Reductio ad Absurdum is preserved on glut theory

https://www.youtube.com/watch?v=xkMTC5TfpiE

One objection to dialetheism, or any kind of glut theory, is that by giving up the Law of Non-Contradiction* we give up the truth seeking tool of reductio ad absurdum. But this isn't true. In a reductio, we start with a conclusion, like God exists, and from there, combined with other necessary truths, conclude that God does not exist. If negativity and positivity cannot overlap, and thus all contradictions are strictly false, then showing that A leads to ~A shows that A is strictly false. Given glut theory, you lose the strictly part of the reductio. Showing that A leads to ~A, on glut theory, shows either that A is strictly false or that A is true and false.

 

But there never has been a case where someone has had orange juice in their fridge and also has not had orange juice in their fridge. There never has been a case where someone is both in Sidney, Australia and not in Sidney, Australia. There has never been a case where someone has both casted a ballot and not casted a ballot. Etc. It seems that for virtually any imaginable contradiction, there's no good reason to think that it's anything other than strictly false. If the only plausible candidates for true contradictions are found in self-reference, law, motion, mathematics, maybe quantum mechanics, and maybe ineffability, then that leaves nearly all imaginable contradictions strictly false. So showing something to be a contradiction is to show that it's a priori almost certainly false. The theist who wants to escape the above reductio's conclusion that God does not exist is forced to refute the reductio or argue that God's existence is a true contradiction – that it's true and false that God exists. But 1) the vast majority of theists would not feel comfortable with this conclusion (probably because their intuitions say that it's impossible for something to be true and false), and 2) this is quite the burden to bear for the theist, considering that God's existence is not found in the list of plausible dialetheias. 
 
So reductio ad absurdum is still a powerful tool on glut theories, and indeed it retains nearly all of its strength. In fact, reductio could be a key tool in discovering further plausible dialetheias. First, use reductio to establish a contradiction. Second, argue that this contradiction is a dialetheia. So while I don't subscribe to any glut theory, the "loss of reductio ad absurdum as a truth seeking tool" objection to glut theory is not a viable objection at all.
 
*Classical logic is typically associated with three laws: The Law of Non-contradiction, the Law of Excluded Middle, and the Law of Identity. You might also include the Law of Bivalence. But why can't we simply reduce all of that to the Law of Excluded Middle? 
 
LEM: All propositions are either true or false.  
 
This is equivalent to: For any proposition, either it is true or its negation is true.
 
∀x(Px⟶Tx∨Fx) "For all x, if x is a proposition then x is true or false."
 
But if something is true or false, then it's a proposition. So:
 
∀x(Tx∨Fx⟶Px)

 

So: ∀x(Px⟷Tx∨Fx)
 
But this means that something is a proposition if and only if it is true or false. But that's not right. A proposition is more than just being something that is true or false. A proposition will include other elements like bundling references to properties and relationships between properties (or whatever your favorite theory of propositions says).

 

Easy fix: ∀x(Px⟷Hx∧(Tx∨Fx))

Something is a proposition if and only if it has [insert preferred theory of propositions here] and is true or false. This assumes acceptance of LEM and separates out the LEM from the other parts of the theory of propositions.

 

On the theory I'm playing around with, a proposition is true when it is comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences.

 

So: ∀x(Rx⟷Tx) = If and only if when something is comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences, then it is true.

 

∀x(~Rx⟷~Tx) = If and only if when something is not comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences, then it is not true. 

 

∀x(Qx⟷Fx) = If and only if when something is comprised of references to properties and relationships between properties that we do not experience and that do not explain our experiences and cannot explain anyone's experiences, then it is false.

 

∀x(Fx⟶~Tx) = When something is false, it is not true.

 

∀x(Fx⟶~Rx) = When something is false, it is not comprised of references to properties or relationships between properties that we experience or that explains our experiences or can explain someone's experiences.
 
A rock is not true, because rocks are not propositions and only propositions can be true, but a rock is not false, because only propositions can be false. So 'not true' and 'false' are not equivalent, but are mutually entailing when it comes to propositions.  
 
This by itself entails a) there are no gaps, as no propositions can be neither true nor false; b) there are no gluts, as no propositions can be both true and false; c) that A=A (because truth and falsity are assumed to be different things) and d) that there are only two truth values: true and false (otherwise, the law would read: Propositions are true or false or a secret third thing, with that third [or fourth, fifth, etc.] thing spelled out by a many-valued logic).