Symbolic logic equivalency

Question

Is there any difference between these two statements, given the following language??

1. Ben likes dogs and either John or Mary likes dogs.
2. Ben likes dogs and John or Mary likes dogs.

Using:

B: Ben likes dogs.
J: John likes dogs.
M: Mary likes dogs.

I have B & (J V M) for both...

I'm limited to & () V ~ as my symbols

Answer 1

1. B & ((J & ~M) V (~J & M))
2. B & (J V M)

Answer 2

You haven't defined what 'either' means. My guess is that 'either' modifies 'or' to exclusive-or, in which case the two statements are different.

Answer 3

I agree with Mr. Garrison. It's been a long time since I took symbolic logic, but I'd suspect "either" as meaning exclusive-or. So:

1. B & ((J V M) & ~(J & M))
2. B & (J V M)

See Exclusive or for some transformations.

Answer 4

It's probably not a good idea to entitle this question "Symbolic logical equivalency", since it might compound the potential confusion of terms, as there is a logic connective for equivalency. As the case may be, if Mary likes dogs is a true proposition, and if John likes dogs also is true (and so forth), then since both propositions have the same truth values, they are equivalent: M <-> J. But that's not the real question here - just want to clarify a potential point of confusion.

Rather, the two example sentences above are about and/or...but specifically, about "or". The first sentence is an example of exclusive "or", since the example follows a conjunction and means either/or. Exclusive "or" is: (a v b) & ~(a & b). That translates to A or B, but not both are true. The second sentence is inclusive "or", as the disjunction follows the conjunction and does not suggest that both only one disjunct must be true; rather, it's an inclusive "or", where either or both could be true.

Therefore, here's how to do the two sentences:

1. b & ((j V m) & ~(j & m))
2. b & (j V m)

TrueWill's answer is correct, but I am providing more explanation and confirming the fact that TrueWill's answer is correct.