﻿ [L01] What is logic?

The term "logic" came from the Greek word logos, which is sometimes translated as "sentence", "discourse", "reason", "rule", and "ratio". Of course, these translations are not enough to help us understand the more specialized meaning of "logic" as it is used today.

So what is logic? Briefly speaking, we might define logic as the study of the principles of correct reasoning. This is a rough definition, because how logic should be properly defined is actually quite a controversial matter. However, for the purpose of this tour, we thought it would be useful to give you at least some rough idea as to the subject matter that you will be studying. So this is what we shall try to do on this page.

## §1. Logic is not the psychology of reasoning

One thing you should note about this definition is that logic is concerned with the principles of correct reasoning. Studying the correct principles of reasoning is not the same as studying the psychology of reasoning. Logic is the former discipline, and it tells us how we ought to reason if we want to reason correctly. Whether people actually follow these rules of correct reasoning is an empirical matter, something that is not the concern of logic.

The psychology of reasoning, on the other hand, is an empirical science. It tells us about the actual reasoning habits of people, including their mistakes. A psychologist studying reasoning might be interested in how people's ability to reason varies with age. But such empirical facts are of no concern to the logician.

## §2. The principles of logic

So what are these principles of reasoning that are part of logic? There are many such principles, but the main (not the only) thing that we study in logic are principles governing the validity of arguments - whether certain conclusions follow from some given assumptions. For example, consider the following three arguments :

If Tom is a philosopher, then Tom is poor.
Tom is a philosopher.
Therefore, Tom is poor.

If K>10, then K>2.
K>10.
Therefore, K>2.

If Tarragona is in Europe, then Tarragona is not in China.
Tarragona is in Europe.
Therefore, Tarragona is not in China.

These three arguments here are obviously good arguments in the sense that their conclusions follow from the assumptions. If the assumptions of the argument are true, the conclusion of the argument must also be true. A logician will tell us that they are all cases of a particular form of argument known as "modus ponens" :

If P, then Q. P. Therefore, Q.

We shall be discussing validity again later on. It should be pointed out that logic is not just concerned with the validity of arguments. Logic also studies consistency, and logical truths, and properties of logical systems such as completeness and soundness. But we shall see that these other concepts are also very much related to the concept of validity.

## §3. Topic neutrality

Modus ponens might be used to illustrate two features about the rules of reasoing in logic. The first feature is its topic-neutrality. As the four examples suggest, modus ponens can be used in reasoning about diverse topics. This is true of all the principles of reasoning in logic. The laws of biology might be true only of living creatures, and the laws of economics are only applicable to collections of agents that enagage in financial transactions. But the principles of logic are universal principles which are more general than biology and economics. This is in part what is implied in the following definitions of logic by two very famous logicians :

[Logic is] ... the name of a discipline which analyzes the meaning of the concepts common to all the sciences, and establishes the general laws governing the concepts.
Alfred Tarski (1901-1983). From his Introduction to logic and to the methodology of deductive sciences, Dover, page xi.
To discover truths is the task of all sciences; it falls to logic to discern the laws of truth. ... I assign to logic the task of discovering the laws of truth, not of assertion or thought.
Gottlob Frege (1848-1925). From his 1956 paper "The Thought : A Logical Inquiry" in Mind Vol. 65.

## §4. Necessity in logic

A second feature of the principles of logic is that they are non-contingent, in the sense that they do not depend on any particular accidental features of the world. Physics and the other empirical sciences investigate the way the world actually is. Physicists might tell us that no signal can travel faster than the speed of light, but if the laws of physics have been different, then perhaps this would not have been true. Similarly, biologists might study how dolphins communicate with each other, but if the course of evolution had been different, then perhaps dolphins might not have existed. So the theories in the empirical sciences are contingent in the sense that they could have been otherwise. The principles of logic, on the other hand, are derived using reasoning only, and their validity does not depend on any contingent features of the world.

For example, logic tells us that any statement of the form "If P then P." is necessarily true. This is a principle of the second kind that logician study. This principle tells us that a statement such as "if it is raining, then it is raining" must be true. We can easily see that this is indeed the case, whether or not it is actually raining. Furthermore, even if the laws of physics or weather patterns were to change, this statement will remain true. Thus we say that scientific truths (mathematics aside) are contingent whereas logical truths are necessary. Again this shows how logic is different from the empirical sciences like physics, chemistry or biology.

## §5. Formal and informal logic

Sometimes a distinction is made between informal logic and formal logic. The term "informal logic" is often used to mean the same thing as critical thinking. Sometimes it is used to refer to the study of reasoning and fallacies in the context of everyday life. "Formal logic" is mainly concerned with formal systems of logic. These are specially constructed systems for carrying out proofs, where the languages and rules of reasoning are precisely and carefully defined. Sentential logic (also known as "Propositional logic") and Predicate Logic are both examples of formal systems of logic.

There are many reasons for studying formal logic. One is that formal logic helps us identify patterns of good reasoning and patterns of bad reasoning, so we know which to follow and which to avoid. This is why studying basic formal logic can help improve critical thinking. Formal systems of logic are also used by linguists to study natural languages. Computer scientists also employ formal systems of logic in research relating to Aritificial Intelligence. Finally, many philosophers also like to use formal logic when dealing with complicated philosophical problems, in order to make their reasoning more explicit and precise.

## §6. Further readings

• Entries on logic in the Stanford Encyclopedia of Philosophy. These are difficult articles on the philosophy of logic.
• Suppose we find out that people reason better when they are not under stress. Should this be counted as a principle of logic?
• How is science different from logic?

• How is mathematics different from logic?