Question $1$

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Take a look at the rules for constructing trees on page $388.$ Rules $($e$)$ and $($f$)$ look familiar. Which of our $20$ old natural deduction rules do they remind you of$?$

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Group of answer choices

$1.$ DS

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$2.$ DN

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$3.$ DeM

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$4.$ Dup

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Question $2$

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What about rule $($i$)?$

Group of answer choices

$1.$ DS

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$2.$ DN

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$3.$ DeM

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$4.$ Dup

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Question $3$

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The branches of completed trees are considered either open$$or$$closed$.$ What causes a branch to be closed?

Group of answer choices

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$1.$ using rule $($i$)$

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$2.$ the appearance of a contradiction

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$3.$ it's no longer possible to apply any rules

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$4.$ frustration

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$$Question $4$

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What does a tree have to look like to show that an argument is valid?

Group of answer choices

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$1.$ all branches open

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$2.$ all branches closed

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$3.$ at least one open branch

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$4.$ at least one closed branch

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Question $5$

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What does a tree have to look like to show that an argument is invalid?

Group of answer choices

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$1.$ all branches open

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$2.$ all branches closed

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$3.$ at least on open branch

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$4.$ at least one closed branch

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Question $6$

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Let's test this argument$-$form using a proof tree:

$($p $$ q$)$ r$,$ p $/$ r

We put $($p $$ q$)$ r on the first line, and p on the second line. What goes on line $3?$

Group of answer choices

$1.$ r

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$2.$ ~r

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$3.$ ~p

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$4.$ ~$(($p $$ q$)$ r$)$

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Question $7$

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Continuing with our tree from the last question. What rule should we use to generate line $4?$

Group of answer choices

$1.$ b

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$2.$ c

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$3.$ f

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$4.$ g

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$$Question $8$

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Which formula$($s$)$ would be on line $4?$

Group of answer choices

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$1.$ p $$ q and ~r

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$2.$ p $$ q

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$3.$ ~r

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$4.$ ~$($p $$ q$)$ and r

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Question $9$

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Which formulas go on lines $5$ and $6?$

Group of answer choices

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$1.$ ~p and ~q

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$2.$ p and q

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$3.$ r and ~r

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$4.$ p and ~p

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$$Question $10$

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Is the argument$-$form valid?

Group of answer choices

$1.$ yes

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$2.$ no