Formal Logic

where the universal (all) has been replaced with an existential (some/exists) then the argument would not be valid. In Ancient Greece, Aristotle exhaustively considered all possible combinations of universals and existentials in syllogisms, allowing also for the possibility of negations, and collected those corresponding to valid inferences in a classification theorem. For many centuries, that classification (slightly enhanced by the scholastics during the Middle Ages) was all there was to know about logic. In its origin, the term “formal” logic used to be a reference to the form of the arguments: the validity of an argument depends exclusively on the form of the premises and the conclusion, not on whether these are true or false. In the previous example, if we were to replace “horses” with “unicorns” the argument would still be valid, regardless of the fact that unicorns do no exist. Nowadays, however, “formal” refers to the use of the formal and rigorous methods of mathematics in the study of logic that began to be put into practice in the second half of the 19th century with George Boole and, especially, Gottlob Frege (see [1]). This trend started with a shift to a symbolic notation and artificial languages, and gradually evolved until, in 1933 with Tarski [2], it culminated with the withdrawal from an absolute notion of Truth and instead focused on the particular truths of concrete structures or models.