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Identify the premises and conclusions of the following deductive arguments and analyze their logical forms. Do you think the reasoning is valid?

Either John or Bill is telling the truth. Either Sam or Bill is lying. Therefore, either John is telling the truth or Sam is lying.

I have identified the premises:

  1. Either John or Bill is telling the truth.
  2. Either Sam or Bill is lying.

I have identified the conclusion as: Therefore, either John is telling the truth or Sam is lying.

My problem is determining whether the reasoning is valid (apparently it is) but I can't understand why.

I am told the argument is valid if the premises cannot be all true without the conclusion being true as well.

From this, I have one question:

Q: Why exactly is the argument true? I am confused. The conclusion has to force both of the premises to be true, but it doesn't. Suppose both John is telling the truth and Sam is telling the truth. This satisfies the conclusion and forces premise 1 to be true, but says nothing about premise 2. Either Bill is telling the truth and the argument is false, or Bill is lying and it is true.

EDIT: Sorry if my question is confusing/wrong. I am confused basically about why the argument is true, and how I can verify if it is true

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  • $\begingroup$ I think you have to exploit the fact that "lying" is the same as "not telling the truth". Thus, if we symbolize "Bill is telling the truth" as Truth_telling(Bill), then "Bill is lying" will be not-Truth_telling(Bill). In this way, you can use the Excluded Middle principle : Truth_telling(Bill) or not-Truth_telling(Bill) ... $\endgroup$ Jul 22, 2015 at 8:44
  • $\begingroup$ But there is a possibility the conjunction is false (Truth(John) or Truth(Bill)) and (not Truth(Sam) or not Truth(Bill)), even if the conclusion is true. When John is telling the truth, Bill is telling the truth, and Sam is telling the truth. So then isn't the argument invalid? $\endgroup$
    – Jason
    Jul 22, 2015 at 9:00
  • $\begingroup$ Exactly : in the "usual" case of an argument with a finite number of premises, the argument is equivalent to a new one with as (single) premise the *conjunction of the original premises and the same conclusion. I.e., having as original argument : "if $p_1, p_2, \ldots, p_n$, therefore $q$", we can rephrase it as : "if ($p_1$ and $p_2$ and ... $p_n$), therefore $q$". $\endgroup$ Jul 22, 2015 at 9:02
  • $\begingroup$ No : a valid argument "works" for true premises; in this case it licenses to assert the truth about the conclusion. When one or all the premises are false, we simply cannot conclude nothing. The definition is : it is not possible that all the premises are true and the conclusion is false. $\endgroup$ Jul 22, 2015 at 9:07
  • $\begingroup$ Thanks, the above definition solved my problems. $\endgroup$
    – Jason
    Jul 22, 2015 at 9:13

2 Answers 2

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You identified correctly the premises:

  1. Either John or Bill is telling the truth.
  2. Either Sam or Bill is lying.

Now, either Bill is lying or (exclusive) Bill is telling the truth, right?

If Bill is lying, then in order for Premise 1 to be true, John must be telling the truth.

If Bill is telling the truth, then in order for Premise 2 to be true, Sam must be lying.

Altogether, we get that either John is telling the truth or Sam is lying.

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Your example says something about premise 2: if Sam is telling the truth, the second premise is true only if Bill is lying and the conclusion is true.

You have two possibilities : John tells the truth or is lying.

If John tells the truth : the conclusion is true.

If John is lying : Bill tells the truth because of premise 1, so Sam must be lying because of premise 2 : the conclusion is true.

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