Theorem about Diaz's truth-relevant logic

[X.Y. Newberry]

Is this theorem correct?

Let *A*, *B* be propositional formulas. The equal sign ‘=’ shall be
interpreted according to Kleene’s strong tables. (Kleene’s strong tables
have three values: true, false, unknown, denoted as *T*, *F*, *x*.)

*Definition:* A propositional formula *L* is a *tautology* iff it evaluates
as true on all combinations of *T*, *F* assignments to the propositional
variables occurring in *L* (i.e. x is not considered.)

There are tautologies such that they evaluate as true on all combinations
of *T*, *F* assignment to a *proper subset* of the variables occurring in
*L*. These propositional variables are called *truth-determining*. The rest
of the propositional variables (*truth-redundant* variables) could be *x*
yet *L* will still evaluate as *T*.

*T**heorem 4.1* Let *L*(*P**1**, P**2**, … P**n*) be any tautology such that
*P**1**, P**2**, … P**n* are all the variables occurring in *L*, and let
our connectives be *interpreted **according to *Kleene’s strong tables.
Then *L* is equivalent to

*(**R**1 **⋁ ~**R**1**) & (**R**2 **⋁ ~**R**2**) **&** … **& **(**R**m* *⋁
~**R**m**)*                              (4.1)

if and only if {*R**i*} is a set of truth-determining variables occurring
in *L*. Naturally {*R**i*} ⊆ {*P**i*}.

*Proof:* First we prove that {*R**i*} contains all the truth-determining
variables. So suppose that there exists a truth-determining variable
*G* occurring
in *L* but not in {*R**i*}. Suppose further that for all *R**i*, |*R**i*| =
*T* or |*R**i*| = *F*, and |*G**|** = **x*. Then the truth value of (4.1)
is *T* but the truth value of *L* is *x*. Now suppose that {*R**i*} contains
a redundant variable *H*, and |*H**|** = **x,* all other variables in {*R*
*i*} are *T* or *F*. Then the truth value of (4.1) is *x*, but the truth
value of *L* is *T*.


Truth-relevant Logic – Propositional Calculus

https://www.researchgate.net/publication/330844403_Truth-relevant_Logic-Propositional_Calculus

p. 14


-- 
X.Y. Newberry

*There are two ways to be fooled. One is to believe what isn't true; the
other is to refuse to believe what is true.*
― Søren Kierkegaard
<https://www.goodreads.com/author/show/6172.S_ren_Kierkegaard>
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