‘The claim that I had an affair with Miss A and that I didn’t father her child is false’

Question

The prosecutor asks, ‘Did you father the child of the murdered victim Miss A?’. Mr. N replies, ‘The claim that I had an affair with Miss A and that I didn’t father her child is false’. In response to the prosecutor’s request for a clarification of this statement, Mr. N says, ‘What I mean is that there is at least one falsehood in the claim that I had an affair with Miss A and that I didn’t father her child.’ The prosecutor then asks, ‘Did you have an affair with Miss A?’ Mr. N replies, ‘Yes, I did.’ Assuming that Mr. N uses language in the usual way, but without knowing whether Mr. N is truthful in answering these questions, what can you deductively infer from Mr. N’s answers? (Note: When my statement X implies statement Y, whether I intend to state Y is irrelevant. X implies Y just when Y follows logically from X. For example, when I say that I’m a man, my claim implies another claim that I’m a human being, whether or not I intend to make the latter claim.)

I. Mr. N had an affair with Miss Α.

II. Mr. N did not father Miss A’s child.

III. Mr. N implied the claim that he had fathered Miss A’s child.

IV. Mr. N implied the claim that he had not fathered Miss A’s child.

A. I, II

B. Only III

C. Only IV

D. II, III

So this question has a few premises, and it asks us to deductively infer the "conclusion" if I am not wrong. So the premises say that

1. Mr N says that there is at least one false statement in the claim that he had an affair with Miss A and that he didn’t father her child.

2. Mr N claimed he did had an affair with Miss A.

Hence to me it logically follows that Mr N fathered the child since Mr N says that at least one of the statements he made were false. So if having an affair with Mr N is not false, then he did not father her child must be false.

So is the answer just simply B)? I am not sure why the question mentioned without knowing whether Mr. N is truthful in answering these questions. Does that have an impact in our deductively inferring of Mr N's answers?

Answer 1

The question is:

So is the answer just simply B)? I am not sure why the question mentioned without knowing whether Mr. N is truthful in answering these questions. Does that have an impact in our deductively inferring of Mr N's answers?

Construct the following symbolization key based on Conifold's suggestion:

• A: Mr. N had an affair with Miss A.
• F: Mr. N fathered Miss A's child.

The premises are the following statements that Mr. N has made:

‘The claim that I had an affair with Miss A and that I didn’t father her child is false’

1. ¬(A ∧ ¬F)

‘What I mean is that there is at least one falsehood in the claim that I had an affair with Miss A and that I didn’t father her child.’

2. ¬A V ¬¬F

‘Did you have an affair with Miss A?’ Mr. N replies, ‘Yes, I did.’

3. A

The above are the three premises. They may be false, because Mr. N might be lying, but assuming they are true what can we conclude from them, that is, what can we deductively infer?

We can eliminate option "D" because II and III are contradictory.

We can eliminate option "C" because Mr. N claimed he had an affair and at least one of the two statements that he had an affair and did not father the child is false. It seems that he did imply the claim that he fathered the child.

We can eliminate option "A" for the same reason as "C". At least one of I and II are false and we know that I is true.

That leaves only option "B" which would be a good one to select under time pressure.

However, the OP's question is about deductively inferring F, that Mr. N is the father of Miss A's child. That would be possible using rules of inference and assuming the premises are true. If we can provide a proof that from the premises we can derive F then, we can deductively infer F assuming the premises are true.

With the symbolization key and the premises we can put this in a proof checker as follows:

So we have deductively inferred that Mr. N fathered Miss A's child. If he was lying or if there were other evidence that needed to be taken into consideration, then the premises would change and we might deductively infer some other conclusion.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Answer 2

Here's the crucial distinction: We don't know if Mr. N fathered Miss A's child, because we don't know if he's truthful. But we certainly do know he IMPLIED he fathered Miss A's child because it's a logical entailment of what he said.

In my opinion we can also infer that he had an affair with Miss A, because he came right out and said so, but since no answer combines I and III, I'm going to assume they don't consider that a "deductive inference" since he just straight admitted to it. This, despite the fact that you can always deductively infer X given X as a premise.

I'd imagine this example is introduced as a foreshadowing of how sure conclusions can be derived from unsure premises --in this case the conclusion "N -> F" where N = "Mr. N is truthful" and F = "Mr. N fathered the child," which is an example of arrow introduction. However, it does puzzle me that they didn't include the more obvious trick answer "Mr. N fathered the child" (which we don't know, given that we can't trust Mr. N's testimony) as one of the options.