An exceptional fact about the ancient Greeks is that they were the first group of thinkers to move beyond the particulars of their own culture to search for universal truths that transcend time and place. This is why they remain perennially relevant and valuable to us today.
The search for universal truth began with the Presocratics and their quest to discover the fundamental, unchanging nature of reality. Socrates would then carry on this tradition in the realm of ethics, where he sought to pin down universal definitions of moral virtues, while Plato, seeking to cover all bases, developed his theory of universal Forms.
But it was Aristotle, Plato’s star pupil, who was the first to turn the mind in on itself to discover the universal principles of reasoning. Unlike the Sophists, who focused on rhetorical tricks of memory and persuasion, Aristotle would define the universal rules by which every argument is constructed. In doing so, he unlocked the potential for the development of general intelligence—the ability to form valid and insightful conclusions regardless of the subject matter.
Foundations: the three laws of thought
Aristotle’s discovery of the three laws of thought may seem self-evident, but that’s exactly the point. There must be a set of propositions that cannot be false for there to be any possibility of acquiring knowledge at all. Without a foundation of universal consent, the possibility of discovery and communication is impossible. What Aristotle did was establish this foundational set of propositions, which represents the starting point for all reasoning. The three laws of thought are as follows:
- The law of identity states that each thing is identical with itself. It takes the form “A is A.” To say, for example, that a tree is not a tree is entirely nonsensical.
- The law of noncontradiction states that two contradictory propositions cannot be true in the same sense at the same time. For example, it cannot be true that both “A is B” and “A is not B.”
- The law of the excluded middle states that for any proposition, either that proposition is true or its negation is true. For example, either “A is B” or “A is not B.”
This shows, among other things, that reason is non-negotiable, if we are to have any possibility of knowledge and discourse at all (and that if you argue against the use of reason, you are using the very thing that you are arguing against). It also demonstrates that reasoning has a universal structure that is independent of specific content, and that we can capture the formal structure of any argument to analyze whether or not the reasoning is valid. This will help us to not only build better arguments ourselves, but to also catch and identify the fallacious reasoning in the deceptive arguments of others. This is, in essence, the foundation of general intelligence.
All three laws of thought essentially roll up into the principle of noncontradiction. You cannot maintain that something is and is not at the same time without losing your grip on reality. And from this basic idea we can build a system of abstract logic that can show each of us how to reason more effectively—and can reveal where our own beliefs are contradictory.
Aristotle’s four categorical propositions
A proposition is a sentence that expresses a statement of fact. Building on the three laws of thought, Aristotle maintained that all propositions must either confirm or deny something, and that this affirmation or denial can apply to either all members of a group or to only some members of the group. This creates four possibilities, or four categorical propositions:
- Universal Affirmation: All A are B
- Universal Negation: No A are B
- Particular Affirmation: Some A are B
- Particular Negation: Some A are not B
From these four propositions, we can derive additional truths. For example, if the proposition “all A are B” is true, that means that “some A are B” is also true, and that the propositions “no A are B” and “some A are not B” must be false.
We can use this in actual arguments by substituting content for the notation. If we say that all humans are mortal (All A are B), but that some humans are immortal (Some A are not B), we’ve contradicted ourselves by combining a universal affirmation with a particular negation.
The idea that we can translate our arguments into a structural form and analyze them is the beginning of logic and the path toward greater mastery of our minds.
Deductive and inductive reasoning
The four categorical propositions were derived from the three laws of thought. Likewise, the four categorical propositions can be combined in various ways to form arguments. In an argument, a series of propositions called premises are offered in support of the truth of another proposition called the conclusion.
Aristotle noticed that all arguments can be divided into two complementary types: inductive and deductive. Inductive reasoning is the derivation of general principles from specific observations. An example of induction would be deriving the general principle that all humans are mortal from observing the specific instances of human beings growing old and dying. The problem with induction, however, is that it is never completely certain: it could always be the case that we have simply yet to meet a human that does not or cannot die, thereby refuting the general claim that all humans are mortal. But until that happens, we can assert with a high degree of confidence that all humans are mortal. Inductive reasoning, therefore, while never certain, is more or less probable based on the number and quality of the specific observations.
Deductive reasoning, in contrast, is the derivation of logically certain conclusions from one or more premises assumed to be true. With deductive reasoning, if the premises are true, the conclusion must be true. In practice, however, deduction is far from certain, because the premises are always the result of induction, which itself always has some degree of uncertainty.
Notice how the following deductive argument uses the inductive premise we established earlier:
Premise 1: All humans are mortal
Premise 2: Socrates is a human
Conclusion: Therefore, Socrates is mortal
Notice that premise 1 and 2 are inductive inferences used to derive the deductive conclusion that Socrates is mortal. Premise 1 is a highly probable but still uncertain inductive inference, and Premise 2, which states that Socrates is a human, is also an uncertain inductive inference based on the interpretation of historical information. Socrates may, in fact, be a fictional character, in a sense making Socrates immortal so long as he is remembered.
This insight into the connection between induction and deduction is important because it suggests two things:
- A deductive argument first has to be logically valid before it can be accepted as true. It must take the form that if the premises are true, the conclusion must be true.
- However, even if the argument is deductively valid, the conclusion may be false if the premises—which were arrived at via induction—are false.
This means that there are two primary ways in which a conclusion can be false or at least improbable. The argument can be either deductively invalid or else the premises can be false or improbable based on poor inductive reasoning. The careful thinker evaluates both aspects of an argument.
For example, here is a deductively invalid argument that contradicts itself by violating the law of non-contradiction:
Premise 1: All humans are mortal
Premise 2: Socrates is a human
Conclusion: Therefore, Socrates is immortal
The argument derives a particular negation from a universal affirmation, making the conclusion invalid.
But it’s important to remember that one can be both logical and wrong at the same time. For example:
Premise 1: All humans have wings
Premise 2: Socrates is a human
Conclusion: Therefore, Socrates has wings
This argument is deductively valid, but its conclusion is false. It is based on a faulty inductive inference that claims all humans have wings. Obviously, the premises in most arguments are not this blatantly false, but the careful reasoner must learn to scrutinize all premises that claim universality. Also, notice how the problem of induction has crept into deduction, creating an underlying level of uncertainty in all arguments.
Aristotle—by equipping us with the proper concepts and vocabulary—gave us the tools to see through fallacious or questionable reasoning and to construct better arguments ourselves. (But we must discover this on our own because, amazingly, these skills and concepts are not taught in public schools.)
Our path to better reasoning begins with our consent to the laws of thought and the principle of noncontradiction. Then, by paying better attention to the structure of arguments, we can evaluate the inductive inferences—which become the premises used in deductive reasoning—in addition to the evaluation of deductive validity itself. In the process, we will come to realize that most of the problems associated with weak arguments result from faulty inductive inferences based on small or non-representative samples, or else result from hidden premises that oversimplify the issues.
Here’s a simple example:
Premise: I’ve met three people from Vermont and they were all socialists
Conclusion: everyone from Vermont is a socialist
Obviously, the three people I met from Vermont are not necessarily a representative sample of everyone from Vermont. It could be the case that the three people I met were the only socialists in Vermont, and that everyone else is conservative. The point is, you must always question the size and representativeness of any sample used in support of a general or universal conclusion (this is the error underlying many racist views).
Here’s another example:
Premise 1: when it rains, the sidewalk gets wet
Premise 2: the sidewalk is wet
Conclusion: it recently rained
This argument appears to be deductively valid, but the conclusion is not necessarily true. You must always question the premises and the connection between the premises and conclusion. Just because A can cause B doesn’t mean that it did cause B. While rain is one cause of a wet sidewalk, so is a garden house, for instance. We must investigate the situation more carefully before jumping to a conclusion.
Sometimes the problem with an argument lies in its hidden assumptions. Consider the following argument:
Premise 1: taxes decrease the amount of money available to corporations
Premise 2: corporations require money to invest, grow, and create jobs
Conclusion: we should therefore cut corporate taxes to stimulate growth and create jobs
The above argument has apparent deductive validity, which makes the argument superficially sound and likely persuasive to many people. But a careful consideration of the premises—and particularly the unstated premises—reveals the weakness of the argument.
Here’s one hidden premise:
Premise 3: corporations can only grow through tax cuts.
While it’s true that taxes decrease available corporate funds, and that lower taxes increase funds, it is not the case that this is the only way to increase corporate income or to stimulate the economy. Taxes used to redistribute money into the hands of consumers—or to free up disposable income via the public provision of critical services such as healthcare—can increase the amount of money consumers have to spend and can stimulate the economy from the bottom up.
And so it is not a given that lower corporate taxes are always good for the economy. It could be the case that higher corporate taxes, redistributed as cash or services for the middle class, creates higher consumer demand and a happier, more productive and healthier workforce.
Also notice that the conclusion may largely be irrelevant depending on the context of the discussion. It could be the case that economic growth is not or should not be the main priority, particularly if only a small percentage of the population benefits from that growth.
Here’s another hidden premise:
Premise 4: corporations, by growing and creating jobs, can provide all of the products and services we value.
Even if it is the case that corporate tax cuts stimulate growth and create jobs (although this is not a foregone conclusion), it does not follow that we should prioritize corporate growth. If, for example, corporations cannot provide healthcare to all members of society (in part by providing only part-time or freelance jobs), then perhaps we should raise taxes to fund this valuable public provision.
The bottom line is that arguments that appear deductively valid and superficially persuasive almost always have additional unstated premises that oversimplify the matter at hand. The skillful reasoner, equipped with the concepts and vocabulary originally developed by Aristotle, will always challenge these assumptions and, like Socrates, will always find exceptions to simple, universal rules.