semistrongform

Semistrongform is a term used in the field of logic and mathematics, particularly in the study of modal logic and provability logic. It refers to a specific type of logical system that combines elements of both classical logic and modal logic. In semistrongform, the logical connectives and quantifiers are interpreted in a classical manner, while the modal operators (such as possibility and necessity) are interpreted in a non-classical way.

The term "semistrongform" was introduced by Solomon Feferman in the context of provability logic. In this context, semistrongform is used to describe a system that captures the logical properties of provability in arithmetic. The system is called "semistrong" because it is stronger than classical logic but not as strong as the full modal logic system known as "strongform."

Semistrongform is characterized by the following properties:

1. It includes all the axioms and rules of classical logic.

2. It includes a modal operator for provability, which is denoted by "Pr."

3. It includes the axiom schema of necessitation, which states that if a formula is provable, then it is necessarily true.

4. It includes the axiom schema of provability, which states that if a formula is provable, then it is true.

Semistrongform has been studied extensively in the context of provability logic, where it is used to analyze the logical properties of provability in arithmetic. It has also been used in the study of other modal logics, where it is used to analyze the interaction between classical logic and modal logic.