The standard interpretation is that the symbols “∧”, “∨”, “¬”, and “⊃” for “and”, “or”, “not” and “implies” mean what you think they do, assuming you think “or” is always inclusive and you interpret “⊃” the way mathematicians and logicians do, i.e. that the expression is true if the hypothesis is false or the conclusion is true. As we discussed in the introduction \({ ( P ⊃ Q ) }\) has the same meaning as \({ ( ( ¬ P ) ∨ Q ) }\).
Like “⊃” the more exotic symbols are all expressible in terms of “∧”, “∨”, and “¬”. \({ ( P ⊼ Q ) }\) has the same meaning as \({ ( ¬ ( P ∧ Q ) ) }\). \({ ( P ⊻ Q ) }\) has the same meaning as \({ ( ¬ ( P ∨ Q ) ) }\). \({ ( P ≡ Q ) }\) has the same meaning as \({ ( ( P ∧ Q ) ∨ ( ( ¬ P ) ∧ ( ¬ Q ) ) ) }\). \({ ( P ≢ Q ) }\) has the same meaning as \({ ( ( P ∧ ( ¬ Q ) ) ∨ ( ( ¬ P ) ∧ Q ) ) }\). It’s the exclusive or which we discussed earlier. \({ ( P ⊂ Q ) }\) has the same meaning as \({ ( P ∨ ( ¬ Q ) ) }\).
The variables are Boolean variables. They can take the values true or false. Technically every possible assignment of values to the variables is a different interpretation of the language.