Motivation:
I'm interested in how different people resolve the Liar paradox and other, related phenomena, like the revenge Liar paradoxes, and so on.
I have a copy of "Formal Theories of Truth," by Beall et al. and recently I got a digital copy of "Logic Without Gaps or Gluts", by Burgis. However, I have difficulties getting off the ground with them.
The Question:
What exactly are capture and release in (nonclassical) logic?
The Details:
Simply put, my understanding is that, for each statement $\alpha$, there is some other statement ${\rm Tr}(\ulcorner \alpha\urcorner)$ that means, "$\alpha$ is true" . . . somehow . . . where $\ulcorner \alpha\urcorner$ is the "name" of $\alpha$, which is perhaps the first place my understanding breaks down.
Now:
- Capture:$$\alpha \vdash {\rm Tr}(\ulcorner \alpha\urcorner).$$
- Release:$${\rm Tr}(\ulcorner \alpha\urcorner)\vdash \alpha.$$
Thoughts:
My intuition fails to grasp the notion that a statement $\alpha$ can entail "$\alpha$ is true" or vice versa.
Maybe this is due to me being used to first order logic. Does that make sense?
I don't know how to articulate this exactly but, from what I remember of a proof of Gödel's Incompleteness Theorems (I taught myself), something similar goes on, but with natural numbers and the use of the Fundamental Theorem of Arithmetic to describe some $\varphi$ then make $\varphi$ about the number you get; for example:
$$\exists x(Px\to \forall yPy)\tag{$\alpha$}$$
would be something like
$$G:=2^13^25^37^411^213^517^619^723^429^731^8$$
because the symbols map to indices like so:
$$\begin{align}\exists &\mapsto 1,\\x&\mapsto 2,\\( &\mapsto 3,\\P&\mapsto 4,\\\to &\mapsto 5,\\\forall &\mapsto 6,\\y&\mapsto 7,\\)&\mapsto 8.\end{align}$$
Is ${\rm Tr}(\ulcorner \alpha\urcorner)$ (or perhaps $\ulcorner \alpha\urcorner$) like $G$?
That isn't my question; my question is what is highlighted above. This is just me trying to describe my understanding.
NB: I have included the intuition tag because I suppose what I'm getting at is, what is the intuition behind capture and release?