Posted: October 8th, 2023

Introduction to Formal Logic

Problem Set 6

 

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Part 1 (two points total). For each of the following instances of derived rules, provide a proof that demonstrates its validity. Make sure to use SL and primitive rules (no derived rules). Remember that with equivalencies (i.e., those sequents with double turnstile “┤├”) you must provide a proof in each direction.

 

Conjunction Commutation (&Comm) (For extra practice prove ÚComm)

P&Q ┤├ Q&P

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Modus Tollendo Tollens (MTT)

~P®~Q, Q├ P

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Wedge Arrow (Ú®)

PÚQ ┤├~P®Q

PÚQ┤├~Q®P

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Negated Arrow (Neg.®)

~(P®Q) ┤├P&~Q

~(P®~Q) ┤├ P&Q

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Biconditional Ponens (BP)

P«Q, P ├Q

 

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Biconditional Tollens (BT)

P«~Q, ~P ├ Q

 

Transposition (Trans)

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P®Q ├~Q®~P

 

Hypothetical Syllogism (HS)

A®B, B®C├ A®C

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Principle of Excluded Middle (PEM)

├ PÚ~P

 

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DeMorgan’s Law (DM)

P&Q ┤├~(~PÚ~Q)

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PÚQ ┤├~(~P&~Q)

 

Part 2 (one point total). For each of the following sequents, provide a proof that demonstrates its validity. Make sure to use SL and you may use derived rules from pages 29-30 of Allen and Hand).

 

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1. ~(~R®~P)├ P
2. ~(~R®~P)├ P&~R
3. P®QÚR, ~(~R®~P)├ P&Q
4. P®QÚR, (~Q&S)Ú(T®~P), ~(~R®~P)├T®~P
5. P®QÚR, (~Q&S)Ú(T®~P), ~(~R®~P)├~T&Q
6. E®~(JÚH), J├~E
7. E®~(JÚH), ~S, JÚS├~E
8. C®E, E®~(JÚH), ~S, JÚS├~C
9. B®(C®E), E®~(JÚH), ~S, JÚS ├ B®~C
10. B®(C®E), E®~(JÚH), ~S, JÚS ├~BÚ~C

 

 

 

Part 3 (one point total). For each of the following theorems, provide a proof that demonstrates its validity. Make sure to use SL and you may use derived rules from pages 29-30 of Allen and Hand).

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1. ├ P®P
2. ├ P® PÚQ
3. ├ P® PÚ(P&Q)
4. ├ PÚ(P&Q)®P
5. ├P « PÚ(P&Q)
6. ├P&Q®Q&P
7. ├~(~PÚ~Q)®Q&P
8. ├~(~PÚ~Q)®~(Q®~P)
9. ├~(Q®~P)®~(~PÚ~Q)
10. ├~~(Q&P)«~(P®~Q)

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