Classical Tautologies in a Deviant Logic of Paradox

Question

I am stuck with the following problem. The Logic of Paradox (LP) has the truth values {f,p,t}, designated values {p,t} and the following truth tables:

   A  ~A   A -> B   A & B   A v B
   t  f    t t t    t p f   t t t
   p  p    p p t    p p f   t p p 
   f  t    f p t    f f f   t p f 

A known theorem between LP and classical logic (M2) is that the Logic of Paradox can reproduce the classical tautologies as follows:

$$\vDash_{LP}\varphi\iff\vDash_{M2}\varphi$$

Now I am experimenting with a deviant Logic of Paradox (LP') where negation and implication is replace by the following truth tables, but conjunction and disjunction are the same as above:

   A  ~A   A -> B
   t  f    t t t 
   p  f    p p t 
   f  t    f f t  

Does $\vDash_{LP'}\varphi\iff\vDash_{M2}\varphi$ still hold?

Remark:
To prove $\vDash_{LP}\varphi\iff\vDash_{M2}\varphi$, first observe that the direction $\vDash_{LP}\varphi\Rightarrow\,\vDash_{M2}\varphi$ is tivial. For the direction $\vDash_{LP}\varphi\Leftarrow\,\vDash_{M2}\varphi$ one can go observe that we have the following valuation identities:

 A -> B = ~A v B
 ~(~ A) = A
 ~(A v B) = ~A & ~B
 ~(A & B) = ~A v ~B

So from a formula $\varphi(x_1,\dots,x_n)$ one can obtain a negation normal form formula $\varphi'(x_1,\dots,x_n)$. And then a monotonic formula $\varphi'(x_1,\dots,x_n,y_1,\dots,y_n)$ and the side conditions $x_1\neq y_1,\dots,x_n\neq y_n$. Since the formula is monotonic it will be also satisfied when side condition is violated via $x_j\&y_j$, the satisfaction corresponds to the designation {p,t} and the pair to the truth value p.

For the deviant Logic of Paradox the same negation normal form proof is not anymore so easily obtained, since already the identity ~(~ A) = A fails:

 A ~(~ A)
 t  t
 p  t
 f  f

So what would be proof idea for $\vDash_{LP'}\varphi\iff\vDash_{M_2}\varphi$? Or is there a counter example?

Answer 1

$LP'$ has still the same theorems as classical logic.

Intuitively, the reason is that $LP'$ cannot distinguish between the designated truth values: Given a three-valued interpretation $I$, if we change some values $p$ to $t$ or $t$ to $p$ in $I$, then the value of $I$ on a given formula never switches from designated to undesignated (it might switch from $p$ to $t$ or vice versa, while value $f$ remains fixed).

This property can be readily verified for the truth tables. (Do it graphically: if you switch between a $t$-column and a $p$-column in the same row, you either have two designated values or $f$ twice. Same goes for rows.)

By induction this property follows for all formulas. Therefore any $LP'$-countermodel can be reduced to a classical countermodel (just identify the truth values $t$ and $p$). Conversely, every classical countermodel is also a $LP'$-countermodel as the truth tables of $LP'$ extend that of classical logic.