaction must be used, instead of unary truth
Source
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 3.3)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.7
A Reaction
I take it this is an important idea for understanding why Tarski developed his account of truth based on satisfaction.
| 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] |