Deck 11: Quantifier Rules Theory

Question 1

Explain, using a concrete example, why we should not allow the following inference when using $\textbf{ EI }$ . What restriction then is required on the use of $\textbf{ EI? }$

Answer 1

It could allow us to move from a true statement to a false one. For example, if Lxy = "x loves y", then from the fact that x loves someone it does not follow that x loves him or herself. EI requires that the variable replacing the variable made free with the removal of the existential quantifier be one that is not free earlier in the proof (restriction 2).

Question 2

Explain, using a concrete example, why we should not allow the following inference when using $\textbf{ UG. }$ What restriction then is required on the use of $\textbf{ UG? }$

Answer 2

It could allow us to move from a true statement to a false one. For example, if Lxy = "x loves y", then from the fact that x loves someone it does not follow that x loves everyone. UG requires that the variable that is universally quantified be one that does not occur previously in a line obtained by EI (restriction 2)

Question 3

Explain, using a concrete example, why we should not allow the following inference when using $\textbf{ EG. }$ What restriction then is required on the use of $\textbf{ EG? }$

Answer 3

It could allow us to move from a true statement to a false one. For example, if Lxy = "x loves y", from the fact that someone loves someone it does not follow that anyone loves him or herself. EG requires that the variable that comes to be existentially quantified be one that does not already occur free in the formula we are quantifying.

Question 4

Carefully explain why we forbid inferences like this one from line 2 to line 4:

Question 5

Carefully explain why we forbid inferences like this one from line 1 to line 2:

Question 6

Do the same for the inference from line 3 to line 4:

Question 7

Suppose we require that in using $\textbf{ UG }$ and $\textbf{ EG }$ there should be a one-to-one correspondence between free u's in (. . .u. . .) and bound w's in . What errors or omissions (incompleteness), if any, would result

Question 8

Suppose we similarly require that in using $\textbf{ UI }$ and $\textbf{ EI }$ there should be a one-to-one correspondence between bound w's in and free u's in (. . .u. . .). What errors or omissions, if any, would result?