Exam 11: Quantifier Rules Theory

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

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Question 1

Correct Answer:

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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)

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

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Question 2

Correct Answer:

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The point of dropping a quantifier is to free the variables it bound. In this case, the variables "freed" by dropping the (x) quantifier are "snatched up" by the quantifier. Suppose l states that every handsome person is loved by someone. Then 2 states that there is someone who, if handsome, loves himself. Clearly 1 could be true when 2 is false.

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

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Question 3

Correct Answer:

Verified

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.

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

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Question 4

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

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Question 5

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

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Question 6

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

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Question 7

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

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Question 8

Showing 1 - 8 of 8

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

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

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

Introduction

Symbolizing in Sentential Logic

Truth Tables

Proofs Without CP or IP

Proofs With CP or IP

Sentential Logic Truth Trees

Predicate Logic Symbolization

Predicate Logic Semantics

Predicate Logic Proofs

Relational Predicate Logic

Predicate Logic Truth Trees

Identity

Syllogistic Logic

Informal Fallacies

Inductive Logic

Axiom Systems

Filters

Exam 11: Quantifier Rules Theory

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

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

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

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

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

Suppose we require that in using UG \textbf{ UG } UG and EG \textbf{ EG } 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

Suppose we similarly require that in using UI \textbf{ UI } UI and EI \textbf{ EI } 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?

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

Introduction

Symbolizing in Sentential Logic

Truth Tables

Proofs Without CP or IP

Proofs With CP or IP

Sentential Logic Truth Trees

Predicate Logic Symbolization

Predicate Logic Semantics

Predicate Logic Proofs

Relational Predicate Logic

Predicate Logic Truth Trees

Identity

Syllogistic Logic

Informal Fallacies

Inductive Logic

Axiom Systems

Filters