# Checking the validity of the logical conclusion gleaned from a heated conversation

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:

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
– user38026
Commented Apr 4, 2019 at 18:21
• 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. Commented Apr 4, 2019 at 18:39
• This is more of a fun little exercise than anything.
– Meta
Commented Apr 4, 2019 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
– user38026
Commented Apr 4, 2019 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. Commented Apr 5, 2019 at 1:48

Aahh.... I see what happened.

There is a little known fact which explains the entire argument. Mathematicians are almost the only people in this world who would understand the logical fallacy committed.

Humans have more than one "equality relation" in our models of thought.

An "equality relation" is a definition of what it means for `x` and `y` to be:

• "the same" or

• "not the same."

For example, if you say...

Joe's car is the same as Sarah's. The cars are both 2019 Ford Mustangs

... then you just invoked an equality relation.

An "equality relation" is a definition of the equal sign "`=`"

Suppose Joe and Sarah buy two separate cars such that:

• the cars have have the same manufacturer (Honda or Toyota or Ford, etc...)
• the cars are of the same model (e.g. Nissan Altima)
• the cars have the same year manufacture (e.g. 2020)

Maybe the cars even have the same paint color.
In one sense they are NOT the same car, because at this very split-second moment of time, Joe's car is parked at Joe's house. Sarah's Car, however, is parked at Sarah's place of work.
If `x` and y are the same car, then `x` and `y` cannot be at two different places at the same time!!!
The cars are slightly different in other ways too.
Maybe Joe's car has a small, almost imperceptible tear in one of the leather seat cushions.
Sarah's seat cushions look brand new.
Also, the new cars have different license plates attached to them.

Guess what mathematicians do? They make two different definitions of what it means for things to be "equal" In order to tell the difference between the two, mathematicians use different symbols

• ≈ (approximately equal)
• ≡ (triple bar.... means really REALLY equally strongly equal.)

CHECK IT OUT!

DEFINITION OF ≡ (triple bar)

For all `C`, `K` in (the set of all cars),
`C ≡ K`
if and only if
For all `t` elements of the set of all nano-second precise date/time stamps (for example, maybe `t` is `2017-10-18 13:47:15.388551`) cars C and K contain the same occupy the same points in space

Definition number two:

DEFINITION OF = (equal sign)

For all `C`, `K` in (the set of all cars),
`C = K` if and only if cars C and K have the same make, model, and paint color (e.g navy blue 2019 Ford Mustangs)

So, we have `(Joe's car) = (Sarah's Car)` and `NOT [(Joe's car) ≡ (Sarah's Car)]`

• Joe and Sarah's cars are the "same" because they are both navy blue 2019 Ford Mustangs
• Joe and Sarah's cars are the `NOT` the "same" because they occupy two different places in space-time.

As a second example, consider symbolic logic:

• "`it is false that (P and Q)`"
• "`it is true that [(not P) or (not Q)]`"

The two strings are not the same.
- One string contains the sub-string "`or`"
- The other string does NOT contain the sub-string "`or`"

However, the strings are "logically equivalent."

"not (P and Q)" = "`(not P) or (not Q)`"
"not (P and Q)" ≢ "`(not P) or (not Q)`"

Is the person you were 5 years ago the same person that you are now, or a different person?
Is the person you were 5 years ago dead, or still alive?
Answer: The person you were 5 years ago double-bar equals you, but the person you were 5 years ago does not triple bar equals you. Two different definitions of "same"/"equal" Two different equality relations.

Most people assume there is one, and ONLY one definition of "`=`"
THAT IS A FALLACY!!!
In your story, that is the fallacy John committed.

John: "The truth" is "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.

Let us symbolize it, shall we?

DEFINITION of equality relation tilde (~)

``````For any two statements, `P` and `Q`, `P ~ Q` {
If {
there exists set `S` and property `\$` such that {
and {
`P` claims that there does not exist `x` in `S` such that `\$(x)`
`Q` claims that for more than 50% of `x` in `S`, `\$(x)`
}
}
} then {
`P ~ Q`
} end implication conclusion
} end "for all" block
``````

The following two statements are equal by equality relation tilde (`~`):

• "Not a single sentence, `P`, in the set {sentences in this book which made a point on the topic of `t`} had the property that `P` was coherent and `P` was substantiated and `P` was stand-alone/independent.

• "More than 50% of sentences, `P`, in the set {sentences in this book which made a point on the topic of `t`} failed to be all of (coherent, substantiated, stand-alone/independent).

You could define `=` to mean "agree for the most part."

The argument looks like:

John: I believe X. Do you agree?
Jane: I believe Y and Y~X
John: Ah! So What you believe has the property that Y=X! Jane: No, that is not what I said! I said Y~X , not Y=X

Basically, john conflates "I partially agree with you" with "I fully agree with you."

John: I think X
Jane: I, at least, very slightly disagree with X
John: Ah! So you agree with X! Jane: No, that is not what I said! I said partially disagree!

Another analogy:

John: I am black
Jane: I am grey
John: Ah! So you are are black too!
Jane: No, that is not what I said! I said grey not black!

Note.... John lives in a black and white world which has no shades of grey.

John believes: If you are less than 100% against him, then you are 100% with him.

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).

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
Commented Apr 4, 2019 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
– user38026
Commented Apr 4, 2019 at 23:24