Prove the following argument is invalid by finding a quantificational model in which all the premises are true and the conclusion is false. ∀x ( ( S ( x ) ∧ P ( x ) ) → R ( x ) ), Premise ∀x ( T ( x ) → ( ¬ R ( x ) ∧ ¬ P ( x ) ), Premise ¬ ∃x ( R ( x ) ), Conclusion
Formal Proofs
In order to prove an quantificational argument invalid it is only necessary to find a quantificational model that shows it is possible for all the premises to be true and the conclusion false.
A quantificational model specifies a universe (domain of discourse), which can be empty, can consist of a finite number of entities, or can consist of an infinite number of entitites. Further, it must specify the set of entities that satisfy every propositional function (both predicates and relations) used in the argument.
Example: U = { 1, 2, 3, 4 }, P ( x ) (a predicate) = { 1, 2, 3 }, R ( x, y ) (a relation) = { ( 1, 1 ), ( 3, 2 ), ( 2, 3 ) }, and so on for every predicate and relation used in the argument.
(Note: a valid quantificational argument is one in which the conclusion is true in all possible models in which all the premises are all true. Using this definition is rather daunting, which is why we use the rules of inference to show quantificational arguments are valid.)
Prove the following argument is invalid by finding a quantificational model in which all the premises are true and the conclusion is false.
- ∀x ( ( S ( x ) ∧ P ( x ) ) → R ( x ) ), Premise
- ∀x ( T ( x ) → ( ¬ R ( x ) ∧ ¬ P ( x ) ), Premise
- ¬ ∃x ( R ( x ) ), Conclusion
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