## LOGIC Course

# Semantic proof

## 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.

I hope you enjoy this one!