Lesson test: Assumptions, conditional proof and reductio

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Assumptions, conditional proof and reductio

1.[2p]
To prove $φ \to ψ$ by conditional proof you

assume $φ$ , derive $ψ$ , then close the subproof and write the conditional

Correct

assume $ψ$ and derive $φ$

The answer is: assume $\varphi$, derive $\psi$, then close the subproof and write the conditional

assume $\neg φ$ and derive a contradiction

The answer is: assume $\varphi$, derive $\psi$, then close the subproof and write the conditional

show $φ$ and $ψ$ are both true

The answer is: assume $\varphi$, derive $\psi$, then close the subproof and write the conditional

2.[3p]
Why may a line inside a closed subproof never be cited afterwards?

It depends on an assumption that is no longer in force

Correct

It might be false

The answer is: It depends on an assumption that is no longer in force

Subproofs are only sketches

The answer is: It depends on an assumption that is no longer in force

The rule that produced it has been used up

The answer is: It depends on an assumption that is no longer in force

3.[2p]
What is the freely made first line of a subproof called? One word.

CorrectNot quite: assumption

4.[2p]
From a contradiction, any formula whatever may be derived.

True

Correct

False

The answer is: True

5.[2p]
Argument by cases uses a disjunction $φ \lor ψ$ by

deriving the same conclusion from each disjunct in turn

Correct

discarding the disjunct that is false

The answer is: deriving the same conclusion from each disjunct in turn

assuming both disjuncts at once

The answer is: deriving the same conclusion from each disjunct in turn

turning it into a conditional

The answer is: deriving the same conclusion from each disjunct in turn

6.[3p]
Match each metatheorem to what it says.

Soundness
Completeness
Decidability

everything entailed is derivable
everything derivable is entailed
a procedure settles every case in finite time

Show the answer

Soundness: everything derivable is entailed Completeness: everything entailed is derivable Decidability: a procedure settles every case in finite time

7.[3p]
Which of these become provable only once assumptions are allowed?

Select all that apply

$P \to P$

Correct

$P \lor \neg P$

Correct

$\neg P \lor \neg Q$ from $\neg (P \land Q)$

Correct

$Q$ from $P \to Q$ and $P$

The answer is: $P \rightarrow P$, $P \vee \neg P$, $\neg P \vee \neg Q$ from $\neg(P \wedge Q)$

8.[3p]
Which rule does intuitionistic logic reject, so that $P \lor \neg P$ is not provable in it?

Double negation elimination

Correct

Conjunction introduction

The answer is: Double negation elimination

Conditional elimination

The answer is: Double negation elimination

Disjunction introduction

The answer is: Double negation elimination

9.[2p]
Soundness and completeness together mean that $Γ ⊢ φ$ and $Γ ⊨ φ$ hold in exactly the same cases.

True

Correct

False

The answer is: True

1.[2p] To prove φ→ψ by conditional proof you

2.[3p] Why may a line inside a closed subproof never be cited afterwards?

3.[2p] What is the freely made first line of a subproof called? One word.

4.[2p] From a contradiction, any formula whatever may be derived.

5.[2p] Argument by cases uses a disjunction φ∨ψ by

6.[3p] Match each metatheorem to what it says.

7.[3p] Which of these become provable only once assumptions are allowed?

8.[3p] Which rule does intuitionistic logic reject, so that P∨¬P is not provable in it?

9.[2p] Soundness and completeness together mean that Γ⊢φ and Γ⊨φ hold in exactly the same cases.

1.[2p]
To prove $φ \to ψ$ by conditional proof you

2.[3p]
Why may a line inside a closed subproof never be cited afterwards?

3.[2p]
What is the freely made first line of a subproof called? One word.

4.[2p]
From a contradiction, any formula whatever may be derived.

5.[2p]
Argument by cases uses a disjunction $φ \lor ψ$ by

6.[3p]
Match each metatheorem to what it says.

7.[3p]
Which of these become provable only once assumptions are allowed?

8.[3p]
Which rule does intuitionistic logic reject, so that $P \lor \neg P$ is not provable in it?

9.[2p]
Soundness and completeness together mean that $Γ ⊢ φ$ and $Γ ⊨ φ$ hold in exactly the same cases.