In a comment to Vallicella here I mentioned an article on the pro-sentential theory in the Internet Encyclopedia of Philosophy, and said that the theory has 'some affinity' with the 'assertoric' theory of truth I have sketched out here (as well as fiercely defended at Vallicella's place.
Here, I will take some of the claims made for the prosentential theory in that article and compare them with the corresponding claims of the assertoric theory, to see how close the affinity is.
The Assertoric Theory
The assertoric theory of truth is that the copula 'is' of a simple declarative subject-predicate sentence, such as
Tom is running
contains two components. One is the copula, whose function is to join subject and predicate to form the 'content' of the sentence. The content we can signify by a noun phrase such as 'that Tom is running' or the verbal noun 'Tom's running'. The other is the assertoric component, whose function is to signal assertion, i.e. to signal that the person uttering the sentence is saying something capable of being true or false. We can make the internal structure of such a declarative sentence transparent by parsing it as follows:
It is true / that Tom is running
The Prosentential Theory compared to the Assertoric Theory
1. "According to the prosentential theory of truth, whenever a referring expression (for example, a definite description or a quote-name) is joined to the truth predicate, the resulting statement contains no more content than the sentence(s) picked out by the referring expression."
Although the theories are similar, the fundamental difference is that, according to the assertoric theory, the 'truth predicate' (actually the truth operator) operates on a noun phrase signifying the content of the sentence, not the sentence.
2. "The central claim of the prosentential theory is that ‘x is true’ functions as a prosentence-forming operator rather than a property-ascribing locution."
The central claim of the assertoric theory is that 'x is true' functions as an operator on a 'that'-clause that forms a normal sentence (not a 'prosentence'). It claims also that this operator is implicit, though not lexically visible, in a sentence that does not use the words 'is true'.
3. "According to the prosentential theory, the statement ‘p is true’ says no more than the statement ‘p,’
The corresponding claim in the assertion theory is that 'it is true that p' says no more than that p. I.e. 'It is true that Tom runs' says that Tom runs. However, the theory distinguishes between the operator 'it is true that', which operates on a sentence to produce another sentence (of equivalent meaning), the operator 'that' which operates on sentences to produce a noun phrase, and the operator 'it is true' which operates on 'that' clauses to form sentences. This is a crucial feature of the theory which is necessary to prevent the 'substitution problem'. If we substitute the noun phrase 'that Tom runs' for 'p' in 3 above, we get the nonsensical
(*) The statement 'that Tom runs is true' says no more than the statement 'that Tom runs'
which is nonsensical because 'that Tom runs' is not a statement, but a noun phrase. If by contrast we substitute the sentence 'Tom runs', we get
(**) The statement 'Tom runs is true' says no more than the statement 'Tom runs'
which is nonsensical because 'Tom runs is true' is not a well-formed sentence (it would be well-formed if we read it as 'that Tom runs is true', i.e. read the sentence as a that-clause, but then we are not substituting the same thing salva significatione. (To be fair, the IEP article does note the problem of substitution, but does not sufficiently explain how the prosentential theory overcomes it).
4. " The prosentential theory of truth counts as a ‘deflationary’ theory because it denies that any analysis of truth of the form "(x)(x is true iff x is F)" can be given, where ‘x is F’ expresses a property that is conceptually or explanatorily more fundamental than ‘x is true.’ "
The assertoric theory is also a deflationary theory on this criterion. According to the assertoric theory, 'is true' operates on a noun phrase (a that-clause) to form a sentence. This operator is present, explicitly or implicitly, in the main verb of every declarative sentence. It is fundamental to the semantics of the declarative sentence, and no more fundamental explanation is available.
5. "The prosentential theory of truth implies a solution to the liar paradox. Consider the following sentence.- This sentence is false - if it says something true, it is false ... if it says something false ... it is true. [... ] Some attempts to solve the liar paradox involve extreme measures. Tarski, for example, thought that the paradox could be avoided only by eschewing ‘semantically closed languages’ [...] Tarski succeeds in avoiding the basic form of the liar paradox—but only at a very high price. He must content himself with providing an account of ‘true-in-Li’ rather than an account of truth. And, since natural languages like English are semantically closed, Tarski’s theory also has the weakness of applying only to artificial languages. ... According to the prosentential theory, (43) is neither true nor false because it fails to pick up an anaphoric antecedent. Just as I cannot inherit my own wealth, a prosentence cannot inherit its content from itself. "
The assertoric theory solves the Paradox in essentially the same way. 'is true' is fundamentally an operator, not a predicate. When attached to a 'that'-clause, it does not predicate anything or attribute anything to anything named by the clause. It merely signals that the speaker is asserting the content signified by the clause. In the case of the Liar sentence, there is no content to be asserted. According to the assertoric theory, we must analyse every declarative sentence into a content-part, and an assertoric part. Thus the Liar sentence must be analysed as
What this sentence says is false.
where the content part is signified by 'What this sentence says', and the assertoric part to the denial 'is false'. But the sentence says nothing, because the attempt to locate a referent for it fails.