# many valued logic question

i have problem solving the following exrecise in many valued logic given the many valued logic $L_5$ defined as follows

$S = \{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\}$ the possible truth values
$D = \{1\}$ the designated set

and the following truth function
$\to (a,b) = \begin{cases} 1 & a \leq b \\ b & a>b \end{cases}$

a) is $((p\to q)\to p) \to p$ a tautology?
this is false using the followin assignment $v[p] = \frac{1}{4} ,v[q] = 0$
$v[((p\to q)\to p) \to p] = \frac{1}{4}$
thus it isn't a tautology

the problem is in the following section:
b) prove that using the following proof system:
axioms:
I1. $A \to (B \to A)$
I2. $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$

inference rule
$A, A \to B$ infer $B$
is not enough to prove any tautology in the classic propositional calculus (true and false possible truth values) use section a

i have no idea how to prove b

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Hint: I1 and I2 are tautologies in $L_5$. Also modus ponens (the inference rule in you system) when applied to tautologies in $L_5$, produces a tautology in $L_5$. Hence everything you derive in your proof system is a tautology in $L_5$. So you need only to find a formula that is a tautology in the classical propositional logic, but not a tautology in $L_5$.