Why Does Logical Negation Change And to Or?

[Bashyboy]

The proposition that I am to negate is, "The summer in Maine is hot and sunny," so I performed the negation and wrote, "The summer in Maine is not hot and sunny." This is incorrect, and the answer is, "The summer in Maine is not hot or it is not sunny." Why did the author change the logical connective from the conjunction (and) to the disjunction (or)?

 

[Bacle2]

The formal negation of (A and B) is NotA or NotB.

One way of seeing it is that a case of NotA or NotB is a counterexample to "A and B".

Maybe it would be better to see it as : Not ( Hot and Sunny) . If it is not both (hot and sunny), then it is either

not hot or not sunny.

 

[Bashyboy]

Okay, I see: $a \wedge b$ has the opposite truth values as $\neg a \vee \neg b$? Would the way I write it be incorrect? Or should I write the "formal" way as my answer?

 

[Bacle2]

Yes, notice a/\b is true exactly one both a,b are true, and this is the only case

when ~a\/~b is false. IOW, (A/\B) & (~A\/~B) is a contradiction

You can even do a short derivation. What do you mean by the way you write is

incorrect?

 

[HallsofIvy]

What you wrote, "the summer in Maine is not hot and sunny" is, at best, ambiguous. It could be interpreted as "the summer in Maine is not hot but it is sunny" which is NOT the negation of the original statement.

 

[Bacle2]

What you wrote, "the summer in Maine is not hot and sunny" is, at best, ambiguous. It could be interpreted as "the summer in Maine is not hot but it is sunny" which is NOT the negation of the original statement.

Yes, of course, but that is almost a necessary tradeoff between a formal language ( of logic)

and a non-formal one like everyday English: accuracy in exchange for flexibility .