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I have two friends - call them John and Jane.

I was recently privy to an argument concerning a book between John and Jane that went like this:

John: This book did not make a single coherent, substantiated point on the topic of t. Do you agree?

Jane: That is not a stance that I hold, given the phrasing used, so I do not agree.

John: Ah, so you do agree with my claim.

Jane: No, that's not what I said.

(the topic itself is not important)

You can probably imagine how the argument devolved from this point, but something seemed off to me about John's conclusion. I decided to try formulating their exchange as a series of logical steps to check whether John was correct to come to the conclusion that he did after hearing Jane's statement.

  1. John's claim:

This book did not make a single coherent, substantiated point on the topic of t.

I translated this as follows:

Let P be the set of points raised in the book that is the subject of discussion.
Let Coherent(x) be the statement "x was a coherent point".
Let Substantiated(x) by the statement "x was a substantiated point".

John's claim then becomes:

For all p in P, it is not the case that Coherent(p) and Substantiated(p).

  1. Jane's rebuttal:

That is not a stance that I hold, given the phrasing used, so I do not agree.

I translated this as follows:

Let c be John's claim.
Let Hold(x) be the statement "x is a statement that is consistent with my views."
Let Phrased(x) be the statement "the statement x is phrased in such a way that I can either agree or disagree with it."

Jane's rebuttal then becomes:

It is not the case that Phrased(c), therefore, it is not the case that Hold(c).

As Jane is speaking about her personal view on whether she agrees with c or not, we can assume that her statement is true.

Jane tells us that Phrased(c) is false in her rebuttal.

After substitution, her rebuttal becomes:

It is not the case that Phrased(c) is false, therefore, it is not the case that Hold(c).

Which simplifies to:

The inverse of Phrased(c) is true, therefore it is not the case that Hold(c).

Now it seems to me John should not be able to come to the conclusion that Hold(c) is true. Looking at the truth table for inverse logical implication:

truth table for inverse logical implication

There is exactly one case where ~Phrased(c) is true and ~Phrased(c) -> ~Hold(c) is true: row 1 of the truth table. Therefore, John should have come to the opposite conclusion.

We can even plug these values into Jane's rebuttal and get an English sentence that makes sense intuitively.

If it is not the case that c is phrased in such a way that I can either agree or disagree with it, then it is not the case that c is a statement that is consistent with my views.

The statement "It is not the case the c is phrased in such a way that I can either agree or disagree with it" is true, therefore, the statement "it is not the case that c is a statement that is consistent with my views" must also be true.

Did John make the wrong conclusion from Jane's rebuttal? Did I check (and refute) his reasoning correctly? Did I translate Jane's rebuttal into logical expressions correctly?

  • can you rewrite this sentence "As Jane is speaking about her personal view on whether she agrees with c or not, we can assume that her statement is true": are you saying the entire statement "It is not the case that Phrased(c), therefore, it is not the case that Hold(c)." is true? if so i don't see the point of disagreeing – another_name Apr 4 at 18:21
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    Using formal logic to judge informal arguments is not always helpful. Many informal conclusions rely on context-specific judgments and only heuristically valid inferences that would not show formally. In this case, John's "did not make a single" is clearly a rhetorical exaggeration, which would not be taken literally by any competent speaker. Then Jane's "given the phrasing used" is what is called equivocation. It is not unreasonable to interpret it as her disagreeing only with the phrasing used, but not with the underlying (not literally expressed) opinion, as John does in his response. – Conifold Apr 4 at 18:39
  • This is more of a fun little exercise than anything. – Meta Apr 4 at 19:41
  • depends on what the book and topic t is. many books make no substantiated coherent point on many topics. i guess i seem pedantic, but this all feels artificial and off topic – another_name Apr 4 at 23:40
  • Jane says she does not agree with John's claim. I don't see why one is analyzing the structure of John's claim. Just call it "c". Perhaps the problem is John is claiming that all points were incoherent and unsubstantiated. To negate that Jane simply needs one point that was coherent or substantiated. – Frank Hubeny Apr 5 at 1:48
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John: This book did not make a single coherent, substantiated point on the topic of t. Do you agree?

Jane: That is not a stance that I hold, given the phrasing used, so I do not agree.

John: Ah, so you do agree with my claim.

Jane: No, that's not what I said.

Is Jane's rebuttal, as you construe it, true? That,

It is not the case that Phrased(c), therefore, it is not the case that Hold(c).

I read this as

  1. if she can neither agree nor disagree with the statement as it is phrased then the statement is not consistent with her views, and

  2. she can neither agree nor disagree with the statement.

For what it's worth, there do seem to be statements that I can neither agree nor disagree with but are consistent with my views. A 19th century astronomer could, I think, neither agree nor disagree that Pluto is a planet, even if they're looking for more planets.

One might guess that Jane is saying that to judge a phrase as true or false we need to understand what that phrase means. That seems reasonable, but I have no idea what it has to do with "consistent views" "holding" etc..

  • The definition of Phrased(x) is my attempt to translate Jane's statement in English into a logical one; one facet of my question is whether I have done this reasonably correctly, or not. When she says "That is not a stance that I hold with the phrasing chosen", I take that to mean "Because the claim is not even phrased in a way that I can either agree or disagree with it, then it is not valid to claim that I agree with it. however, I may agree with it were it phrased differently." I then translated this to logical implication: ~Phrased(c) -> ~Hold(x). Can you suggest a better translation? – Meta Apr 4 at 23:18
  • "I may agree with it were it phrased differently" you could look at how to notate whatever you mean by "may". do you mean in some sense possible?? it's obvious that if jane can't agree with it so phrased, and isn't just saying that, then the claim she does is false, so not sure why you'd want to notate it to convince someone @Meta – another_name Apr 4 at 23:24

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