14. Tautologies

Question 1

Tautologies (4)

Answer 1

A) true under every possible valuation on their atoms
B) true in all situations bc of its propositional connectives
C) notation: if α is a tautology, then ⊨ α
D) Vacuously true, useless to describe the real world
EX: P ∨ ¬P all evaluations are true
P | P ∨ ¬P
1 | 1
0 | 1
EX: it’s raining or it’s not raining

Question 2

Contradictions (3)

Answer 2

A) false under every possible valuation on their atoms
B) notation: if α is a contradiction, then ⊨ ¬α
C) Vacuously true, useless to describe the real world
EX: P ∧ ¬P all evaluations are true
P | P ∧ ¬P
1 | 0
0 | 0

Question 3

Law of Excluded Middle

Answer 3

P ∨ ¬P

Question 4

Law of Non-Contradiction

Answer 4

¬(P ∧ ¬P)

Question 5

Contingency (3)

Answer 5

A) Not a tautology nor a contradiction
B) TRUE under a valuation on their atoms
FALSE under a valuation on their atoms
C) Notation: ⊭ α and ⊭ ¬α
EX: : P, P ∧ Q, P ∨ Q

Question 6

Simple Tautologies, Contradictions and Contingencies (3)

Answer 6

A) if α is a tautology, then ¬α is a contradiction
B) if α is a contradiction, then ¬α is a tautology
C) if α is a contingency, then ¬α is a contingency

Question 7

Complex Tautologies, Contradictions and Contingencies (3)

Answer 7

A) if α is a tautology and β is a tautology,then α ∧ β is a tautology
B) if α is a tautology and β is a contingency,then α ∧ β is a contingency
C) if α is a contingency and β is a contingency,then α ∧ β is either a contingency or a contradiction: P ∧ Q or P ∧ ¬P

Question 8

Tautological Form

Answer 8

A) A form which every instantiation is a tautology
EX: α ∨ ¬α
B) Non tautological forms can still instantiate a tautology
EX: α ∨ β is not a tautological form, but instantiates α ∨¬α