Elementary Logic by Garrett Brian

Author:Garrett, Brian.
Language: eng
Format: epub
ISBN: 9781317547495
Publisher: Taylor & Francis (CAM)

We can apply ∨O to lines (1), (7) and (11), lines that either are, or depend only on, premises:

1,2,3 (12) C ∨ D 1, 7, 11 ∨O

A fair copy of the proof:

A ∨ B, A → C, B → D Ⱶ C ∨ D

1 (1) A ∨ B P

2 (2) A → C P

3 (3) B → D P

4 (4) A A

2,4 (5) C 2,4 →O

2,4 (6) C ∨ D 5 ∨I

2 (7) A → (C ∨ D) 4–6 →I

8 (8) B A

3,8 (9) D 3, 8 →O

3,8 (10) C ∨ D 9, ∨I

3 (11) B → (C ∨ D) 8–10 →I

1,2,3 (12) C ∨ D 1, 7, 11 ∨O

PROVING LOGICAL TRUTHS

Lines (4) and (7) in our proof of A ∨ B h B v A are tautologies or logical truths. Logical truths are true come what may: they rest on no premises or assumptions. They are absolutely and unconditionally true.

Any tautology is provable on the basis of no premises. For example, A → (A ∨ B) is a logical truth. We should then be able to prove the sequent:

Ⱶ A → (A ∨ B)

[i]

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