In more an English sense than a Puzzling one...
In the question there is the keyword:
That means if 100% true, then the statement is true. else it is false, even it is true 99%.
In other words:
Hope this helps!
I think this is just a matter of understanding the language used in logic.
In the implication
If A, then B
you seem to be arguing that, since there are cases where A is true but B can be either true or false, we should say "the implication is neither true nor false".
However, every mathematician I know would say that the implication is false. In order ...
A lot depends on exactly how we move from (ambiguous) English to (unambiguous) logic.
If some Smaugs are Thors and some Thors are Thrains, then some Smaugs are definitely Thrains
The first way we can do it is to treat it as a syllogism.
Some Smaugs are Thors.
Some Thors are Thrains.
Therefore some Smaugs are definitely Thrains.
Syllogisms aren't true ...
The word "definitely" is ambiguous as to what it modifies. Taken literally, it modifies "Thrains"; according to the standard rules of English grammar, the default is that modifiers modify the next word. Under this interpretation, the statement is saying that there exists a nonempty set (and if we take the plural literally, the set needs to not only be ...
This is a classic paradox and, to quote the Wikipedia article about it,
Despite significant academic interest, there is no consensus on its precise nature and consequently a final correct resolution has not yet been established.
So if you're expecting anyone here to provide something that once you see it is obviously The One True Explanation Of What's ...
Consider the 1st part:
If some Smaugs are Thors
The following diagram shows how some Smaugs are Thors. The violet shaded part is the one that represents the few Smaugs that are Thors.
some Thors are Thrains
This can have 2 possibilities:
1. In the above case, as seen by the red shaded part, some Smaugs are Thrains.
2.But then, the other ...
The rules leaves it open ended whether Smaugs are Thrains.
So it could describe a relationship like :-
Some Humans are Female, Some Females are Mothers
in this case, some human females are mothers.
Or it could be
some Humans are Female, some Females are Kangaroos
in this case, No Human is a Kangaroo
The last case clearly shows that Smaugs ...
The 3 over-laying circles don't always make sense.
Smaugs may be touching into Thrain territory, or it may not.
50% chance it's the 3 over-laying rings theory above; 50% chance it looks more like 3 serial links- as only the Smaugs and Thors must intersect, and only the Thors and Thrain must intersect. There is no data to intersect Smaugs with Thrains (or ...
I already agree with some answers presented. Let me provide a graphical one. This is the 3-set venn diagram of all posible sets an element can be a member of:
I assume "some" means "1 or more". Thus, let's interpret the question:
some Smaugs are Thors
means elements of Green + Red are 1 or more
some Thors are Thrains
means elements of Blue + Red are ...
The statement could only be true if ALL of the Thors were also Smaugs or all of the Thors were Thrains
The statement is FALSE because you cannot guarantee that some Smaugs are DEFINITELY Thrains (perhaps only the Thors that are not Smaugs are the ones that are Thrains)
The only way it could be NEITHER is if there is sufficient ambiguity in the facts or the ...