Consider the following invalid argument: Invalid argument (3x) Jx Hx) Complete a

Question

Consider the following invalid argument: Invalid argument (3x) Jx Hx) Complete an indirect truth table for the interpretation of the given argument in a finite universe with only one member (a): Ja Ha Ja Ra Ja Ha Ra Enter truth values as you determine them. Click x to mark a contradiction if you find one. Check Answer Start over Assumptions The premises were assumed to be and the conclusion to be Results Since this results in a finite universe with only one member is to demonstrate that the given argument is invalid.

Explanation / Answer

Answer: For a indirect truth table

1) Assumption: The premises are assumed to be true and the conclusion to be false.

Ja . Ha / Ja . Ra // Ja . ( Ha . Ra )

F T F     F T F      F F F F F

For the given argument to be invalid, premises should be true and conclusion be false. For the argunment, given operator is OR which is false when any one of its arguments or both of its arguments are false. In conclusion, we assume that all the arguments (Ja, Ha, Ra) are false. This also implies that for the assumed values of Ja, Ha and Ra, premises will be false, but they are required to be true. This raises a contradiction because it need to be true and false both at the same time.

Results: Since this results in a contradiction, a finite universe with only one member is not sufficient to demonstrate that the given argument is invalid.

2. Assumption: The premises were assumed to be true and the conclusion to be false

For the given operation "OR", if we look at the assumed values of arguments in conslusion so that it is false, it becomes clear that premises will also need to be false. But as per assumption, they should be true when the conclusion is false for the statement to be invalid. This leads to contradiction.

Results: Since this results in a contradiction, a finite universe with two members is also not sufficient to demonstrate that the given argument is invalid.

(Ja . Ha) V (Jb . Hb) / (Ja . Ra) V (Jb . Rb) // [Ja . (Ha . Ra)] V [Jb . (Hb . Rb)] F T F T F T T F F T T F F F F F F F T F F T F F

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