The Foundation of Logic

There are a lot of different ways to approach understanding what mathematics truly is in modern times. And these different approaches are fundamentally different. Does one come from the angle of discussing the beauty of patterns? Or perhaps of the power of the human mind to move from specific examples to general conclusions (similar to science)? Or maybe the historical development of mathematics and how different cultures used it? Or an exposition of the different kinds of things that mathematicians study? All of these I find quite reasonable approaches, and I have and will adopt each of these when I write and speak on my beloved subject, and I’d love to write on each of them.

But for now, I wish to take a different angle. Instead, I wish to emphasize the patterns of thought and of writing that mathematicians most heavily rely on; I think adjectives like “rigorous” and “logical” are probably accurate. What I hope I can do here is to develop a series of posts which brings a clearer picture of what mathematics is about through discussing how mathematics is communicated. I have tried to bring in as few assumptions as possible o make this accessible, and I hope that the approach I take here will make certain “abstract” ideas easier to process by first explaining the framework out of which the abstract ideas arise.

Without further ado, then, we must begin by understanding the absolute, rock-bottom concept that underlies all of mathematics. Even more deeply that the idea of number itself, mathematics has as its foundation one word – truth. It may seem an odd question, but we must begin by asking what truth actually is. What does it mean for something to be true or false? This is surprisingly difficult to answer when you have to be precise, but we need not go into philosophical debates here. For our purposes, it is enough to say that a statement is true if, and only if, the information conveyed by that statement corresponds to the way reality actually is. A false statement has precisely the opposite definition. Truth is a fundamental concept, and is vital in every human endeavor. Even in the arts, truth is at the center. Take music, for instance. We know that certain combinations of musical notes sound pleasant, and others do not. This pleasantness I taught as not merely experiential, but as something true. Reality in fact is a certain way, and we tell the truth when we accurately reflect that reality in our speech.

The next step towards mathematics is the idea of an argument, the basis of the study which has the ‘fancy’ name of propositional calculus. The term ‘calculus’ really just denotes some form of calculation, and the adjective ‘propositional’ means ‘dealing with truth and falsehood.’ And when we say ‘argument’, we don’t mean angry bickering back and forth. Rather, any train of thought that is attempting to prove a point of some kind counts as an argument. So, the ‘propositional calculus’ is really nothing more than understanding the truth or falsehood of various combinations of true or false statements. For simplicity, I will give all of this the umbrella term of logic.

What then, exactly, does logic encompass? As we are beginning at the very foundations of logic, our goal is to discover what kinds of statements are undeniably true given other kinds of true statements. We aim to know in what manner we can use truths we already know to learn more truths. This study has been undertaken in depth for thousands of years, as long as intellectual discussions have been around. To begin this study, we use as an example probably the most common form of argument ever used, and one that we all know and use in daily life and thought.

Suppose you have invited a friend over to your house. You also know it happens to be raining outside today. You are in another room, and you hear your friend enter your front door and exclaim “I forgot my umbrella and raincoat!” If I were in this situation, I would immediately start thinking “Oh, my friend is wet, let me go get him a towel.” If we slow down for a bit, isn’t that strange? Your friend never actually told you they are wet. The reason we all assume he is wet is because we have an internal understanding that, if it is raining outside, then a person outside without rain gear will get wet. We combine this truth we already know with what we heard our friend tell us, and conclude that our friend must be wet.

The previous paragraph counts as an argument. We can summarize the general form of this argument by using the italicized letters P and Q to be placeholders for generic statements. The logical argument of the following paragraph has two premises, that is, two initial facts available to us. We know that “If P, then Q” and “P“, where P represents “it is raining outside” and Q represents “you will get wet without rain gear.” From these two facts, we understand that Q is also true. In the form of a list,

  1. If P, then Q,
  2. P,
  3. Therefore, Q.

This is the starting point of logic – the idea of fusing together certain kinds of truth (like “If P, then Q” and “P“) to obtain another truth (“Q“). The question we now ask is, which basic building blocks can be combine to form other, related true statements? Not just any two truths will do, we must be absolutely careful in thinking about how our statements match up with reality. We are to be as careful as possible, because our goal is that whatever we say, it must be absolutely impossible that we are incorrect. This is the case with out first example: for if we know that “If P, then Q” and “P” are true statements, then then the very meanings of these statements inform us that “Q” is undeniably true.

For those of you that are interested in such things, the technical name for the argument form we have just discussed is modus ponens, Latin for “mode that by affirming affirm.”. It is called this because, by affirming “P“, we end up affirming “Q“, so one affirmation leads us to another affirmation. The next natural argument form to bring up is given the Latin name modus tollens, “mode that by denying denies.” This argument enables us to, by denying one statement, conclude that we can deny another statement. Let us reconsider your visiting friend. For simplicity, I think it is fair to say that “if it is raining, then the ground is wet.” Suppose your friend, upon entering the house, tells you that the ground is dry. In your head, you will recognize then that it must not be raining, for if it were raining, the ground would be wet and not dry. Using the letters P and Q for shorthand once again, we have the following presentation of our new argument form:

  1. If P, then Q.
  2. Not Q.
  3. Therefore, not P.

If you take the time to think about it, this is also airtight. If (1) and (2) are true, to deny (3) would be utterly irrational. Formulating sequences of abstract, airtight reasoning is the goal of logic.

What more then can we say? This can sometimes feel like beating a dead horse, after all we can all use this kind of logic without thinking. And yet I still think it is important, as we all screw this up from time to time and don’t think through what we say, do, or believe. To end this post, I will present what are generally considered the nine fundamental rules of logic which, when combined with suitable definitions for how to understand words like ‘not’, ‘or’, and ‘and’, forms the complete basis of the fundamentals of logic, upon which more sophisticated thought can be built.

Briefly, before these are discussed, a final point must be made about the definition of truth. There are certain laws of logic, which are taken as even more basic than the ones just discussed, that I will lay out. These are called the Law of Non-Contradiction and the Law of the Excluded Middle. These can be described as follows:

Law of Non-Contradiction: A statement cannot be both true and false.

Law of the Excluded Middle: Any proposition (roughly speaking, any matter-of-fact claim) is either true or false.

These are absolutely essential to all human thought; so much so that it becomes impossible to deny either of them without assuming that they are true. (For instance, if the Law of Non-Contradiction is false, it could also be true, and then it cannot be false, but it is false… spend a few minutes trying to think through that mess!)

To close an initial discussion of logic, for those who are interested, here is a list of the 9 basic building blocks used for a rational, logical argument: (Throughout, the letters P, Q, R, and S are symbols that represent some statement, any line without the word ‘therefore’ is a premise, and any line with ‘therefore’ is a conclusion)

One can safely ignore all the fancy names; these arguments are given names just so that we can refer to them in sentences without having to write them out every time; the ideas behind them are things that can be understood by the usual meanings of the words ‘and’, ‘or’, ‘if’, and ‘then’.

Modus Ponens

  1. If P, then Q,
  2. P,
  3. Therefore, Q.

Modus Tollens

  1. If P, then Q,
  2. Not Q,
  3. Therefore, not P.

Hypothetical Syllogism (stringing together two if-then statements)

  1. If P, then Q,
  2. If Q, then R,
  3. Therefore, if P, then R.

Conjunction (“True and True = True”)

  1. P,
  2. Q,
  3. Therefore, P and Q.

Simplification (If both are true, then each is true on its own)

  1. P and Q,
  2. Therefore, P.
  3. (Similarly, therefore Q)

Absorption (If P implies Q, then Q can be thought of as ‘contained inside’ P)

  1. If P, then Q,
  2. Therefore, if P then P and Q.

Addition (If P is true, then P or Q is also true)

  1. P,
  2. Therefore, P or Q.

Disjunctive Syllogism (If an ‘or’ statement is true, at least one part of it is true)

  1. P or Q,
  2. Not P,
  3. Therefore, Q.

Constructive Dilemma (Combining two if-then statements with ‘and’)

  1. “If P, then Q” and “If R, then S“,
  2. P or R,
  3. Therefore, Q or S.

To close, we ask why are these rules are considered fundamental, and other potential rules of logic not? Firstly, these rules serve the purpose of making fully clear the meanings of the terms and, or, not, and if-then. No extra rules are needed, because any other rules using these words could be built by combining these in different ways.

But more than this, there are other important logical words that are not part of this system (words like necessary, every, and some are examples). But it would be exceedingly difficult, if not impossible, to use these additional words without also using the simpler ones. So the system just described is in a sense the “smallest” system of logic. Everything else, including mathematics, is an extension of this system, with new rules and ideas added.

Transforming this 9 rule system into genuine mathematics is actually quite tedious, doing so would probably need a whole new series of posts. For my purpose here, let’s just say that when you define words like numbers, addition, every, and some to your language, you can start to call this math.

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