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?
In general, $\text T \ulcorner \phi \urcorner$ is a "name" for expression $\phi$; in the classical Gödelian approach to Arithmetization of Syntax the name can be produced via encoding.
According to Deflationsists; "there is some strong equivalence between a statement like ‘snow is white’ and a statement like “‘snow is white’ is true,” and this is all that can significantly be said about that application of the notion of truth."
You can compare with JC Beall and David Ripley's Nonclassical theories of truth, Ch.2 Reasoning with truth for the "name" of a sentence $A$ and transparency: "the principle that $A$ and $\text T \ulcorner A \urcorner$ are intersubstitutable with each other in all non-opaque contexts."
And see page 3: "Capture and release are argument forms or 'rules of inference'. Capture is the rule going from $A$ to $\text T \ulcorner A \urcorner$, the idea being that the truth predicate 'captures' the content of $A$, and release is the converse, the rule from $\text T \ulcorner A \urcorner$ to $A$."
The theory supporting so-called Tarski's T-schema: "$\text T \ulcorner \phi \urcorner \leftrightarrow \phi$, where $\text T$ is the truth predicate, $\phi$ is a sentence and $\ulcorner \phi \urcorner$ is a name for the sentence $\phi$", is the analysis of "truth talk" in terms of bi-conditionals like: "The proposition that Brutus killed Caesar is true if, and only if, Brutus killed Caesar."
Tarski's formalization of this intuition is reminiscent of so-called correspondence theory, the theory "that what we believe or say is true if it corresponds to the way things actually are – to the facts."
See also Aristotle's Metaphysics, Book IV, 1011b25: "we define what the true and the false are. To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true."