In a logical system, a judgement is a statement that is either true or false. So far, you are most familiar with the type of judgement "A is true", which is often simply abbreviated as "A" when this is the only kind of judgement being considered. However, it is often benecial to define judgement forms in addition to "truth" to construct logics. For example consider the following truth concept "Set":
WFF's = {{,},',',0}, i.e., all strings made of left curly braces, right curly braces, commas, and 0's.
A WFF is true if it represents a valid set of strings of 0's, possibly with repeated elements, i.e. it is a comma separated list of strings of 0's surrounded by a pair of braces.
For example, the WFF {0; 000; 0; 00} is true, while each of the following WFFs are false: {0}, {0,} {0,,0} 0,00,000
It is possible to construct a sound and complete logic for this truth concept directly, however, it is perhaps more intuitive if we define the additional judgement forms "Number" and "List".
Axiom: 0 is a "Number".
Axiom: {} is a theorem.
Inference Rule: If x is a "Number", then x0 is a "Number".
Inference Rule: If x is a "Number", then x is a "List".
Inference Rule: If x is a "List" and y is a "List", then x, y is a "List".
Inference Rule: If x is a "List", then {x} is a theorem.
Prove that this logic is sound and complete for "Set"