LOGIC Course

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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