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 \({ ( ( ¬ P ) ∨ Q ) }\). The meaning of “⊃” can sometimes confuse people. Under the interpretation above \({ [ ( p ⊃ q ) ∨ ( q ⊃ p ) ] }\) is true for any \({ p }\) and \({ q }\). If this seems odd to you then you are probably thinking in terms of causality rather than logical implication.
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. Statements which are true in any of these interpretations, i.e. for any assignment of truth values to the variables occurring in them, are called tautologies. Statements which are true in some interpretation, i.e. for some assignment of truth values to the variables, are said to be satisfiable. Note that it’s only interpretations of the kind described above which are relevant. In judging whether a statement is a tautology or is satisfiable we don’t consider, for example, interpretations where “∨” means exclusive or.