3.5.xa. Exercise answers

1.

The rules that may be applied are indicated in the annotations for these derivations by bracketed subscripts on elements of the proximate argument, on the sentence to which the rule is applied in the case of development rules and on the slash between resources and goal in the case of closure rules.

  a.
│¬ A (3)
├─
││B ∧ A 2
│├─
2 Ext ││B
2 Ext ││A (3)
││●
│├─
3 Nc ││⊥ 1
├─
1 RAA │¬ (B ∧ A)
  1. One open gap.
    ¬ A / ¬ (B ∧ A)[RAA].
    Developled.
  2. One open gap.
    ¬ A, B ∧ A[Ext] / ⊥.
    Developed.
  3. One open gap.
    ¬ A, B, A /[Nc] ⊥.
    Closed gap.
  4. No gaps open.
  b.
│A ∧ B 1
├─
1 Ext │A (3)
1 Ext │B (4)
│
││●
│├─
3 QED ││B 2
│
││●
│├─
4 QED ││A 2
├─
2 Cnj │B ∧ A
  1. One open gap.
    A ∧ B[Ext] / B ∧ A[Cnj].
    Developed by Ext.
  2. One open gap.
    A, B / B ∧ A[Cnj].
    Developed.
  3. Chose first open gap.
    A, B /[QED] A.
    Closed gap.
  4. One open gap.
    A, B /[QED] B.
    Closed gap.
  5. No gaps open.
  c.
│B (2)
├─
││●
│├─
2 QED ││B 1
│
│││¬ A
││├─
│││○ B, ¬ A ⊭ ⊥
││├─
│││⊥ 3
│├─
3 IP ││A 1
├─
1 Cnj │B ∧ A
  1. One open gap.
    B / B ∧ A[Cnj].
    Developed.
  2. First open gap.
    B /[QED] B.
    Closed gap.
  3. One open gap.
    B / A[IP].
    Developed.
  4. One open gap.
    B, ¬ A / ⊥.
    Dead end.
  d.
│¬ (A ∧ B) 2
│A (4)
├─
││B (5)
│├─
││││●
│││├─
4 QED ││││A 3
│││
││││●
│││├─
5 QED ││││B 3
││├─
3 Cnj │││A ∧ B 2
│├─
2 CR ││⊥ 1
├─
1 RAA │¬ B
  1. One open gap.
    ¬ (A ∧ B), A / ¬ B[RAA].
    Developed.
  2. One open gap.
    ¬ (A ∧ B)[CR], A, B / ⊥.
    Developed.
  3. One open gap.
    A, B / A ∧ B[Cnj].
    Developed.
  4. Chose first open gap.
    A, B /[QED] A.
    Closed gap.
  5. One open gap.
    A, B /[QED] B.
    Closed gap.
  6. No gaps open.
  e.
│¬ (A ∧ B) 2
│¬ (B ∧ C) 6
├─
││B (4), (8)
│├─
│││││¬ A
││││├─
│││││││●
││││││├─
8 QED │││││││B 7
││││││
││││││││¬ C
│││││││├─
││││││││○ B, ¬ A, ¬ C ⊭ ⊥
│││││││├─
││││││││⊥ 9
││││││├─
9 IP │││││││C 7
│││││├─
7 Cnj ││││││B ∧ C 6
││││├─
6 CR │││││⊥ 5
│││├─
5 IP ││││A 3
│││
││││●
│││├─
4 QED ││││B 3
││├─
3 Cnj │││A ∧ B 2
│├─
2 CR ││⊥ 1
├─
1 RAA │¬ B
  1. One open gap.
    ¬ (A ∧ B), ¬ (B ∧ C) / ¬ B[RAA].
    Developed.
  2. One open gap.
    ¬ (A ∧ B)[CR], ¬ (B ∧ C)[CR], B / ⊥.
    Developed by first CR.
  3. One open gap.
    ¬ (B ∧ C), B / A ∧ B[Cnj].
    Developed.
  4. Chose second open gap.
    ¬ (B ∧ C), B /[QED] B.
    Closed gap.
  5. One open gap.
    ¬ (B ∧ C), B / A[IP].
    Developed.
  6. One open gap.
    ¬ (B ∧ C)[CR], B, ¬ A / ⊥.
    Developed.
  7. One open gap.
    B, ¬ A / B ∧ C[Cnj].
    Developed.
  8. Chose first open gap.
    B, ¬ A /[QED] B.
    Closed gap.
  9. One open gap.
    B, ¬ A / C[IP].
    Developed.
  10. One open gap.
    B, ¬ A, ¬ C / ⊥.
    Dead end.
2. The stages at which choices are made are indicated by references to notes below that describe the choices that were made.
  a.
│A ∧ B 1*
├─
1* Ext │A (3†)
1* Ext │B (4)
│
││●
│├─
3† QED ││B 2
│
││●
│├─
4 QED ││A 2
├─
2 Cnj │B ∧ A
│A ∧ B 2†, 5
├─
2† Ext ││A
2† Ext ││B (3‡)
││●
│├─
3‡ QED ││B 1*
│
│││¬ A (6)
││├─
5 Ext │││A (6)
5 Ext │││B
│││●
││├─
6 Nc │││⊥ 4§
│├─
4§ IP ││A 1*
├─
1* Cnj │B ∧ A
   

* Chose Ext instead of Cnj

† Chose first of 2 gaps

* Chose Cnj instead of Ext

† Chose first of 2 gaps
and Ext instead of IP

‡ Chose first of 2 gaps

§ Chose IP instead of Ext

  b.
│¬ (A ∧ B) 3
│B ∧ C 1*
├─
1* Ext │B (6)
1* Ext │C
│
││A (5†)
│├─
││││●
│││├─
5† QED ││││A 4
│││
││││●
│││├─
6 QED ││││B 4
││├─
4 Cnj │││A ∧ B 3
│├─
3 CR ││⊥ 2
├─
2 RAA │¬ A
│¬ (A ∧ B) 2†
│B ∧ C 5**
├─
││A (4§)
│├─
││││●
│││├─
4§ QED ││││A 3‡
│││
5** Ext ││││B (6)
5** Ext ││││C
││││●
│││├─
6 QED ││││B 3‡
││├─
3‡ Cnj │││A ∧ B 2†
│├─
2† CR ││⊥ 1*
├─
1* RAA │¬ A
   

* Chose Ext instead of RAA

† Chose first of 2 gaps

* Chose RAA instead of Ext

† Chose CR instead of Ext

‡ Chose Cnj instead of Ext

§ Chose first of 2 gaps

** Chose Ext instead of IP

  c.
│¬ (A ∧ B) 2*
│¬ (B ∧ C) 6
├─
││B (4†), (8‡)
│├─
│││││¬ A
││││├─
│││││││●
││││││├─
8‡ QED │││││││B 7
││││││
││││││││¬ C
│││││││├─
││││││││○ B,¬ A,¬ C⊭⊥
│││││││├─
││││││││⊥ 9
││││││├─
9 IP │││││││C 7
│││││├─
7 Cnj ││││││B ∧ C 6
││││├─
6 CR │││││⊥ 5
│││├─
5 IP ││││A 3
│││
││││●
│││├─
4† QED ││││B 3
││├─
3 Cnj │││A ∧ B 2*
│├─
2* CR ││⊥ 1
├─
1 RAA │¬ B
│¬ (A ∧ B) 5‡
│¬ (B ∧ C) 2*
├─
││B
│├─
││││
│││├─
││││B 3
│││
│││││¬ C
││││├─
│││││││
││││││├─
│││││││B 6**
││││││
││││││││¬ C
│││││││├─
││││││││○ B,¬ A,
││││││││ ¬ C⊭⊥
│││││││├─
││││││││⊥ 7††
││││││├─
7†† IP │││││││C 6**
│││││├─
6** Cnj ││││││A ∧ B 5‡
││││├─
5‡ CR │││││⊥ 4†
│││├─
4† IP ││││C 3
││├─
3 Cnj │││B ∧ C 2*
│├─
2* CR ││⊥ 1
├─
1 RAA │¬ B
   

* Chose CR on 1st premise instead of 2nd

† Chose second of 2 gaps

‡ Chose first of 2 gaps

* Chose CR on 2nd premise instead of 1st

† Chose second of 2 gaps

‡ Chose second of 2 gaps

§ Chose second of 2 gaps

** Chose second of 2 gaps

†† Chose third of 3 gaps

(Notice that two gaps remain incomplete at the end. They would close if attention were turned to them, but the procedure ends work on a derivation as soon as any open gap has reached a dead end.)

Glen Helman 01 Aug 2011