how would i formalize the following arguments using predicate logic where px means x is a potato head and a means adam,
there is at most 2 potato heads
adam is the only potato head
there are no potato heads
thanks for any help
how would i formalize the following arguments using predicate logic where px means x is a potato head and a means adam,
there is at most 2 potato heads
adam is the only potato head
there are no potato heads
thanks for any help
First, note that existence of anything is not claimed. If some world is completely devoid of potato heads (as a result of war or pestilence...), then this statement is true in that world. The statement claims that the existence of three or more potato heads is impossible.there is at most 2 potato heads
Therefore, an equivalent, but, perhaps, less intuitive, was to say this is: It is not the case that there exist three p.h.'s and all of them are different. Note the last part because the formula $\displaystyle \exists x_1\,\exists x_2\,\exists x_3.\,p(x_1)\land p(x_2)\land p(x_3)$ is true in a world with just one potato head.
For the second statement, as a hint, look at the definition of the uniqueness quantifier.
Yes, e.g., to say that $\displaystyle x_1$, $\displaystyle x_2$ and $\displaystyle x_3$ are all different you can say $\displaystyle x_1\ne x_2\land x_1\ne x_3\land x_2\ne x_3$, where $\displaystyle a\ne b$ is a contraction for $\displaystyle \neg(a=b)$.
Note that since $\displaystyle \ne$ is not a transitive relation, $\displaystyle x_1\ne x_2\land x_2\ne x_3$ does not imply $\displaystyle x_1\ne x_3$.