Human beings have been thinking logically (and sometimes illogically) since the earliest era of human existence. However, they have not always been aware of the general principles that distinguish logical from illogical forms of thought. Logic, as an academic subject, is the systematic study of those principles. The logician asks, Which rules should we follow if we want our reasoning to be the best possible?

The rules of logic are guides to correct reasoning just as the rules of arithmetic are guides to correctly adding, subtracting, multiplying, and dividing numbers, the principles of photography are guides to taking good photos, and so on. You can improve your reasoning by studying the principles of logic, just as you can improve your number-crunching abilities by studying the principles of mathematics. Because correct reasoning can be applied to any subject matter whatsoever, the number of potential applications of logical theory is practically unlimited.

The Greek philosopher Aristotle (384–322 BC) wrote the first book on the standards of correct reasoning and later wrote four additional treatises on the subject. Thus, in five highly original (and extremely complex) works, collectively known as the Organon (Greek for “tool,” as in “general tool of thought”), Aristotle launched the study of the principles of correct reasoning and earned the title historians have conferred on him: founder of logic. [i] The noted twentieth-century logician and philosopher Benson Mates writes:

[W]e can say flatly that the history of logic begins with the Greek philosopher Aristotle . . . Although it is almost a platitude among historians that great intellectual advances are never the work of only one person (in founding the science of geometry Euclid made use of the results of Eudoxus and others; in the case of mechanics Newton stood upon the shoulders of Descartes, Galileo, and Kepler; and so on), Aristotle, according to all available evidence, created the science of logic absolutely ex nihilo. [ii]

Logic was first taught as an academic subject in the universities of ancient Athens, Greece during the fourth century BC, making it one of the oldest of all academic subjects. For twenty-five hundred years, it has been considered a core academic requirement at institutions of higher learning around the world. Logic remains part of the core curriculum around the world today because the principles of correct reasoning can help anyone reason more accurately, no matter what subject, making it an all-purpose “tool kit” for your mind.

Major Divisions of Logic

Formal logic studies the abstract patterns or forms of correct reasoning. Here the focus is on form rather than content, that is, on the logical structure of reasoning apart from what it is specifically about. Since ancient times, logicians have used special symbols and formulas, similar to those used in mathematics, to record the abstract logical forms they have discovered. This is why formal logic is sometimes also called “symbolic logic” or “mathematical logic.”

Informal logic studies the non-formal aspects of reasoning—qualities that cannot be accurately translated into abstract symbols. This is why informal logic for the most part dispenses with special symbols and formulas. In this division of logic, the focus is often reasoning expressed within everyday language.

Elements

Logical theory begins with the notion of an argument, which is defined as one or more statements, called “premises,” offered as evidence, or reason to believe, that a further statement, called the “conclusion,” is true. In plain terms, an argument is reasoning offered in support of a conclusion. Arguments are part of everyday life. You present one every time you put your reasoning into words to share it with others. In the following example, the premises are marked P1 and P2, and the conclusion is labeled C.

1. P1: All songwriters are poets.
2. P2: Bob Dylan is a songwriter.
3. C: Therefore, Bob Dylan is a poet.

The second building block of logical theory is the distinction, first noted by Aristotle, between deductive and inductive reasoning. A deductive argument aims to establish its conclusion with complete certainty, in such a way that if its premises all are true, then its conclusion must be true. Put another way, the underlying claim in the case of a deductive argument is that it is not even possible the premises all are true and the conclusion is false. For example:

1. P1. Tiny Tim played the ukulele.
2. P2. Anyone who plays the ukulele is a musician.
3. C. Consequently, Tiny Tim was a musician.

Deductive arguments aim for certainty and nothing less. If a deductive argument succeeds in its aim, it is a valid deductive argument. If it does not, it is an invalid deductive argument. A deductive argument is said to be sound if it is (a) valid and (b) all of its premises are true. The following deductive argument is clearly valid although it is not sound.

1. P1. All students are millionaires.
2. P2. All millionaires drink vodka.
3. C. Therefore, necessarily, all students drink vodka.

In contrast, the following argument is invalid (and hence also unsound).

1. P1. Ann and Sue are cousins.
2. P2. Sue and Rita are cousins.
3. C. So, Ann and Rita must be cousins.

The following argument hits the target—it is both valid and sound.

1. P1. All whales are mammals.
2. P2. All mammals are warm-blooded.
3. C. Ergo, all whales are warm-blooded.

Deductive logic is the study of the standards of correct deductive reasoning. Here is an example of a law of deductive logic. Let A, B, and C be variables ranging over terms that stand for categories—words such as cats, dogs, people, trucks, and so forth. Aristotle proved that the following form or pattern of reasoning, named Barbara by logicians in Europe during the Middle Ages, is a valid form, meaning that any argument—about any subject—that exactly follows this pattern is valid.

The Barbara Argument Form

1. All B are C.
2. All A are B.
3. Therefore, necessarily, all A are C.

Let’s test Barbara. If we replace the variable A with sparrows, the variable B with birds, and substitute animals for the variable C, we get the following “substitution instance” of the corresponding form:

1. P1. All birds are animals.
2. P2. All sparrows are birds.
3. C. Therefore, necessarily, all sparrows are animals.

This argument is clearly valid. Aristotle proved that any argument that exactly follows this form of reasoning is valid. For instance:

1. P1. All mammals are animals.
2. P2. All cats are mammals.
3. C. Therefore, necessarily, all cats are animals.

To return to Barbara for a moment, notice that the form is not about any particular subject—it is an abstract pattern with no material content. Barbara is all form and no content. Aristotle discovered that an argument’s validity is always a function of its form rather than its content. You can learn a lot about reasoning by studying valid argument forms. Logicians have catalogued hundreds of them. The study of logical forms is valuable, for if your argument follows a valid form, then it is guaranteed to be valid and therefore your conclusion must be true if your premises are true. As you may have guessed, formal logic and deductive logic overlap in the study of valid patterns of reasoning, of which there are many.

An inductive argument, on the other hand, does not aim to show that its conclusion is certain. Rather it aims to show that its conclusion is probably, though not definitely, true so that if its premises are true, it is likely that its conclusion is true. This argument aims to establish its conclusion with a probability less than one:

1. P1. Joe has eaten a Dick’s Deluxe burger for lunch every day for the past month.
2. C. So, it is very probable that he will have a Dick’s Deluxe for lunch tomorrow.

If an inductive argument achieves its aim, it is a strong argument. An inductive argument that does not achieve its aim is a weak argument. An inductive argument is said to be cogent if it is (a) strong, and (b) all of its premises are true. The following inductive argument is strong although it is surely not cogent:

1. P1. We interviewed one thousand people from all walks of life and every social group all over Seattle over a ten-week period, and 90 percent said they do not drink coffee.
2. C. Therefore, probably about 90 percent of Seattleites do not drink coffee.

The following argument is clearly weak:

1. P1. We interviewed one thousand people from all walks of life as they exited coffee shops in Seattle, and 98 percent said they drink coffee.
2. C. Therefore, probably about 98 percent of Seattleites drink coffee.

The following argument is better—it is strong as well as cogent:

1. P1. NASA announced that it found evidence of water on Mars.
2. P2. NASA is a scientifically reliable agency.
3. C. Therefore it is likely there is or was water on Mars.

Inductive logic is the study of the standards of good inductive reasoning. One inductive standard pertains to analogical arguments—arguments that take the following form:

1. A and B have many features in common.
2. A has attribute x and B is not known not to have attribute x.
3. Therefore, B probably has attribute x as well.

For instance:

1. P1. Monkey hearts are very similar to human hearts.
2. P2. Drug X cures heart disease in monkeys.
3. P3. Drug x is not known to not cure heart disease in humans.
4. C.Therefore, drug X will probably cure heart disease in humans.

Analogical arguments can be evaluated rationally. Here are three principles commonly used to judge their strength:

• The more attributes A and B have in common, the stronger the argument, provided the common features are relevant to the conclusion.
• The more differences there are between A and B, the weaker the argument, provided the differences are relevant to the conclusion.
• The more specific or narrowly drawn the conclusion, the weaker the argument. The more general or widely drawn the conclusion, the stronger the argument.

Informal and inductive logic overlap in the study of the many non-formal aspects of inductive reasoning, which include guides to help us improve our assessments of probability.

Information Spillover

The history of ideas is fascinating because often one idea leads to another which leads to a completely unexpected discovery. Economists call this “information spillover” because freely traded ideas tend to give birth to new ideas that give birth to still more ideas that spill from mind to mind as the process cascades into ever widening circles of knowledge and understanding. Aristotle discovered logical principles so exact they could be expressed in symbols like those used in mathematics. Because they could be expressed so precisely, he was able to develop a system of logic similar to geometry. Recall that geometry begins with statements, called “axioms,” asserted as self-evident. With the addition of precise definitions, the geometer uses precise reasoning to derive further statements, called “theorems.” Aristotle’s system began in a similar way, with precise definitions and exact formulas asserted as self-evident. With the base established, he derived a multitude of theorems that branched out in many directions. When he was finished, his system of logical principles was as exact, and proven, as any system of mathematics of the day.

Some observers thought the rules of his system were too mechanical and abstract to be of any practical use. They were mistaken. Aristotle’s system of logic was actually the first step on the path to the digital computer. The first person to design a computing machine was a logician who, after reflecting on the exact and mechanical nature of Aristotle’s system of logical principles, raised one of the most seminal questions ever: Is it possible to design a machine whose gears, by obeying the “laws” of Aristotle’s logic, compute for us the exact, logically correct answer every time?

The logician who first asked the question that connected logic and computing was Raymond Lull (1232–1315), a philosopher, Aristotelian logician, and Catholic priest. Lull has been called the “father of the computer” because he was the first to conceive and design a logical computing machine. Lull’s device consisted of rotating cogwheels inscribed with logical symbols from Aristotle’s system, aligned to move in accord with the rules of logic. In theory, the operator would enter the premises of an argument by setting the dials, and the machine’s gears would then accurately crank out the logically correct conclusion.

Lull’s design may have been primitive, but for the first time in history someone had the idea of a machine that takes inputs, processes them mechanically on the basis of exact rules of logic, and outputs a logically correct answer. We usually associate computing with mathematics, but the first design for a computer was based not on math but on logic—the logic of Aristotle.

Ideas have consequences, and sometimes ideas that seem impractical have consequences that are quite practical. Lull was the first in a long succession of logical tinkerers, each seeking to design a more powerful computing machine. You have a cell phone in your hand right now thanks to the efforts of these innovators, each trained in logical theory. In addition to Lull, the list includes computer pioneers Leonardo da Vinci (1452–1519), Wilhelm Schickard (1592–1635), William Oughtred (1574–1660), Blaise Pascal (1623–1662), Gottfried Leibniz (1646–1716), Charles Babbage (1791–1871), Vannevar Bush (1890–1974), Howard Aiken (1900–1973), and Alan Turing (1912–1954).

Thus, a continuous line of thought can be traced from Aristotle’s logical treatises to the amazing advances in logic and computing theory of the nineteenth and twentieth centuries which led to the completion of the world’s first digital computer (at Iowa State College in 1937) and from there to the much smaller yet more powerful devices of today. It is no coincidence that the circuits inside every digital computer are called “logic gates.” In the logic classroom, this is my answer to those who suppose that abstract logical theory has no practical applications.

Computer science is only one spin-off of logical theory. The subject Aristotle founded remains as vital today as it was in ancient Athens. Aristotle probably had no idea how important his new subject would be—or how long the spillover and information overflow would continue.

What does all of this have to do with anything? In everyday life as well as in every academic subject, reason is our common currency. It follows that the ability to reason well is an essential life skill. But skills require knowledge as well as practice. Since logic is the study of the principles of correct reasoning, a familiarity with elementary logic and its applications can help anyone improve his or her life. Some people suppose logic is a useless subject; the truth may be the reverse—it may be the most useful subject of all.

[i] An editor applied the name Organon (“tool”) to Aristotle’s logical works after his death. The name reflects Aristotle’s claim that logic is an all-purpose tool of thought, a guide to the precise thinking needed to attain solidly proven truth on any subject.

[ii] Benson Mates, Elementary Logic, 2nd ed. (New York: Oxford University Press, 1972), 206. Ex nihilo is Latin for “out of nothing” and means “from scratch” in this context.

For a deeper look at the fundamentals of this subject, check out the free course “Short Little Lessons in Logic” published by Philosophy News. This course will teach you the fundamentals of logic in bite-sized lessons that you can learn at your own pace.

About the author

Paul Herrick received his Ph.D in philosophy from the University of Washington. Since 1983 he has taught philosophy at Shoreline Community College, in Shoreline, Washington, near Seattle. He is the author of Reason and Worldview. An Introduction to Western Philosophy , Think with Socrates: An Introduction to Critical Thinking, The Many Worlds of Logic, and Introduction to Logic .

Other articles by Paul Herrick

Who is Socrates ? In this article in the “What is” series written for Philosophy News, Paul Herrick describes Socrates both as a thinker and as a model. One of the three major early Greek thinkers, Socrates not only lived what he believed, he died for the principle that by thinking critically we can create a life worth living.

Books by Paul Herrick

Introduction to Logic , Oxford University Press, 2012

This is a comprehensive introduction to the fundamentals of logic (both formal logic and critical reasoning), with exceptionally clear yet conversational explanations and a multitude of engaging examples and exercises. Herrick’s examples are on-point and fun, often bringing in real-life situations and popular culture. And more so than other logic textbooks, Introduction to Logic brings in the history of philosophy and logic through interesting boxes/sidebars and discussions, showing logic’s relation to philosophy.

Think with Socrates: An Introduction to Critical Thinking, Oxford University Press, 2014

Brief yet comprehensive, Think with Socrates: An Introduction to Critical Thinking uses the methods, ideas, and life of Socrates as a model for critical thinking. It offers a more philosophical, historical, and accessible introduction than longer textbooks while still addressing all of the key topics in logic and argumentation. Applying critical thinking to the Internet, mass media, advertising, personal experience, expert authority, the evaluation of sources, writing argumentative essays, and forming a worldview, Think with Socrates resonates with today’s students and teaches them how to apply critical thinking in the real world. At the same time, it covers the ancient intellectual roots and history of the field, placing critical thinking in its larger context to help students appreciate its perennial value.

Reason and Worldview , Harcourt College Publishers, 1999

A comprehensive look at major movements in philosophy and how those movements helped shape the way we think and behave.