Symbolic logic equivalency
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19-09-2019 - |
Question
Is there any difference between these two statements, given the following language??
- Ben likes dogs and either John or Mary likes dogs.
- 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
Solution
B & ((J & ~M) V (~J & M))
B & (J V M)
OTHER TIPS
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.
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:
- B & ((J V M) & ~(J & M))
- B & (J V M)
See Exclusive or for some transformations.
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:
- b & ((j V m) & ~(j & m))
- 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.