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

Amiya Melendez

Answered question

2022-10-22

A vs. A
As far as I understand the notion of semantic consequence (denoted by ), 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".
However, when we take a look at A , 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

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:
A 1 , . . . , A n B
means
"[For all valuations, ] if A 1 and ... and A n 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 A 1 and ... and A n 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 A 1 or not ... or not A n , or B".
Step 2
Now
A
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
A
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.

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