Tautological entailment

Question 1

Definition of tautological entailment

Answer 1

A1, A2, . . . , An tautologically entail C iff no valuation of the relevant atomic sentences makes all of A1, A2, . . . , An true and C false.

Question 2

Checking T.E on a truth table

Answer 2

To check whether ‘¬L → (J ∨ L)’ and ‘¬L’ tautologically entail ‘J’, we simply need to check whether there is ANY valuation which makes both ‘¬L → (J ∨ L)’ and ‘¬L’ true whilst making ‘J’ false
- this SINGULAR line would prove that there is not T.E

Question 3

P,P →Q⊨Q - what does this abbreviate?

Answer 3

The TFL sentences ‘P ’ and ‘P → Q’ tautologically entail ‘Q’

Question 4

⊨ C

Answer 4

there is no valuation which makes all the sentences mentioned on the left side of ‘⊨’ true whilst making C false - C is a tautology

Question 5

A ⊨

Answer 5

this says that no valuation makes A true - A is a contradiction

Question 6

How would we represent this:
it is not the case that A1,…,An ⊨ C

Answer 6

A1,A2,…,An ⊭ C