propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed. The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the truth values true and false. This function is known as a truth assignment or valuation function. In many presentations, it is literally a truth value that is assigned, but some presentations assign truthbearers instead. For a language with n distinct propositional variables there are 2n distinct possible interpretations. For any particular variable a, for example, there are 21=2 possible interpretations: 1) a is assigned T, or 2) a is assigned F. For the pair a, b there are 22=4 possible interpretations: 1) both are assigned T, 2) both are assigned F, 3) a is assigned T and b is assigned F, or 4) a is assigned F and b is assigned T. Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above. First-order logic Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a signature. The signature consists of a set of non-logical symbols and an identification of each of these symbols as a constant symbol, a function symbol, or a predicate symbol. In the case of function and predicate symbols, a natural number arity is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables. For example, in the language of rings, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.) Again, we might define a first-order language L, as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols. Formal languages for first-order logic Given a signature σ, the corresponding formal language is known as the set of