Deduction Method and Other Rules

Covers the deduction method, hypothetical syllogism, and shortened proof sequences.

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QUESTION OF

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Views #: 144

Questions #: 16

Time: 20 minutes

Pass Score: 80.0%

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When to Use the Deduction Method

1 pts

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When should the deduction method be used?

When the conclusion is a negation.

When the conclusion is a conjunction.

When the conclusion of what you want to prove is an implication.

When there are no hypotheses.

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Deduction Method Action

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What does the deduction method allow?

Add the antecedent \(R\) as an additional hypothesis and then derive \(S\).

Remove a hypothesis from the argument.

Replace the conclusion with a hypothesis.

Use a truth table instead of a proof sequence.

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Deduction Method Transformation

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Given \(P_1∧...∧P_n→(R→S)\), what can we instead prove?

\(P_1∧...∧P_n∧R→S\)

\(P_1∧...∧P_n→R\)

\(R→S\)

\(S→R\)

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Advantage of Deduction Method

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Why is the deduction method advantageous?

It avoids all derivation rules.

It gives one more hypothesis and simplifies the desired conclusion.

It makes the conclusion false.

It removes all hypotheses.

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Deduction Method Hypotheses

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In proving \([A→(A→B)]→(A→B)\) using the deduction method, what are the hypotheses?

\(A→(A→B)\) and \(A\)

\(A→B\) and \(B\)

\(A\) and \(B\)

\(A→B\) only

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Deriving A→B

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In the proof of \([A→(A→B)]→(A→B)\), what rule derives \(A→B\) from \(A→(A→B)\) and \(A\)?

Addition

Modus ponens

Conjunction

Modus tollens

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Deriving B

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In the proof of \([A→(A→B)]→(A→B)\), what rule derives \(B\) from \(A\) and \(A→B\)?

Addition

Simplification

Modus ponens

Modus tollens

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Hypothetical Syllogism

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What is the rule of hypothetical syllogism?

From \(P→Q\) and \(Q→R\), derive \(P→R\).

From \(P\) and \(Q\), derive \(P∧Q\).

From \(P\) and \(P→Q\), derive \(Q\).

From \(P→Q\) and \(Q'\), derive \(P'\).

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Hypothetical Syllogism Wff

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What wff does hypothetical syllogism claim is a valid argument?

\((P→Q)∧Q'→P'\)

\(P∧Q→P\)

\((P→Q)∧(Q→R)→(P→R)\)

\((P→Q)∧P→Q\)

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Deriving A→B from A'∨B

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In proving \((A'∨B)∧(B→C)→(A→C)\), what rule derives \(A→B\) from \(A'∨B\)?

De Morgan

Addition

Implication

Conjunction

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Deriving A→C

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In proving \((A'∨B)∧(B→C)→(A→C)\), what rule derives \(A→C\) from \(B→C\) and \(A→B\)?

Modus ponens

Simplification

Modus tollens

Hypothetical syllogism

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Proof Without Hypothetical Syllogism

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Without hypothetical syllogism, how could the proof continue after deriving \(A→B\)?

Add \(C\) as a hypothesis and stop.

Add \(A\) as a hypothesis, derive \(B\) by modus ponens, then derive \(C\) by modus ponens.

Use addition to derive \(A∨C\).

Use simplification to derive \(B\).

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Equivalence for C'∨A

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In proving \((A→B)∧(C'∨A)∧C→B\), what equivalence applies to \(C'∨A\)?

Implication, giving \(C→A\)

De Morgan, giving \(C∧A\)

Commutative, giving \(A'∨C\)

Double negation, giving \(C∨A\)

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Using C→A and A→B

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After \(C'∨A\) becomes \(C→A\), what can be derived from \(C→A\) and \(A→B\) by hypothetical syllogism?

\(C→B\)

\(C∧B\)

\(A→C\)

\(B→C\)

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Deriving B from C

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From \(C\) and \(C→B\), what rule derives \(B\)?

Modus ponens

Modus tollens

Addition

Conjunction

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Effect of Additional Rules

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What is the effect of adding additional proved inference rules?

Invalid arguments become provable.

The formal system becomes incorrect.

The original rules are replaced.

Proof sequences may be shorter, but nothing more can be proved.

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Year 11 Year 12

When to Use the Deduction Method

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Explanation

Deduction Method Action

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Explanation

Deduction Method Transformation

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Explanation

Advantage of Deduction Method

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Explanation

Deduction Method Hypotheses

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Explanation

Deriving A→B

Correct Answer

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Deriving B

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Hypothetical Syllogism

Correct Answer

Explanation

Hypothetical Syllogism Wff

Correct Answer

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Deriving A→B from A'∨B

Correct Answer

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Deriving A→C

Correct Answer

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Proof Without Hypothetical Syllogism

Correct Answer

Explanation

Equivalence for C'∨A

Correct Answer

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Using C→A and A→B

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Deriving B from C

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Explanation

Effect of Additional Rules

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Explanation

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