1.1.7. Formal logic

The subject we will study has traditional been given a variety of names. Deductive logic is one. Another is formal logic, and this term reflects an important aspect of the way we will study deductive reasoning. Even among the inferences that are deductive, we will consider only ones that do not depend on the subject matter of the data. This means that these inferences will not depend on the concepts employed to describe particular subjects, and it also means that they will not depend the mathematical structures (systems of numbers, shapes, etc.) that might be employed in such descriptions. This can be expressed by saying that we will limit ourselves to inferences that depend only on the form of the claims involved.

The distinction between form and content is a relative one. For example, the use of numerical methods to extract information can be said to depend on content by comparison with the sort of inferences we will study. However, it can count as formal by comparison with other ways of extracting information since all that matters for much of the numerical analysis of data is the numbers that appear in a body of measurements, not the nature of the quantities measured.

Our study is formal in a sense similar to that in which numerical methods are formal, but it is formal to a greater degree. What matters for formal logic is the appearance of certain words or grammatical constructions that can be employed in statements concerning any subject matter. Examples of such logical words are and, not, or, if, is (in the sense of is identical to), every, and some. While this list does not include all the logical words we will consider, it does provide a fair indication of the forms of statements we will study. Indeed, these seven words could serve as titles for chapters 2-8 of this text, respectively. The way in which a statement is put together using words like these (and using logically significant grammatical constructions not directly marked by words) is its logical form, and formal logic is a study of reasoning that focuses on the logical forms of statements.

So the subject we will study will be not only deductive logic but formal logic. That means that the norms of deductive reasoning that we will study will be general rules applying to all statements with certain logical forms. It happens that we can give an exhaustive account of such rules in the case of the logical forms that we will consider, so the content of the course can be defined by these forms. Truth-functional logic, which will occupy us through chapter 5, is concerned with logical forms that can be expressed using the words and, not, or, and if while first-order logic (with identity) is concerned with the full list above, adding to truth-functional logic forms that can be expressed by the words is, every, and some.

Another traditional label for the subject we will study is the term symbolic logic that appears in the course title. Most of what this term indicates about the content of our study is captured already by the term formal logic because most of the symbols we use will serve to represent logical forms. Certain of the logical forms that appear in the study of truth-functional logic are analogous to patterns appearing in the symbolic statements of algebraic laws. Analogies of this sort were recognized by G. W. Leibniz (1646-1716) and by others after him, but they were first pursued extensively by George Boole (1815-1864), who adopted a notation for logic that was modeled after algebraic notation. The style of symbolic notation that is now standard among logicians owes something to Boole (though the individual symbols are different) and something also to the notation used by Gottlob Frege (1848-1925), who noted analogies between first-order logic and the mathematical theory of functions. This interest in analogies with mathematical theories distinguished logic as studied by Boole and Frege from its more traditional study, and the term symbolic has often been used to capture this distinction. The phrase mathematical logic would be equally appropriate, and it has often been used as a label for the subject we will study. However, it has also been used a little more narrowly to speak of an application of logic to mathematical theories that makes these theories objects of mathematical study in their own right. That application of logic in a mathematical style to mathematics itself produces a kind of research that is also known as metamathematics (which means, roughly, ‘the mathematics of mathematics’).