How are you intended to interpret x.R and x.S in Davidson "Truth and Meaning"?

Question

I'm entirely unfamiliar with the syntax and corresponding semantics Davidson uses for the four equations at the beginning of Truth and Meaning(https://www.jstor.org/stable/20114563). He states that the following sentences are supposed to be equivalent in some invalid theory, but I'm not sure what two of them are even supposed to mean in the naive theory. The four sentences are:

(1) R
(2) x̂(x=x.R)=x̂(x=x)
(3) x̂(x=x.S)=x̂(x=x)
(4) S

His conclusion is that the naive (possibly Fregean?) theory doesn't work because it says that all sentences with the same truth value must be synonymous. I get that that is an odd result, but I don't understand what these four equations are communicating to get that result. Thanks for the help!

Answer 1

The phrase x̂(x=x) means 'the x such that x=x'. This is just a way of forming a singular term that refers to something. The . is conjunction ('and').

So x̂(x=x.R)=x̂(x=x) is logically equivalent to R. And, since R is logically equivalent to S (by assumption), both x̂(x=x.R) and x̂(x=x.S) refer to the same thing. And then x̂(x=x.R)=x̂(x=x) and x̂(x=x.S)=x̂(x=x) refer to the same thing. This is how Davidson gets to the conclusion that any two sentences have the same reference if they are logically equivalent.