Hey could somebody please check the following "proof". I am using generalized valued models for set theory in order to build a model for paraconsistent set theory. The approach i am following is very similar to Loewe and Tarafder (2016), but i am trying to verify some additional axioms a part from the ZF axioms.
Fix a model of set theory $$V^A$$ (we define the $A$-names as usual by transfinite recursion) and consider the following reasonable implicative deductive algebra $$A={<A,\wedge,\vee,\to,1,0>}$$ . An algebra $A$ is called a reasonable implicative deductive algebra if the following axioms hold :
- $A={<A,\wedge,\vee,\to,1,0>}$ is a complete distributive lattice,
- $x \wedge y \leq z$ implies $x \leq y \to z$,
- $x \leq y $ implies $z \to x \leq z \to y$, and
- $x \leq y $ implies $y \to z\leq x \to z$
- $(x \wedge y) \to z=x \to x\to (y \to z)$
Then we will use the following three valued logic $PS_3$=$<\{1,\frac12,0\},\wedge, \vee, \to , \neg >$ where the negation is a paraconsistent one so $\neg 1=0, \neg 0 =1, \neg \frac12 =\frac12$. The truth values of $\wedge $= min(x,y), for $\vee$=max(x,y) and implication is only false when x=1, y=0 and $x=\frac12$ and $y=0$(the rest of truth values are 1). This model is called $V^{PS_3}$ and is Loewes and Tarafder proposed model. But now i want to add a unary operator $o \varphi$ which reads "$\varphi$ is consistent" to our object language which is true when x receives classical truth values and false when $x=\frac12$ (Additionally we have also a consistency predicate $C$, which i propose should be evaluated similarly, but i am not sure about this). Now my claim is the following : $$\parallel \neg C(x)\to x \in x \parallel^{PS_3}=0 $$
Proof. Notice that given the truth table of the implication our axiom can only receive truth values $0$ or $1$. An easy inductive arguments shows that the consequent is false (theorem 12 of Loewes and Tarfaders paper) , so $$\parallel x \in x \parallel =0$$. Then we want to show that $$\parallel\neg C(x) \parallel >0$$. Therefore we can just take a name that does not receive a classical truth value as $$x =<\emptyset, \frac12>$$. Does this imply $$\parallel \neg C(x)\parallel=1$$ and therefore $\parallel \neg C(x)\to x \in x \parallel=0 $ ? Furthermore do you think that $$\neg o \varphi \to (\varphi \wedge \neg \varphi)$$ would hold in $V^{PS_3}$?