Exercise 6.4 Prove the remaining MN-theorems.

I, NEC Example 6.11: Proof of OP-OQ-(PAQ)): 1. Q-(P-(PAQ)) PL 2. (Q-(P-(PAQ))) 3. OQ-(P-(PAQ)) K, 2, MP 4. (P-(PAQ))(OP-(PAQ)) KO 5. OQ-OP-(PAQ)) 3, 4, PL (syllogism) 6. OP-OQ-(PAQ)) 5, PL (permutation) Let's derive another helpful shortcut involving the o, the following "modal negation" (MN) theorem schemas: tk~00 ond ~- (MN) Fx06-0~ FO~6--08 I I'll prove one of these; the rest can be proved as exercises. Example 6.12: Proof of ~00+0~0, i.e. ~0~0~~$ (for any 6): 1. ~~60 PL 2. O~~$$ 1, NEC, K, MP 3. ~04-0~~ 2, PL (contraposition) The MN theorems let us "move" ~s through strings of Os and Os. Example 6.13: Show that FK 000~P~ODOP: 1. O~P~OP MN 2. 00~P-OOP 1, NEC, KO, MP 3. O-OP-OOP MN 4. 00~P~OOP 2, 3, PL (syllogism) 5. 000~P~0~O0P 4, NEC, K, MP 6. D-DOPODOP MN 7. 000-P-00OP 5, 6, PL (syllogism) It's important to note, by the way, that this proof can't be shortened as follows: I It's important to note, by the way, that this proof can't be shortened as follows: 1. 000-P-00~OP MN 2. DO~OPO-DOP MN 3. O~DOPODOP MN 4. 000~PODOP 1, 2, 3, PL Steps 1 and 2 of the latter proof are mistaken. The MN-theorems say only that particular wffs are provable, whereas steps 1 and 2 attempt to apply MN to the insides of complex wffs. I, NEC Example 6.11: Proof of OP-OQ-(PAQ)): 1. Q-(P-(PAQ)) PL 2. (Q-(P-(PAQ))) 3. OQ-(P-(PAQ)) K, 2, MP 4. (P-(PAQ))(OP-(PAQ)) KO 5. OQ-OP-(PAQ)) 3, 4, PL (syllogism) 6. OP-OQ-(PAQ)) 5, PL (permutation) Let's derive another helpful shortcut involving the o, the following "modal negation" (MN) theorem schemas: tk~00 ond ~- (MN) Fx06-0~ FO~6--08 I I'll prove one of these; the rest can be proved as exercises. Example 6.12: Proof of ~00+0~0, i.e. ~0~0~~$ (for any 6): 1. ~~60 PL 2. O~~$$ 1, NEC, K, MP 3. ~04-0~~ 2, PL (contraposition) The MN theorems let us "move" ~s through strings of Os and Os. Example 6.13: Show that FK 000~P~ODOP: 1. O~P~OP MN 2. 00~P-OOP 1, NEC, KO, MP 3. O-OP-OOP MN 4. 00~P~OOP 2, 3, PL (syllogism) 5. 000~P~0~O0P 4, NEC, K, MP 6. D-DOPODOP MN 7. 000-P-00OP 5, 6, PL (syllogism) It's important to note, by the way, that this proof can't be shortened as follows: I It's important to note, by the way, that this proof can't be shortened as follows: 1. 000-P-00~OP MN 2. DO~OPO-DOP MN 3. O~DOPODOP MN 4. 000~PODOP 1, 2, 3, PL Steps 1 and 2 of the latter proof are mistaken. The MN-theorems say only that particular wffs are provable, whereas steps 1 and 2 attempt to apply MN to the insides of complex wffs