3.1: Natural Deduction Proofs (Basic)

Natural Deduction Proofs

Prove:

p

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p V ((q & r) -> (s & t))

Natural Deduction Proofs

Simplifcation:

p & q

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p

 

p & q

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q

 

Natural Deduction Proofs

Conjunction:

p

q

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p & q

 

Natural Deduction Proofs

Modus Ponens:

p -> q

p

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q

 

Natural Deduction Proofs

Modus Tollens:

p -> q

~q

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~p

 

Natural Deduction Proofs

Addition:

p

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p V q

 

Natural Deduction Proofs

Prove:

(p & q) & (r & s)

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q & r

1. (p & q) & (r & s)
2. p & q
3. q
4. r & s
5. r
6. q & r

1. PREMISE
2. SIMPLIFICATION (1)
3. SIMPLIFCATION (2)
4. SIMPLIFCATION (1)
5. SIMPLIFICATION (4)
6. ADDITION (3, 5)

Natural Deduction Proofs

Prove:

p -> q

~q

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~p V r

1. p -> q
2. ~q
3. ~p
4. ~p V r

1. PREMISE
2. PREMISE
3. MODUS TOLLENS (1, 2)
4. ADDITION (3)

Natural Deduction Proofs

Prove:

p -> ~(q V r)

q

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~p & q

1. p -> ~(q V r)
2. q
3. q V r
4. ~p
5. ~p & q

1. PREMISE
2. PREMISE
3. ADDITION (2)
4. MODUS TOLLENS
5. CONJUNCTION (2, 4)

Key Concepts

• Natural Deduction
• Simplification
• Conjunction
• Modus Ponens
• Modus Tollens
• Addition