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Determine whether the following sentence is satisfiable, contradictory or valid: P → Q → ~P

A predicate logic sentence w is satisfiable if there exists some interpretation in which w is true. A predicate logic sentence is unsatisfiable (i.e., it is a contradiction ) if it is not satisfiable (in other words, there exists no interpretation in which it is true). It is valid if every interpretation is true.


We know that, P implies Q can be written as,

~P ∨ Q.


Therefore, the sentence becomes,

~P ∨ Q → ~P


Further solving,

~P ∨ Q → ~P

= ~(~P ∨ Q) ∨ ~P

= P ∧ ~Q ∨ ~P

= P ∨ ~P ∧ ~Q

= TRUE ∧ ~Q

= ~Q


Now, ~Q can either be true or false. Hence, the sentence is satisfiable.

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