Lesson test: Rules of inference

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Rules of inference

1.[2p]
From $φ \to ψ$ and $φ$ , which rule licenses writing $ψ$ ?

Conditional elimination

Correct

Conditional introduction

The answer is: Conditional elimination

Disjunctive syllogism

The answer is: Conditional elimination

Conjunction elimination

The answer is: Conditional elimination

2.[3p]
Match each rule to what it needs on earlier lines.

Modus tollens
Disjunctive syllogism
Hypothetical syllogism
Conjunction introduction

two conditionals that share a middle formula
both conjuncts, on any two lines
a disjunction and the negation of one disjunct
a conditional and the negation of its consequent

Show the answer

Modus tollens: a conditional and the negation of its consequent Disjunctive syllogism: a disjunction and the negation of one disjunct Hypothetical syllogism: two conditionals that share a middle formula Conjunction introduction: both conjuncts, on any two lines

3.[2p]
From $P$ alone you may write $P \lor Q$ for any $Q$ at all.

True

Correct

False

The answer is: True

4.[3p]
Why is disjunction introduction sound even though its conclusion can look absurd?

A disjunction is true whenever either side is true, so no true line can yield a false one

Correct

Absurd conclusions are harmless in logic

The answer is: A disjunction is true whenever either side is true, so no true line can yield a false one

The added disjunct is always false

The answer is: A disjunction is true whenever either side is true, so no true line can yield a false one

It is a derived rule, so it needs no justification

The answer is: A disjunction is true whenever either side is true, so no true line can yield a false one

5.[3p]
Put these lines in the order of a derivation of $R$ from $(P \land Q) \to R$ , $P$ , $S \to Q$ and $S$ .

$R$ , by conditional elimination
$Q$ , from $S \to Q$ and $S$
$P \land Q$ , by conjunction introduction

Show the answer

a, b, c

6.[2p]
What is Gentzen's 1934 system of introduction and elimination rules called? Two words.

CorrectNot quite: natural deduction

7.[3p]
Which connectives have no introduction rule in the system of this lesson?

Select all that apply

The conditional

Correct

Negation

Correct

Conjunction

The answer is: The conditional, Negation

The biconditional

The answer is: The conditional, Negation

8.[3p]
Why can this system not derive $P \to P$ ?

With no premises there is nothing to eliminate, and no rule builds a conditional

Correct

$P \to P$ is not a tautology

The answer is: With no premises there is nothing to eliminate, and no rule builds a conditional

It needs a truth table instead

The answer is: With no premises there is nothing to eliminate, and no rule builds a conditional

Conditional elimination cannot be applied twice

The answer is: With no premises there is nothing to eliminate, and no rule builds a conditional

9.[1p]
A derivation can be checked line by line without considering the argument as a whole.

True

Correct

False

The answer is: True

1.[2p] From φ→ψ and φ, which rule licenses writing ψ?

2.[3p] Match each rule to what it needs on earlier lines.

3.[2p] From P alone you may write P∨Q for any Q at all.

4.[3p] Why is disjunction introduction sound even though its conclusion can look absurd?

5.[3p] Put these lines in the order of a derivation of R from (P∧Q)→R, P, S→Q and S.

6.[2p] What is Gentzen's 1934 system of introduction and elimination rules called? Two words.

7.[3p] Which connectives have no introduction rule in the system of this lesson?

8.[3p] Why can this system not derive P→P?

9.[1p] A derivation can be checked line by line without considering the argument as a whole.

1.[2p]
From $φ \to ψ$ and $φ$ , which rule licenses writing $ψ$ ?

2.[3p]
Match each rule to what it needs on earlier lines.

3.[2p]
From $P$ alone you may write $P \lor Q$ for any $Q$ at all.

4.[3p]
Why is disjunction introduction sound even though its conclusion can look absurd?

5.[3p]
Put these lines in the order of a derivation of $R$ from $(P \land Q) \to R$ , $P$ , $S \to Q$ and $S$ .

6.[2p]
What is Gentzen's 1934 system of introduction and elimination rules called? Two words.

7.[3p]
Which connectives have no introduction rule in the system of this lesson?

8.[3p]
Why can this system not derive $P \to P$ ?

9.[1p]
A derivation can be checked line by line without considering the argument as a whole.