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.
- 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).
- 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:
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?