This usually gets on an exam:
if α is a tuautology, then -α is a contradiction and so on.
(everything relating to this)
α is a tautology and β is a tautology, their ^ is also a tautology.
α is a tautology and β is a contingency, then their ^ is a contingency
α is a contingency and β is a contingency, their ^ is either a contingency or a contradiction.
α is a tautology and β can be whatever, their v is always a tautology
(We cannot conclude that v’s parts are tautologies based on the v itself being a tautology, eg. pv-p)
for any formula, αv-α is a tautological form.
α is deductively necessary.
α is true in all situations
all widows are womens
all widows are widows
it’s raining or it’s not raining.
α is logically necessary
α is true in all situations in virtue of its logical vocabulary
all widows are widows
it’s raining or it’s not raining
all widows are womens
α is tautologically necessary
α is true in all situations in virtue of its propositional connectives (^,v,-)
it’s raining or it’s not raining
all widows are womens
all widows are widows
P v Q
-P
Q
PQ | PvQ | -P | Q
1 0 1 0 0
1 1 1 0 1
0 0 0 1 0
0 1 1 1 1
P v Q
P v -Q
Q v R
→ (P ^ Q) v (-Q ^ R)
”A tautological valid argument happens when the conclusion behaves like a tautology, in light of the premises.”
An argument is tautologically valid iff all valuations which make all its premises true, also make its conclusion true.
An argument is tautologically valid iff there does not exist a valuation which makes all its premises true, yet its conclusion false.
An argument is tautologically valid iff there exists a valuation which makes all its premises true, yet its conclusion is false.
α v β, -α, therefore β is a tautologically valid form.
Therefore, (P ^ Q) v -P v -Q → tautologically valid (despite lacking premises)
A tautology can be viewed as the conclusion of a tautologically valid argument with no premises.
α≈β iff α⊨β and β⊨α
eg. the De Morgan’s laws.