I have been reading Leitgeb's What Theories of Truth Should be Like, and one of the desiderata for a theory of truth, he argues, is to have unrestricted derivable T-biconditionals. But I am having trouble understanding what he meant exactly by the T-biconditional.
What he described seems to be the equivalent of Tarski's material adequacy, because he explicitly mentioned:
For example, if ‘Tr(‘snow is white’) if and only if snow is white’ were not derivable, then an important aspect of our understanding ‘Tr’ as being applied to the sentence ‘snow is white’ would not be reflected by this definition. At best the definition would specify the actual meaning of ‘Tr’ incompletely, in the worst case it might assign the wrong extension to ‘Tr’.
And this seems to corresponds to Tarski's motivation for the material adequacy condition. In other words, when he is talking about ‘Tr(‘snow is white’) if and only if snow is white’, this appears to be the Convention-T which is the formalised version of material adequacy, not T-schema which is an inductive definition of truth.
However, in a latter part where he describes the unrestricted derivability of T-biconditionals he talks about the T-scheme, which I take it to be the T-schema:
The same seems to hold for all other sentences of the form ‘Tr(‘A’) if and only if A’ where ‘A’ is replaced by a sentence of the very language that we are interested in. Indeed, the derivability of all instances of the T-scheme from a definition of truth guarantees that the latter assigns the ‘right’ extension to ‘Tr’, which at least seems to be a necessary condition for what a ‘good’ definition of truth must be like.
This seems to suggest that when Leitgeb is talking about the T-biconditional, he doesn't have Convention-T - the condition that Tarski argues all theories of truth should comply - in mind; instead he seems to have the T-schema in mind.
But this doesn't seem to make sense. Leitgeb is not trying to come up with a theory of truth in this paper, but a desiderata for a theory of truth. The T-schema is Tarski's theory of truth which satisfies the material adequacy condition. If by the T-biconditional Leitgeb meant the T-schema, not Convention-T, this means he is arguing that all theories of truth must have a unrestricted provable T-schema - which is equivalent to saying that the T-schema is a presumed desiderata for all theories of truth, and obviously that is circular.
For example, intuitionistic logic conforms with Convention-T, but does not comply with T-schema since it is not truth-functional, which is a key characteristic of T-schema. Therefore what Leitgeb suggested would have excluded all theories of truth that are non-classical based)
Could anyone help me interpret what exactly does Leitgeb mean by the T-biconditional please?
PS: This question has some overlaps with my previous question about the distinction between the T-schema and Convention-T, but I feel like this is not quite the same question. Is T-schema just another name for Convention-T, or are they two different things?
The relevant excerpt is as follows:
Sentences of the form ‘Tr(‘A’) if and only if A’, in which the schematic letter ‘A’ is replaced by a declarative sentence of a given language, are called T-biconditionals for this language (‘T’ for truth,‘biconditional’ because of the ‘if and only if ’); ‘Tr(‘A’) if and only if A’ itself is called the T-scheme. The famous ‘Tr(‘snow is white’) if and only if snow is white’ is the paradigm case example of a T-biconditional.
Tarski was the first to notice the methodological importance of the T-scheme: his idea was to define truth, i.e. to state a definition of the form ‘Tr(x) if and only if . . .’, and then to test the semantic adequacy of this definition by checking whether all T-biconditionals for the language for which truth is to be defined are derivable from it. If not, the definition is deficient. For example, if ‘Tr(‘snow is white’) if and only if snow is white’ were not derivable, then an important aspect of our understanding ‘Tr’ as being applied to the sentence ‘snow is white’ would not be reflected by this definition. At best the definition would specify the actual meaning of ‘Tr’ incompletely, in the worst case it might assign the wrong extension to ‘Tr’.
The same seems to hold for all other sentences of the form ‘Tr(‘A’) if and only if A’ where ‘A’ is replaced by a sentence of the very language that we are interested in. Indeed, the derivability of all instances of the T-scheme from a definition of truth guarantees that the latter assigns the ‘right’ extension to ‘Tr’, which at least seems to be a necessary condition for what a ‘good’ definition of truth must be like.
