Principles of Formal Classical Logic and Symbolic Systems

Formal Classical Logic: Bivalent and Truth-Functional

Logic is a science that studies the laws and principles that determine the validity of an argument. It is a philosophical field of knowledge whose foundations were first forged in antiquity. In the 4th Century BC, a thinker named Aristotle from the region of Stagira composed a treatise devoted to logic; it is the first book we have record of devoted entirely to logic.

Now, this science has developed as independent knowledge, especially since the late nineteenth century. Some crucial contributions that made possible the development of formal logic as a science are due to: Gottlob Frege, Bertrand Russell, Alfred North Whitehead, Jan Łukasiewicz, Rudolf Carnap, Ludwig Wittgenstein, and Willard Van Orman Quine.

Formal Logic and Symbolic Languages

Current logic is a science that uses symbolic languages. Formal languages are systems of symbols processed expressly for the purpose of analyzing the logical validity of propositions and reasoning. A formal language is a well-constructed artificial language: it introduces a vocabulary, a set of rules for the formation of syntactically well-formed expressions, and a transformation rule set to produce some expressions from others to make possible what is called a logical calculus. From the time we decided to use a language of these features, an important distinction arises:

Object Language

This is the language we are using for the logical analysis of the validity of reasoning. Therefore, in our case, the object language will be the formal language that we have constructed for the study of the principles and laws of the logical validity of an argument.

• In the case of arithmetic, the object language would be the numerical language that we used for the introduction of the different arithmetic operations.
• In geometry, the object language would be the set of symbols and geometric figures that we used for the demonstration of a particular theorem.
• In the case of a linguistics class, the object language would be the set of sentences belonging to the Castilian language we have chosen as valid examples of sentences that can be analyzed linguistically (in this case, the object language is our own language).

Metalanguage

The metalanguage is the language we use to talk about our object language. In our case, our main metalanguage is the language we use to communicate habitually (e.g., Castilian). Therefore, all sciences require at least one possible metalanguage for understanding and communication between users of the object language of each science.

Use and Mention

• Use: We use a word when we only use it as we were taught when we learned the vocabulary of our everyday language. The function of normally using a word is to refer to something named by that word, and that is its usual meaning.
• Mention: We mention a word when we want to highlight some of its syntactic and semantic properties. In this case, we must use a metalanguage to talk about such properties.