Question

5. Symbolize the following argument and prove it is a valid argument. Let B ( x ) = x is a bear; D ( x ) = x is dangerous, and H ( x ) = x is hungry.

Every bear that is hungry is dangerous.
There is a hungry animal that is not dangerous.
Therefore there is an animal that is not a bear.

6.  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.

1. Prove the following argument is invalid by finding a quantificational model in which all the premises are true and the conclusion is false.
   1. ∀x ( ( S ( x ) ∧ P ( x ) ) → R ( x ) ), Premise
   2. ∀x ( T ( x ) → ( ¬ R ( x ) ∧ ¬ P ( x ) ), Premise
   3. ¬ ∃x ( R ( x ) ), Conclusion

Answer 1

5.

Below 3 are premises

1. Every bear that is hungry is dangerous.
2. There is a hungry animal that is not dangerous.
3. Therefore there is an animal that is not a bear.

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