are given to the other symbols in a formula. This is how we define logical connectives in propositional logic: ¬Φ is True iff Φ is False. (Φ ∧ Ψ) is True iff Φ is True and Ψ is True. (Φ ∨ Ψ) is True iff Φ is True or Ψ is True (or both are True). (Φ → Ψ) is True iff ¬Φ is True or Ψ is True (or both are True). (Φ ↔ Ψ) is True iff (Φ → Ψ) is True and (Ψ → Φ) is True. So under a given interpretation of all the sentence letters Φ and Ψ (i.e., after assigning a truth-value to each sentence letter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logical connectives. The following table shows how this kind of thing looks. The first two columns show the truth-values of the sentence letters as determined by the four possible interpretations. The other columns show the truth-values of formulas built from these sentence letters, with truth-values determined recursively. Now it is easier to see what makes a formula logically valid. Take the formula F: (Φ ∨ ¬Φ). If our interpretation function makes Φ True, then ¬Φ is made False by the negation connective. Since the disjunct Φ of F is True under that interpretation, F is True. Now the only other possible interpretation of Φ makes it False, and if so, ¬Φ is made True by the negation function. That would make F True again, since one of Fs disjuncts, ¬Φ, would be true under this interpretation. Since these two interpretations for F are the only possible logical interpretations, and since F comes out True for both, we say that it is logically valid or tautologous. Interpretation of a theory An interpretation of a theory is the relationship between a theory and some subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a full interpretation, otherwise it is called a partial interpretation. Interpretations for propositional logic The formal language for propositional logic consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, propositional variables) and logical connectives. The only non-logical symbols in a formal language for propositional logic are the