17. Explosion and Absurdity

Question 1

Explosion

Answer 1

A) ex falso sequitur quodlibet = from contradiction, anything [follows]’
B) Inconsistent truth table, not case with all true premises, hence tautologically valid.
C) α, ¬α ⊨ γ

Question 2

Falsum

Answer 2

A) Nullary connective (Symbol ⊥)
B) Truth condition: If anything Conclusion=0
C) α ⊨ ⊥ if and only if ⊨ ¬α
c) Falsum in all rows where α is true

Question 3

Verum

Answer 3

A) Nullary connective (Symbol ⊤)
B) Truth condition: If anything Conclusion=1

Question 4

Equivalences involving ⊤ and ⊥

Answer 4

A) ⊤ ≈ ¬⊥ and ⊥ ≈ ¬⊤
B) ⊤ ≈ α ∨ ¬α and ⊥ ≈ α ∧ ¬α (for any wff α)
C) α ∧ ⊤ ≈ α and α ∨ ⊥ ≈ α

Question 5

Tautologically valid arguments involving ⊤ and ⊥

Answer 5

A) ⊥ ⊨ α
B) α ⊨ ⊤

Question 6

Five ways of saying Falsum!

Answer 6

{α1, . . . , αn} is inconsistentα
1, . . . , αn ⊨ ⊥
α1 ∧ · · · ∧ αn is a contradiction
α1 ∧ · · · ∧ αn ⊨⊥
⊨ ¬(α1 ∧ · · · ∧ αn)