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XI. CHAPTER ELEVEN: QUANTIFIER RULES THEORY
1. Explain, using a concrete example, why we should not
1. ( y)Lxy
allow the following inference
when using EI. What restriction then is required on the use of EI?
2. Lxx 1EI
2. Explain, using a concrete example, why we should not
1. ( y)Lxy
allow the following inference
when using UG. What restriction then is required on the use of UG?
2. Lxy 1EI
3. (y)Lxy 2UG
3. Explain, using a concrete example, why we should not
1. Lxy
allow the following inference
when using EG. What restriction then is required on the use of EG?
2. ( x)Lxx 1EG
4. Carefully explain why we forbid inferences like this one from line 2 to line 4:
1. ( x)(Rx Sx)p
2. ( y)(By Ty)p
3. Rx Sx 1EI
4. Bx Tx 2EI
5. Carefully explain why we forbid inferences like this one from line 1 to line 2:
1. (x) ( y) (Hx Lyx)p
2. ( y) (Hy Lyy) 1 UI
6. Do the same for the inference from line 3 to line 4:
1. (x) ( y) (Rx Byx)p
2. ( y) (Rx Byx) 1 UI
3. Rx Byx 2EI
4. (z) (Rz Byz) 3 UG
5. ( y) (z) (Rz Byz) 4 EG
7. Suppose we require that in using UG and EG there should be a one-to-one
correspondence between free u’s in (. . .u. . .) and bound w’s in (w) (. . .w. . .) or ( w) (. .
.w. . .). What errors or omissions (incompleteness), if any, would result?
8. Suppose we similarly require that in using UI and EI there should be a one-to-one
correspondence between bound w’s in (w) (. . .w. . .) or ( w) (. . .w. . .) and free u’s in (. .
.u. . .). What errors or omissions, if any, would result?
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XI. ANSWERS
1. It could allow us to move from a true statement to a false one. For example, if Lxy =
2. It could allow us to move from a true statement to a false one. For example, if Lxy =
4. It could allow us to move from true statements to a false one. In particular, if we allow
5. The point of dropping a quantifier is to free the variables it bound. In this case, the
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8.We get into similar trouble with UI. That is, we would block valid inferences using UI,