It might be interesting to start a thread on the various logical theories of truth, and related ideas. In my enthusiasm, which might well evaporate, I anticipate discussing Tarski's T-sentences, Kripke's three-valued logic and revision theory.

This all by way of improving my own understanding by paraphrasing the formal stuff in less formal terms.

There's a strategy that Tarski used in his original paper, and that Davidson later mirrored. I'll have a go at paraphrasing it.

We want a theory of truth.

We can ask what such a theory might look like. If it is adequate to its task, it will deliver, for every sentence, something that tells us if that sentence is true.

So it will have the form

For any sentence p, p is true if and only if ϕ

Further, to avoid circularity, the notion of truth cannot occur in ϕ.

And finally, this will not work for a language strong enough to talk about its own sentences, because directly it will be able to generate a sentence of the form

This sentence is false

Putting these together, if we have as one of our sentences

Snow is white

then our theory will produce a sentence in the metalanguage that looks like

"snow is white " is true iff s

where s is a sentence in the metalanguage.

You should be able to see where this is going. All we need to do now is work out what s might be.