Logic

I

Introduction

Logic (Greek logos, “word”, “speech”, “reason”), science dealing with the principles of valid reasoning and argument. The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to a conclusion that is claimed to follow from them. Logical validity is the characteristic of an argument that guarantees that if the premises of the argument are true then the conclusion must necessarily be true.

The validity of an argument should be distinguished from the truth of the conclusion. If one or more of the premises is false, the conclusion of a valid argument may be false. Take, for example:

• All mammals are four-footed animals
• All people are mammals
• Therefore, all people are four-footed animals.

This is a valid argument with a false conclusion because its first premise is false. On the other hand, an invalid argument may by chance have a true conclusion. For example:

• Some animals are two-footed
• All people are animals
• Therefore, all people are two-footed.

This happens to have a true conclusion, but the argument is not valid.

Logical validity depends, therefore, on the form of the argument, not on its content. If the argument were valid, some other term could be substituted for all occurrences of any one of those used and validity would not be affected. By substituting “four-footed” for “two-footed” in the second argument, it can be seen that the premises could both be true and the conclusion false. Thus the argument is invalid, even though it has a true conclusion.

II

Aristotelian Logic

What is now known as Classical or traditional logic was first formulated by Aristotle, who developed rules for correct syllogistic reasoning. A syllogism is an argument made up of statements in one of four forms: “All As are Bs” (universal affirmative), “No As are Bs” (universal negative), “Some As are Bs” (particular affirmative), or “Some As are not Bs” (particular negative). The letters stand for common nouns, such as “dog”, “four-footed animal”, or “living thing”, which are called the “terms” of the syllogism. A well-formed syllogism consists of two premises and a conclusion, each premise having one term in common with the conclusion and one in common with the other premise. A famous example of a well-formed syllogism is:

• All men are mortal
• Socrates is a Man
• Therefore, Socrates is mortal.

In Classical logic, rules are formulated by which all well-formed syllogisms are identified as valid or invalid forms of argument.

III

Kant and Hegel

Traditional logic was often described as the science of the laws of thought, meaning the laws that thinking must obey if it is to be rational, rather than psychological generalizations about the way that people actually think. The 18th-century German philosopher Immanuel Kant relied on this definition in order to introduce a new area of logic. In his Critique of Pure Reason (1781), Kant argued that thinking is not just a matter of deriving one proposition from another. In order to make any sense of its sensations, the mind has to organize them through certain general concepts or categories, and this organizing activity is itself a kind of thinking. Kant said that traditional logic, which he called “general logic”, was the science of the principles of rational thinking in general. In contrast to this, he introduced the idea of a “transcendental logic” that would be the science of the categories and the structural principles associated with them, through which the mind has to “think” its sensations in order to organize them into rational experience. Kant called these structural principles “synthetic a priori” because they are informative and yet are not derived from experience. Instead, they are intrinsic to the nature of the mind, which has to bring them to its experience in order to make rational sense of it. For example, the mind has to organize its sensations through the category of cause and effect, and accordingly it has to experience the world as governed by the synthetic a priori principle that “every event has a cause”, in order to make sense of its sensations.

The science of these categories and synthetic a priori principles is logic, because it elucidates the principles in terms of which the mind must think its sensations in order to make rational sense of them. However, it is also metaphysics, since these categories and principles govern the workings of the world as it has to be experienced, and this is the only world that can possibly be known. Kant also retained the idea of a world of things in themselves, things as they are independent of human experience of them, to which the categories do not apply and about which nothing can be said.

Kant’s successor, G. W. F. Hegel, extended the idea of logic as a science of the necessary categories of rational thinking. However, he abandoned Kant’s notion of the thing in itself, so that for Hegel there is no world at all apart from the world as it is thought. Thereby, Hegel completed the identification of logic and metaphysics begun by Kant. His Science of Logic (1812-1816) is an attempt to expound the complete system of categories that must both govern rational thinking and underpin reality, beginning with the simplest category of all, that of “being” or “is”. Hegel tried to show that these categories necessarily arise out of each other through a process of internal reflection and contradiction. For example, in thinking about the category “being”, one is led necessarily to formulate the category “nothing”, and realizing the contradiction between these two categories must in turn lead one to formulate the category “becoming”. Not only thinking but reality itself is supposed to be essentially characterized by this development of new categories out of old ones through internal contradiction.

This kind of logic, in which categories are derived from each other through internal contradiction, has been called dialectical logic. It was subsequently applied to the analysis of material economic structures by the social and political theorist Karl Marx, and Marxists have since tried to combine the idea of dialectical logic with a materialist metaphysics. However, most logicians did not take up the idea of a “logic of categories” initiated by Kant and Hegel, and instead pursued the traditional notion of logic as a science of the principles governing valid arguments.

IV

Modern Formal Logic

In the mid-19th century, the British mathematicians George Boole and Augustus De Morgan opened a new field of logic, now known as symbolic or formal logic, which was further developed by the German mathematician Gottlob Frege and especially by the British mathematicians Bertrand Russell and Alfred North Whitehead in Principia Mathematica (3 vols., 1910-1913). The logical system of Russell and Whitehead covers a far greater range of possible arguments than those that can be cast into syllogistic form. It introduces symbols for complete sentences and for the conjunctions that connect them, such as “or”, “and”, and “if ... then. ... “. It has different symbols for the logical subject and the logical predicate of a sentence; and it has symbols for classes, for members of classes, and for the relationships of class membership and class inclusion. It also differs from Classical logic in its assumptions as to the existence of the things referred to in its universal statements. The statement “All As are Bs” is rendered in modern formal logic to mean, “If anything is an A, then it is a B”, which, unlike Classical logic, does not assume that any As exist. The basic elements of modern formal logic are called the propositional (or sentential) calculus and the predicate calculus.

Both Classical logic and standard modern formal logic are systems of deductive logic. In a sense, the premises of a valid argument contain the conclusion, and the truth of the conclusion follows from the truth of the premises with certainty. To hold that the premises are true and the conclusion is false would be to contradict oneself. Efforts have also been made to develop formal systems of inductive logic, such that the premises are evidence for the conclusion, but the truth of the conclusion follows from the truth of the evidence only with a certain probability. The most notable contribution to inductive logic is that of the British philosopher John Stuart Mill who, in his System of Logic (1843), formulated the methods of proof that he believed to characterize empirical science. This inquiry developed in the 20th century into the field known as the philosophy of science. Closely related is the branch of mathematics known as probability theory.

Both Classical and modern formal logic in their usual versions assume that any well-formed sentence is either true or false. In recent years efforts have been made to develop systems of so-called many-valued logic, such that an assertion may have some value other than true or false. In some this is merely a third neutral value; in others it is a probability value expressed as a fraction ranging between 0 and 1 or between -1 and +1. Another development in recent years has been the effort to develop systems of modal logic, to represent the logical relations between assertions of possibility and impossibility, necessity and contingency. Still another development is deontic logic, the investigation of the logical relations between commands or between statements of obligation.