A doubt about 12.5 example 2

[yhirshhorn]

In 12.5 example 2 the book provides "Jan is neither bold, nor not-bold" as a limit to validity tests using TFL.

The problem described is that a possible symbolization would be: $'\neg J \wedge \neg \neg J'$ which is a contradiction. It is explained that we could have added "Jan is on the borderline of boldness" and that is why the original sentence is possible.

Does this not mean that our symbolization was faulty to begin with? True, "Jan is bold" would probably be symbolized as some sentence letter, say 'J', but since in reality the phrase "not-bold" is not the complement of "bold", is it not a mistake to symbolize it as $'\neg J'$ to begin with? Shouldn't we have added at least one more sentence letter for "not-bold", say 'N', and get the following symbolization: $'\neg J \wedge \neg N'$ which is not a contradiction?

Now, if this example comes to show that misuse of TFL can bring us faulty test results, it is, of course, true, but it is not a limit of TFL specifically but of any possible logic language, existing or hypothetical. Wouldn't it be clearer to mention that this is a limitation that is not unique to TFL?

[kedarmhaswade]

I believe you mean "bald", and not "bold".

I believe what the text is trying to say is that even though in TFL we adhere to the Law of Excluded Middle, in English that may not always be the case. Trying to TFL-symbolize such English sentences may have unacceptable results. Such adamant TFL-symbolization is what the text is trying to discourage. We have just got to accept that TFL has its limitations. Those limitations are likely to remain even if you symbolize some English sentences differently.

A similar example appears in 5.1 where Jane is in a state of blank indifference.

[yhirshhorn]

I understand what you are saying, yet I still think it is wrong to try to show a limitation of TFL by misusing it.

I believe my above suggestion demonstrates it well:

$J$ := Jan is bald. $N$ := Jan is not bald. $'\neg J \land \neg N'$ := "Jan is neither bald nor not-bald".

I believe this is a legitimate symbolization of the sentence "Jan is neither bald nor not-bald", and it does not contradict the original meaning of the sentence. Also, "Trying to TFL-symbolize such English sentences may have unacceptable results" is true to any English sentence. Every English sentence can be improperly symbolized in TFL. This one is a bit confusing, and one may end up using a wrong symbolization, but this is not the same as a sentence that cannot be symbolized properly in TFL without losing meaning.

In fact this is exactly what the book explains in 5.1:

If we let the TFL-sentence ‘H ’ symbolize ‘Jane is happy’, then we can symbolize sentence 7 as ‘H ’. However, it would be a mistake to symbolize sentence 8 with ‘¬H ’

I claim that if symbolizing "Jane is unhappy" with $~H$ is wrong according to 5.1, then symbolizing "Jan is not-bald" with $~N$ in 12.5 is also wrong.

The point being made in 12.5 is clear enough without this example. Not every English sentence can be symbolized in TFL. But this sentence is not a good example.

[kedarmhaswade]

The symbolization above has formal issues. Following is more formal. Here is the key: 𝒥: Jan is bald. 𝒩: Jan is not-bald.

Then, at the risk of being adamantly TF-logical, we can unambiguously symbolize the English sentence "Jan is neither bald nor not-bald", as

¬𝒥 ∧ ¬𝒩

in TFL.

I believe, however, that we are already down a slippery slope. But I agree, we can provide a clarification in the text. I am not so sure that the example as it stands in the text now is a misuse of symbolization to demonstrate the limitations of TFL. Better examples are always welcome!

[yhirshhorn]

I am not sure I understand now.

1. What is the slippery slope we are in?

2. What is the difference between 5.1 and 12.5? I wanted to infer from their similarity that if $\neg H$ is a mistake there, then $\neg \scriptN$ is a mistake here. Why don't you agree?

[kedarmhaswade]

I am saying we could have symbolized this in any way we want. I am not convinced that the ambiguity goes away because of that. That said, yes, there is a room for improvement in the example. Please consider submitting a pull request!

[yhirshhorn]

I can't submit a PR if I don't understand the problem you are pointing at.

1. What is the slippery slope?

2. Why can we symbolize it any way we want here, but not in 5.1?

[rzach]

I hope the expanded explanation addresses your concern. It's an important point but you can't really do it justice without going into a whole discussion of vagueness and alternative logics