Mastering Basic Symbolic Logic: Understanding Negations Intuitively

[TrevE]

Basic symbolic logic, as it stands, is very straightforward to me, but I don't seem to understand it verbosely, i.e. without constructing truth tables. Actually it's only the negation of statements that I don't seem to grasp intuitively.

For instance, the negation of

John is fat and John is blonde

is simply

John is not fat or John is not blonde

but I don't get why it is so, without truth tables. How can I go around with working with negations intuitively? Should I just find out what negates to what and keep that in mind when working with statements?

 

[0rthodontist]

If it is not true that John is both fat and blonde, that's the same thing as saying that he's not fat or not blonde. For example if John is thin and blonde, then he's not fat and blonde. If he's fat and black-haired, then he's not fat and blonde. If he's thin and black haired, then he's not fat and blonde. If it is not the case that two things are both true, then at least one of the things must be false (when dealing with logical statements).

 

[TrevE]

Thank you for presenting it clearly; I think I get the gist of it now. Cheers!

 

[TrevE]

OK one last thing. I think I'm running into trouble with negating exclusive or statements. So let's say John is fat or John is blonde, but not both. I don't see how I can negate "not both" because after which it looks like "not both" is insignificant in negation. Say we do away with "but not both" and it becomes inclusive, then it follows that the negation is "John is not fat and blonde", and that's what I sort of came up with with exclusive or, which leads me to asking, are negated statements and original statements converses to each other, or is that not necessarily a requirement? Or did I just negate the above exclusive or statement wrongly?

Thanks in advance.

 

[CRGreathouse]

not(a xor b) is just a = b in a logical sense. Exclusive or is true iff the two are different, so its negation is true iff they are the same.

$$\overline{a\oplus b}\Longleftrightarrow a=b$$

"not (John is fat or John is blonde, but not both)" <--> "(John is fat and blonde) or (John is neither fat nor blonde)"

 

[TrevE]

How logical.Heh, many thanks, CRG!