Modal operator in the context of Binary function

In logic, a strict conditional (symbol: ${\displaystyle \Box }$, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while ${\displaystyle \Box (p\rightarrow q)}$ says that p strictly implies q. Strict conditionals are the result of Clarence Irving Lewis's attempt to find a conditional for logic that can adequately express indicative conditionals in natural language. They have also been used in studying Molinist theology.

Doxastic logic is a type of logic concerned with reasoning about beliefs.

The term doxastic derives from the Ancient Greek δόξα (doxa, "opinion, belief"), from which the English term doxa ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation ${\displaystyle {\mathcal {B}}_{c}x}$ to mean "reasoner ${\displaystyle c}$ believes that ${\displaystyle x}$ is true", and the set ${\displaystyle \mathbb {B} _{c}:\left\{b_{1},\ldots ,b_{n}\right\}}$ denotes the set of beliefs of ${\displaystyle c}$. In doxastic logic, belief is treated as a modal operator.