As far as I understand the notion of semantic consequence (denoted by models), models A means A is a semantic consequence of the empty set. So the "empty space" on the left side of the double turnstile means "empty set".
Amiya Melendez
Answered question
2022-10-22
vs. As far as I understand the notion of semantic consequence (denoted by ), means A is a semantic consequence of the empty set. So the "empty space" on the left side of the double turnstile means "empty set". However, when we take a look at , now that means A is a contradiction, i.e., everything is a semantic consequence of A. Now the empty space means "everything". Why is that? Is there any explanation for that difference?
Answer & Explanation
fjaldangi
Beginner2022-10-23Added 9 answers
Step 1 It helps to think of semantic consequence as a disjunction, where the premises are negated and the conclusion is positive:
means "[For all valuations, ] if and ... and are all true [under that valuation], then B is true [under that same valuation], too" which can, using the fact that "If X then Y" is in mathematical use equivalent to "either X is not the case or Y is the case", be reformulated as "either not all of and ... and are true, or B is true" which is in turn, using the equivalence between "not both X and Y" and "not X or not Y", equivalent to "not or not ... or not , or B". Step 2 Now
where there are no (negated) premises and only the (unnegated) conclusion means "[For all valuations, ] (nothing) or A" so A is the only option to make the "or"-statement happen -- A must be true under all circumstances, i.e., A is a tautology. However, if A occurs on the left side of the sequent
it is negated, so we have "[For all valuations, ] not A or (nothing)" so now "not A" is the only option to satisfy the disjunction, meaning that A can only ever be false - i.o.w., A is a contradiction.