Logic Connectives If for example I have two statements, a) I'm not old. b) I'm not gay. I can let O - I'm old and G - I'm gay and negate both. Using the same example in the book that I'm reading I can do it this way, -(O ^ G) which means I'm not both old and gay. However it seems to me that it is more reasonable to say -O ^ -G or I'm not old and (I'm) not gay. However these two are not equivalent expressions since -(O ^ G) = -O v -G. Which of the two should be used?

If I tell you

> I am not both old and gay.

then I could be either old or gay, but not both. That is $$ \
eg ( O \wedge G ) $$ is compatible (meaning , can be simultaneously true) with either $O$ or $G$, but not both.

If I tell you

> I am not old and I am not gay.

then I cannot be old, nor can I be gay. That is $$ \
eg O \wedge \
eg G $$

is compatible neither with $O$ nor with $G$.

Now, de Morgan's Law applied to the sentence $\
eg O \wedge \
eg G$ gives $\
eg (O \vee G)$. That is, it turns "I am not old and I am not gay" into "I am not (either old or gay)." It is a frequent deficiency of spoken language that there is no good way to speak the parentheses, so we use different constructions. For instance, "Neither am I old or gay." This sentence eliminates the possibility of the speaker being old and eliminates the possibility of the speaker being gay. That is, this sentence is compatible neither with $O$ nor with $G$.