Inclusive Disjunction - why can it be "or" and "both" at the same time?

[MathHelp]

I'm really confused by why inclusive disjunctive can also include both parts being true.

I have been googling explanation and reading books on it but to be honest the explanations don't make a lot of sense to me - it is almost like they assert something that seems obviously wrong and then move on.

How can it be that when I say "you are either good at hockey or bad at hockey" that it could also be that I am both good and bad at the same time?

What if someone likes dogs so I use the addition rule of inference and reason:

Greg likes dogs. Therefore either Greg likes dogs or he likes cats. Even though it may be that Greg likes both cats and dogs, I know that logic asserts that this is a correct deduction because logic treats "either... or..." as meaning both or/and but how can that be true in reality?

I don't know how else to explain my confusion but I am hoping the confusion is something everyone else has gone through at some time.

It is like saying "ships can fly" and then declaring the statement to be true because "flying also means floating" (when it obviously doesn't).

 

 

[Lorentz Jr]

36 minutes ago, MathHelp said:

How can it be that when I say "you are either good at hockey or bad at hockey" that it could also be that I am both good and bad at the same time?

That's exclusive, not inclusive. In the geekier circles, it's sometimes written "xor".

36 minutes ago, MathHelp said:

What if someone likes dogs so I use the addition rule of inference and reason:

Greg likes dogs. Therefore either Greg likes dogs or he likes cats. Even though it may be that Greg likes both cats and dogs, I know that logic asserts that this is a correct deduction because logic treats "either... or..." as meaning both or/and but how can that be true in reality?

The word "either" strongly suggests the exclusive meaning, but it doesn't absolutely imply it, at least not in casual conversation. If the condition for being admitted to an animal-lover's club is that you "either" like dogs or like cats, they probably won't mind if you like both dogs and cats. It's just people using the terminology informally. If they really want you to understand that they mean the exclusive version, they may say something like "... but not both". Or they might not say anything because they don't know any better. 🤷‍♂️

Edited December 15, 2022 by Lorentz Jr

[Genady]

8 minutes ago, MathHelp said:

I'm really confused by why inclusive disjunctive can also include both parts being true.

I have been googling explanation and reading books on it but to be honest the explanations don't make a lot of sense to me - it is almost like they assert something that seems obviously wrong and then move on.

How can it be that when I say "you are either good at hockey or bad at hockey" that it could also be that I am both good and bad at the same time?

What if someone likes dogs so I use the addition rule of inference and reason:

Greg likes dogs. Therefore either Greg likes dogs or he likes cats. Even though it may be that Greg likes both cats and dogs, I know that logic asserts that this is a correct deduction because logic treats "either... or..." as meaning both or/and but how can that be true in reality?

I don't know how else to explain my confusion but I am hoping the confusion is something everyone else has gone through at some time.

It is like saying "ships can fly" and then declaring the statement to be true because "flying also means floating" (when it obviously doesn't).

 

 

Saying "either ... or ..." implies exclusive or. Inclusive or is just "... or ...". Like in, e.g., "Single digit numbers divisible by 2 or 3 are 2, 3, 4, 6, 8, and 9." 

[MathHelp]

Quote

The word "either" strongly suggests the exclusive meaning.

So does that mean that the addition rule of implication is only for inclusive disjunctions? So I must first recognise that an argument is using a inclusive disjunction and only then can I apply the addition rule of implication?

That would also mean that the disjunctive syllogism is by default only a rule of implication for inclusive disjunctions:

Either p or q. Not p. Therefore q.

Quote

Saying "either ... or ..." implies exclusive or. Inclusive or is just "... or ...". Like in, e.g., "Single digit numbers divisible by 2 or 3 are 2, 3, 4, 6, 8, and 9." 

Uh, so really the way the textbook explained it was in a confusing way. It explained it like there was some brilliant logical explanation for why "either... or..." could also mean "... and ...".

If it had started off by saying "there are two types of disjunction i logic: Inclusive and exclusive. Inclusive disjunctions are ones where the the relationship between the two simple statements can be "or" and "both". The rules of implication that apply to disjunctions are for inclusive disjunctives only.

Overall it is a very good book. That seems to be the only thing that was explained poorly.

Thank you to you all. This has been a great help!

[Lorentz Jr]

1 hour ago, MathHelp said:

So does that mean that the addition rule of implication is only for inclusive disjunctions?

Correct.

1 hour ago, MathHelp said:

That would also mean that the disjunctive syllogism is by default only a rule of implication for inclusive disjunctions:

Either p or q. Not p. Therefore q.

No, that one works either way. Once one statement is known to be false, the question of whether or not both statements are allowed to be true becomes moot.

[MathHelp]

Quote

No, that one works either way. Once one statement is known to be false, the question of whether or not both statements are allowed to be true becomes moot.

Oh, that is incredibly obvious. I'm shocked I did not realise that myself.

[studiot]

11 hours ago, Genady said:

Saying "either ... or ..." implies exclusive or. Inclusive or is just "... or ...". Like in, e.g., "Single digit numbers divisible by 2 or 3 are 2, 3, 4, 6, 8, and 9." 

6 is divisible by  2 as well as 3 !

Interestingly this example demonstrates the power of the English language compared to Maths or Philosophy     (Which or did I mean ?  I could also have said and/or)

English provides many ways to express something, very often allowing for small differences and gradations of meaning.

Note I said 'as well as' instead of 'or', or 'both 2 and 3'.

 

Edited December 15, 2022 by studiot

[Genady]

24 minutes ago, studiot said:

6 is divisible by  2 as well as 3 !

Interestingly this example demonstrates the power of the English language compared to Maths or Philosophy     (Which or did I mean ?  I could also have said and/or)

English provides many ways to express something, very often allowing for small differences and gradations of meaning.

Note I said 'as well as' instead of 'or', or 'both 2 and 3'.

 

Is 'and/or' a proper or literary English?  

[studiot]

1 hour ago, Genady said:

Is 'and/or' a proper or literary English?  

Unlike French, there is no 'authority' that tries to control what is and what is not good English.

The nearest we come is The Oxford English Dictionary.

The OED, particularly the longer versions, draw from usage stretching back to before Shakespeare up to and including just before the annual updating as examples of
'correct English'

I am sure that at least 60 years ago people were using and/or and that I understood it at that time.
So that is good enough for me.

That is another strength of Ennglish  -  It is not hidebound by stuffy rules and regulations.

[Lorentz Jr]

1 hour ago, Genady said:

Is 'and/or' a proper or literary English?  

No, but it's still a spectacularly useful expression. One of the few in English that are more or less impossible to misinterpret.