My textbook "Discrete and Combinatorial Mathematics, an Applied Introduction" by Ralph P. Grimaldi contains the following definition:
Let $s$ be a statement. If $s$ contains no logical connectives other than $\wedge$ and $\vee$, then the dual of $s$, denoted $s^d$, is the statement obtained from $s$ by replacing each occurrence of $\wedge$ and $\vee$ by $\vee$ and $\wedge$, respectively, and each occurrence of $T_0$ and $F_0$ by $F_0$ and $T_0$, respectively.
(I apologize for not knowing the formatting conventions here)
and the following theorem:
The Principle of Duality . Let $s$ and $t$ be statements that contain no logical connectives other than $\wedge$ and $\vee$. If $s \leftrightarrow t$, then $s^d \leftrightarrow t^d$.
Is anyone familiar with these? Can I use them to reason as follows:
- Start with a premise
- take its dual
- manipulate the dual
- arrive at the dual of our conclusion
- take its dual
This would allow me to do something like:
to prove that $\forall x\ P(x)\vee \forall\ Q(x) \rightarrow \forall x\ [P(x)\vee Q(x)]$
1. Ax,P(x) V Ax,Q(x) premise 2. Ax,P(x) ^ Ax,Q(x) the dual of (1) 3. Ax,P(x) conjunctive simplification 4. P(c) universal specification 5. Ax,Q(x) conjunctive simplification from (2) 6. Q(c) universal specification 7. P(c) ^ Q(c) rule of conjunction from (4) and (6) 8. Ax[P(x) ^ Q(x)] universal generalization 9. Ax[P(x) V Q(x)] the dual of (8)
When I wrote this proof I didn't know that you could use universal specification to go from (1) to the dual of (7), so I needed a way to separate (1) into two parts. Taking the dual seemed to be a thing I can do. I worry though, because it seems like I could prove the converse this way, and the converse is false.
So what's wrong? Why can't I use the concept of the dual of a statement in this way?
PS oh and I solved the question in another way, so this is just my curiosity. PPS thank you for being patient with my ugly post, I'll try to learn your mark up.