# Semantic proof ¶

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## Exercise ¶

Let's consider the following clauses :

• If Diego is angry then Jonathan's father dies.
• If Diego poisons Jonathan's father then Jonathan's father dies.
• Diego poisons Jonathan's father or Jonathan investigates.
• If Jonathan investigates then Diego is angry.

Prove that whatever happens Jonathan's father dies.

$p$, $i$, $a$ and $d$ are propositional variable meaning respectively:

• Diego poisons Jonathan's father
• Jonathan investigates
• Diego is angry
• Jonathan's father dies

So the clauses are represented by the following formulas :

• $(1) \quad a \Rightarrow d$
• $(2)\quad p \Rightarrow d$
• $(3)\quad p \vee i$
• $(4)\quad i \Rightarrow a$

So we want to prove that $\{(1),(2),(3),(4)\} \models d$
We call $I$ an interpretation satisfying $\{(1),(2),(3),(4)\}$

First case : If $I \models i$

Then with $(4)$, $I \models a$ : Indeed (cf. the truth table of the implication),
$I(i \Rightarrow a)=1$ that is $I(i) \Rightarrow I(a)=1$
But we assume that $I(i)=1$, then necessarily $I(a)=1$

In the same way, with $(1)$, $I \models d$

Second case : If $I \not\models i$

Then with $(3)$, $I \models p$ (cf. the truth table of the logical disjunction),
$I(p \vee i)=1$ that is $I(p) \vee I(i)=1$
But we assume that $I(i)=0$, then necessarily $I(p)=1$

And with $(2)$, $I \models d$

In conclusion, no matter what, Jonathan's father will die.
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