When you think about anything, try as an exercise to list all of your assumptions. You will soon discover that so many are tacitly posited and strongly believed in, perhaps without ever occurring to your mind. One such assumption is that the external world exists: that is, the things that you see, hear, and feel are really out there, and you’re not simply imagining them. Another important assumption is one regarding causality: That everything that happens, had a cause — that effects were caused by forces preceding them in time. A puzzling, third assumption is that of constancy: That the rules of the universe don’t change all of a sudden. These three assumptions, and so many more, outline how we think. They are pleasant cognitive biases because without them, it is hard to think coherently at all. You may be surprised to know that these three assumptions are not necessary. We can imagine worlds without them, even though our world is difficult to navigate without them. There are, however, three assumptions that make it possible to think. Without these assumptions, not only would coherent thought be difficult, but we would lose the capability of even the simplest of thoughts in any possible world. These three assumptions are called the three axioms of logic. Before proceeding, let us clarify the two terms just mentioned: Axioms and logic.

Axioms.

An axiom is a special type of assumption that is simple enough to act as a definition or a rule and robust enough to become the basis of many (indeed, all) ideas in a field. Any assumption in a field besides the axiom is either (i) a definition, (ii) a theorem, (iii) a fact, (iv) a conjecture, or (v) a false belief. A definition is a set of relations between words that can be used as a tool to probe other thoughts. Definitions do not tell us anything new about the world; they only inform us how words will be used, going forward. A theorem is a necessary conclusion that follows from a definition. For example, if we define a bachelor to be an unmarried man, then a theorem would look like this: “No woman is married to a bachelor.” This, we know, from the symmetric meaning of ‘married.’ There are two types of facts. Empirical facts are those that are gained through the senses. All scientific facts are empirical facts, but not vice versa. Theorems and definitions are analytical facts. A discussion of facts can be very complicated, and so this is left for a later discussion; sufficient for the time being is the understanding that we call the set of supported beliefs, however verified, facts. Finally, a conjecture is an assumption that is yet to be verified: It might be true or false. If it is verified, it becomes a fact; if proved to be false, a false belief; if neither, it remains a conjecture.

Axioms can be definitions, but they cannot be theorems, facts, conjectures, or false beliefs. For if an axiom (X) is a fact, then it has been verified; furthermore, it has been verified based on simpler concepts (Y1) or assumptions (Y2) — and these (Y1 and Y2), then, would gain the status of axioms, making what we previously thought to be an axiom (X) a theorem or a fact. Axioms cannot be theorems by definition, because a theorem is not the starting block (it follows from axioms and definitions), whereas an axiom is. Finally, to view conjectures or false beliefs as axiomatic is to defeat the whole purpose of formalizing a science.

A proper way to view an axiom is to notice that the negation of an axiom always leads to a contradiction. The simplest assumptions that cannot be negated without resulting in contradiction are axioms of a formal science. Often, by analysis, we may discover that these assumptions can be further simplified and analyzed. Scientists often do that, but they now deem this analysis to be an engagement in a new, different science. Given any point, we have to draw clear lines: Above this point is the science of interest, A, and below it can be one, B, or many sciences (C, D, E, etc.) that are formally distinct from A. It is futile to try to prove an axiom, because it is the simplest idea in a science: Any attempt to prove it, through another idea, simply shifts the status of ‘axiom’ to that new idea. Mathematicians tried for 21 centuries in vain to prove Euclid’s fifth axiom for plane geometry (the parallel postulate), which says that two straight lines that are not parallel would intersect, and the angle of intersection would be less than 180° on that side. Euclidean geometry takes the parallel postulate as an axiom, whereas non-Euclidean geometries do not.

A final note, mostly interesting to mathematicians and logicians, is that there will always be statements or beliefs that cannot be proven using the current axioms. Either an axiom (or a set of axioms) needs to be changed, or some new axioms need to be added. If we remove an axiom, that frees up the science, allowing for more results to be derived, but the trade-off is a loss of order and structure. By the same token, adding an axiom can make the science more rigid, as all results must adhere to more rules. For any choice of axioms, there will always be some statements that are out of reach and that new sciences may tackle. The choosing of axioms is both an art and a science: The ‘art’ comes from the human expertise in that specific field, and the science is the science of logic.

Logic.

As for logic, it is the science of correct, ordered thought. Logic is a science in that it has a proper object of study: Arguments; a list of axioms: Identity, Contradiction, and the Excluded Middle; and a sociology of scientists and practitioners: Logicians, mathematicians, philosophers.

There are two main branches to logic: Formal Logic and Informal Logic. Under formal logic, we have syllogistic logic, which studies the forms of simple arguments (called syllogisms), symbolic logic, which is often employed in the foundations of mathematics, and modal logic, which studies necessity and possibility. Many other subdisciplines exist in formal logic; however, it is best to concentrate on the simplest and most straightforward. Let us consider a simple example from syllogistic logic, the branch founded by Aristotle in his Prior Analytics (a book now extant). A syllogism might look like this:

• All apples are fruits;
• All Granny Smiths are apples;
• Therefore, all Granny Smiths are fruits.

This syllogism bears the symbol (AAA-1), and it is unconditionally true. ‘A’ here means that the proposition is universal (all) and has the positive copula (is/are), and ‘1’ means that the order of the terms appears in this order: M, P; S, M. Another unconditionally true syllogism is (EIO-4):

• All banks are not fruits [All P is not M];
• Some fruits are green [Some M is S];
• Therefore, some green things are not banks [Some S is not P].

Now, a conditionally valid syllogism is (AAI-1). This is identical to our first syllogism with one minor difference:

• All apples are fruits [All M is P];
• All Granny Smiths are apples [All S is M];
• Therefore, some Granny Smiths are fruits [Some S is P].

The universal quantifier (all) is replaced with the existential quantifier (some). This is a conditionally valid statement, since what is missing is ‘existential import.’ ‘All’ and ‘No’ do not require the existence of the term immediately following it, but ‘Some’ does, by definition, and it means that at least one. In the previous syllogism, S (that is, Granny Smiths) has to exist. If it does not, we have committed a formal fallacy called the Existential Fallacy.

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Informal logic combines some of the tools of formal logic with analytic tools and human wisdom to consider the analysis of ideas in a less formal (i.e., less rigorous and arranged) setting. Whereas formal fallacies might consider syllogisms that are invalid or unsound, an informal fallacy can consider a latent error that requires a certain understanding of how humans think or reason. This is not something that we can know using the axioms of logic alone, but requires human intuition and understanding. Informal logic is concerned with (informal) fallacies, critical thinking, persuasion, rhetoric, and so on. The boundary of informal logic is not as clear-cut as that of formal logic, as it contains objects of study that might easily lie in other fields.

Let us consider an example. An informal fallacy, such as ad hominem (attacking the person defending the idea, instead of the idea itself), does not have a specific form. Rejecting someone’s argument because you consider him stupid or ugly, or of a different religious sect, does not help in knowing whether the argument is true or not. Notice that ad hominem does not necessarily produce an error. For suppose that I doubt a claim because who uttered it has lied to me before (as the legal dictum falsus in uno, falsus in omnibus codifies) or because he does not have a degree in the field he is discussing. We have the intuition that an argument presented by someone who is not an expert in a field is less compelling, so ad hominem often makes sense; but it is, in general, not a useful attitude to attack the person since this mode of thinking is heavily prone to error, and it does not address head-on what is wrong with the argument. Even though the expertise of the person is ultimately irrelevant to whether the claim is true or not, the claim is more probable, and we have more grounds to believe it, if it were presented by an expert. It is nevertheless classified as an informal fallacy because it is not conducive to a proper discussion aiming at finding the truth.

Another interesting principle of informal logic is that of charity. The principle of charity tells us that it is better to do as much as possible to present the opponent’s argument in the best light possible. In modern parlance, this principle is called ‘steelmanning the argument,’ as contrasted by the strawman fallacy. Common variants of the principle of charity are the Rogerian Argument (that one must rephrase the opponent’s argument to the satisfaction of the opponent), or Rapaport’s Rule of Ethical Debate (that one must outline all the strengths of the opponent’s arguments and mention the favorable reasons why one may adopt them). These variants are not as strong or as fundamental as the pure principle of charity. There is no logical reason why one ought to follow this principle: It does not necessarily lead to correct arguments. However, it increases our chances of understanding our opponent’s arguments and seeing if it is indeed true. Moreover, the principle of charity helps us eliminate other cognitive biases in that it increases the friendliness between opponents and fosters a healthier discussion. The pure principle’s ultimate concern is not to make us adopt our opponent’s argument, but to ensure that we have responded to the best form their argument can take in order to advance ours. It is a strategy, in the end, and herein lies the ultimate difference between formal and informal logic: Formal logic seeks arguments whose conclusions are necessarily true, whereas informal logic seeks strategies to form more probable (or more convincing) arguments.

The Three Axioms of Logic.

Now, whether someone is working under formal or informal logic, there are three fundamental axioms that should always be observed. These axioms consider an abstract object, call it A. Let us also have an abstract variable and call it X. X can be anything, and is not specified as it stands into any object. But once A is chosen, it stays that thing. Logicians always seek abstractness, because solving an abstract problem produces a family of solutions that applies to as many particular problems as possible in many fields. ‘X’ might stand for a building, a concept, a number, a human, the relation of being in between a McDonald’s and a Starbucks, God, or what have you. Now that we have presented our abstract variable, X, we may now discuss the three axioms:

1. The axiom of identity: For an X that is A, X is A;
2. The axiom of contradiction: No X is both A and not A;
3. The axiom of the excluded middle: Any X is either A or not A.

The first axiom is obvious: Given anything (X), X is itself. The axiom of identity simply says that a thing is identical to itself. This form by itself is not very useful now except as a way of defining variables and giving them values. However, we can extract several tools from this axiom.

Take, for example, the equivalence class. For a set of objects S = {S₁, S₂, …, S_N}, one relation (let us call it ~) between the elements is an equivalence if it satisfies these conditions: If A, B, and C are elements in S, then:

1. A ~ A [Reflexivity];
2. If A ~ B, then A ~ B [Symmetry];
3. If A ~ B, and B ~ C, then A ~ C [Transitivity].

Any elements that satisfy these three conditions are said to be equivalent under the relation ‘~.’ Note that equivalence is like a watered-down form of identity. Let us look at these three examples: (i) Equality among numbers, (ii) having the same birthday for people, and (iii) ‘liking each other’ for friends.

If x, y, and z are numbers, then:

1. x = x;
2. x = y ⇒ y = x;
3. x = y and y = z ⇒ x = z.

This means that equality is an equivalence relation, and the numbers are an equivalent class (set). Now, for birthdays, consider three people, Adam, Kowther, and Ahmad:

1. Adam has the same birthday as himself;
2. If Adam has the same birthday as Kowther, then Kowther has the same birthday as Adam; and,
3. If Adam has the same birthday as Kowther, and Kowther has the same birthday as Ahmad, then Adam has the same birthday as Ahmad.

Adam, Kowther, and Ahmad are equivalent only insofar as we are concerned with their birthdays. The equivalent relation is having the same birthday, and only in that are they equivalent. Most other things may be different between them. Finally, let’s take the class to be friends, and the relation to be ‘likes’:

1. Adam likes Adam (not a necessary state of affairs);
2. If Adam likes Kowther, Kowther likes Adam (not a necessary state of affairs);
3. If Adam likes Kowther, and Kowther likes Ahmad, then Adam likes Ahmad (not a necessary state of affairs).

This fails all three conditions, and is therefore not an equivalence relation, and Adam, Kowther, and Ahmad don’t form an equivalence class.

Two other useful tools that can be extracted from the law of identity are ‘the identity of the indiscernibles’ and ‘the indiscernibility of identities.’ The first states that if two objects, X and Y, are identical, then they share all properties (i.e., it is impossible to discern between them). The latter says that if two objects, X and Y, have the same properties, then they are identical. Two things are identical when there is no difference whatsoever between them.

Finally, the axiom of identity can be used to define existence. Bertrand Russell, G. E. Moore, and Ludwig Wittgenstein all define existence as being identical to an object in the world. So, X exists if it is identical to an object in the world, Y. This definition of existence is wanting, since Y is already thought to exist (so, what was it identical to? And so on.).

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A contradicting set of beliefs leads to the clearest form of cognitive dissonance. Contradictions are to be eliminated from our arguments, lest they lead us to bizarre and shaky results. In any case, they make it impossible to think coherently, as any argument may lead to many different results. For when the rain falls, and the floods come, and the winds blow and beat against our arguments, our arguments will fall, and great will be their fall. [Matthew 7:27]

The axiom of contradiction is very useful for proofs. In fact, one of the most used proofs in mathematics is the proof by contradiction. In it, we assume that X is A, and follow along with inferences until we reach an error, thus demonstrating that X is also not A, which leads to a contradiction, showing therefore that at least one of our assumptions is wrong.

Not all systems of thought adopt the law of noncontradiction as Aristotle proposed it. Two intellectual groups at least come to mind when thinking about the nontraditional adoption of the classical concept of contradiction: Some Eastern philosophies, such as Taoism, and some German philosophies, such as Hegelian and Marxist philosophers. However, both groups cannot defy the axiom of noncontradiction; what they call contradiction is simply an opposite.

Let us look at the Greek Presocratic philosopher, Heraclitus. One of his famous quotes is: “The way up and the way down is one and the same.” Even though up is the opposite of down, up and down are not pure contradictory concepts: ‘Up’ and ‘Not Up’ are. And even so, there’s no reason why the way up and the way down have to be opposites: Imagine a normal staircase: The same staircase is a way up for those down, and a way down for those up, while being the same staircase. Of course, this is a shallow understanding of the Heraclitus excerpt used for illustration.

The axiom of noncontradiction offers us a standard reason for rejecting claims. Once we show that a claim contradicts itself, we need not elaborate anymore on why we reject it, other than to say that it is contradictory. If it contradicts a fact, we may also reject it based on that alone (or maybe reconsider what we thought to be a fact, for it may be a conjecture, or a false belief).

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The final axiom is that of the excluded middle. This axiom is used extensively in categorizing objects; however, the more that is known about an object, the easier our task will be. Let us use it now to classify an egg. An egg is either (pick another object… A book? Alright.) a book or not a book. Books are made of paper, but an egg is not. Therefore, an egg is not a book. This can only get us so far. It is apparent, then, that it is better to use the law of the excluded middle to see if an object belongs to a large group or not. Either an egg is a living entity or it is not. This leads us to discuss what a living entity is, and to see whether an egg fits the definition. Indeed, this axiom is also used in definitions just as we have considered eggs and living entities.

There is a problem, however, with this axiom: Consider the Barber’s Paradox which states that a certain barber cuts the hair of those who do not cut their own hair. He either cuts his own hair, or he does not. If he does, then it is simply not true (a contradiction) that he only cuts the hair of those who do not cut their own hair. If he does not, then he is a prospective client to himself, and will cut his hair — another contradiction. A logical get-around is to acknowledge that no such barber may exist.

The law of the excluded middle is also used in mathematical proofs. Consider this elegant proof that an irrational exponent of an irrational number can lead to a rational number. A rational number can be represented as the ratio between two integers (1/2 = 0.5 or 2/1 = 2 are both examples), whereas an irrational number cannot. Some characteristic rational numbers are √2, π, and e. Now, for the proof: Let x = √2^√2. Either x is rational or it is irrational. If it is rational, then our proof is complete. If it is irrational, then consider x^√2 = (√2^√2)^√2 = √2^{(√2 * √2)} = √2² = 2, which is rational. Thus concludes the proof.

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I hope to have shown you how potent the three axioms of logic are. They can be used for any argument, and in the next article, we will start employing them to study a new topic in logic.