Symbolic logic

The roots of symbolic logic can be traced as far back as to the study of syllogisms by Aristotle. However, it was not until the second half of the nineteenth century that its modern form began crystallizing, whereas solid foundations had to wait until the second decade of the twentieth century. Symbolic logic is concerned with the validity of arguments, for whose study it abstracts away from concrete instances: the validity of an argument depends exclusively on the form of the premises and the conclusion, not on whether these are true or false. For example, if “Anne and Bob are not both clorophastreo” and “Bob is clorophastreo”, then it follows that “Anne is clorophastreo” regardless of the meaning of the adjective “clorophastreo”. Symbolically, concluding “A” from the premises “not (A and B)” and “not B” is a valid inference rule. Among the different logics (sub-disciplines) encompassed in this field, there is one that deserves to be singled out for its relevance: predicate logic. Its interest lies in reasoning about what is universally true and, as such, it is especially suited to reason about mathematical statements and can be considered a generalization of Aristotelian syllogisms. Predicate logic goes beyond syllogisms by introducing predicates with arbitrary numbers of arguments, and quantifiers that allow to refer either to all or to some of the elements in the universe that is under consideration. It has a proof theory, which consists of a set of rules that describe how to mechanically derive sentences from a given set of premises (and such derivations are called “proofs”), as well as a model theory that assigns meaning to the sentences with respect to structures so that a given sentence is either true or false in a given structure. Predicate logic is sound, meaning that every sentence that can be derived using the rules in its proof theory holds in every possible structure. More importantly, a fundamental result by Gödel in 1929 shows that predicate logic is also complete in the sense that if a sentence is true in every structure then it can be derived from the empty set of premises. But this is not the only possibility. Within the discipline of symbolic logic, a veritable cornucopia of different logics has sprung during the last century. Although there is no