2.2.x. Exercise questions

1. Restate the derivation below in two ways: (i) as a tree-form proof, labeling each horizontal line with the number of the stage at which it is entered, and (ii) as its associated argument tree. That is, do with it what is done with the example in 2.2.5 (ignoring the extra decoration, such as colors and dashed lines, that appeared there).
 
│(A ∧ C) ∧ B 1
├─
1 Ext │A ∧ C 2
1 Ext │B (4)
2 Ext │A
2 Ext │C (5)
│
││●
│├─
4 QED ││B 3
│
││●
│├─
5 QED ││C 3
├─
3 Cnj │B ∧ C
2. Use the system of derivations to establish each of the following claims of entailment:
a. A ∧ B ⊨ B ∧ A
b. A ⊨ A ∧ A
c. A ∧ (B ∧ C) ⊨ (C ∧ B) ∧ A
d. A, B ∧ C, D ⊨ (C ∧ (B ∧ A)) ∧ B
[The derivation for d will have three premises above the initial horizontal line.]
e. A ∧ (B ∧ C) ⊨ (B ∧ A) ∧ (C ∧ A)

For more exercises, use the exercise machine.

Glen Helman 04 Sep 2009