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A Crash Course in Logic

Updated: Aug 8, 2021

Few philosophers in all of history are as important, and probably none more, than Aristotle. Living from 384-322 B.C., his work covered a wide array of fields in philosophy, science, and rhetoric. Among his many significant contributions to philosophy is the first systematic treatment of logic. This little crash course, in many ways, owes its existence to Aristotle. You can read about him in this article in the Stanford Encyclopedia of Philosophy if you're interested.

Logic can often seem abstract and difficult to grasp, and because of this, people are often intimidated by it. When I talk about logic, I try to encourage people by telling them that they think in logical terms all the time. Logic, as a field of philosophy, merely puts words and symbols to the conversations that we have every day.

Let me give an example. Bill comes home to his wife after work and asks her, "Did you pick up groceries for this weekend?" His wife answers no, that she didn't pick up groceries for the weekend. Bill responds, "Oh, you must not have gone to the grocery store." Why would he have come to that conclusion? In fact, she could have gone to the grocery store and not picked up anything. He comes to this conclusion because, that morning, they had discussed plans for the evening, and his wife had told him that, if she goes to the grocery store after work, she will pick up groceries for the weekend. Since she didn't pick up any groceries, she must not have gone to the grocery store. This somewhat mundane conversation between a husband and wife is actually an example of deductive logical reasoning. Notice that we tend to intuitively grasp the right conclusion here. Once we know what Bill knows, it makes good sense that he would come to that conclusion.

That is not to say that logic is no more complicated than that conversation. Believe me, logic can and does get incredibly complex. But the complex side of logic does not in any way conflict with the idea that we use logical reasoning every day, in the same way that we use math every day, even though only a few people know how to do calculus. In the same way that knowing simple mathematical principles can help us in everyday life, knowing simple logical principles can help us in everyday life.

Last week, I mentioned that God has created the human mind to comprehend logic and to know His nature, however incompletely, by what He has made. Our minds were created to discover God, and our capacity for understanding logic helps us to come to knowledge of the truth. This capacity, which many philosophers have called reason, allows us to probe into questions of ultimate importance, such as the question, "Does God exist?" Logic helps us to do this more reasonably and to notice when we or others make mistakes in our reasoning. It helps us to probe these questions more carefully and to come to better conclusions.

We are followers of Jesus Christ. Our Lord calls us to love Him with all of our minds (Matthew 22:37). As I alluded to last week, God is the foundation of logic. The exact meaning of that statement is difficult to describe, just as it's difficult to explain how God is the foundation for mathematics, but it rings true nonetheless. It was Johannes Kepler who said that when he studied astronomy, he was "thinking God's thoughts after Him." There is something in probing reality itself that fills (or should fill) the thoughtful Christian with awe. Our cultivation of the mind is an act of worship. Logic, then, is part of this process of thinking God's thoughts after Him, since, in using logic well, we think along the patterns of God Himself.

Logic will also help us in our defense of the faith. As we discuss God's existence and the truth of Christianity with unbelievers, logic will help us to catch logical mistakes and correct them and give us greater confidence in the validity of our own arguments. Finally, logic will also help us to analyze our own beliefs, since sometimes we make logical mistakes in our belief systems. This will further refine our own thinking and help us to know the truth. In many ways, therefore, knowing basic logical principles will be of benefit to us.

I will start by discussing what logic is and what makes a logical argument good. I will then discuss the difference between deductive and inductive logic and discuss two rules of inference, which we will need to know in order to discuss these arguments for God's existence. Finally, I will discuss two common logical fallacies, or mistakes in reasoning.

What is logic? In short, logic is the field of philosophy related to the rules that connect propositions to each other to form conclusions. By proposition, I am referring to the content of statements. An example of a proposition would be, "Roses are red." All propositions have a truth value; they are either true or false. Individual propositions cannot do much, but when they come together, other propositions follow from them. If two true propositions connect to infer a third, then the third proposition is true. The rules by which the two are connected to form the third is logic.

Any logical argument will involve premises and a conclusion. The premises are the propositions whose connection draws a certain conclusion. The conclusion, of course, is that proposition which follows from the premises.

What makes a logical argument good? In general, a logical argument is good if it fulfills two conditions. It must fulfill both. First, a logical argument must be valid. This means that the conclusion follows logically from the premises according to some rule of valid inference. Second, a logical argument must be sound. This means that the premises are true, or at least more probably true than false. This condition is much harder to fulfill than the first, since it involves thinking long and hard about the premises themselves and what arguments and evidence can be offered in support of them. Because of this, when we evaluate arguments to see if they're good arguments, we will always check their validity first. If the argument is invalid, then it makes no difference to us if the premises are true; the conclusion doesn't follow either way.

In general, logical arguments can be divided into two categories: deductive and inductive. A deductive argument is one in which the conclusion follows necessarily from the premises according to a rule of valid inference. There are only a few rules (exactly nine), and a deductive argument must follow these rules in order to be valid. Because of this, as long as a deductive argument is valid and sound, then the conclusion must be true. To deny the conclusion of a good deductive argument is not just mistaken; it is irrational. An inductive argument involves premises that provide support for the conclusion, but the conclusion doesn't necessarily follow from the premises. Because of this, the support that an inductive argument provides for a certain conclusion is somewhat weaker than that for a conclusion in a deductive argument, but that doesn't mean that inductive arguments are somehow worse. They can still be very powerful arguments.

In this series on natural theology, we will mainly discuss deductive arguments. Because of this, we will need to understand at least two main rules of inference. Any deductive argument must follow a rule of inference. A syllogism is an argument that follows these rules in order to derive a conclusion. We write syllogisms by listing the premises followed by their conclusions, such that the rules can be more clearly seen. When we want to know if and how rules are followed in the argument, we will convert the English sentences in the argument to symbols in order to better be able to see the logical structure of the argument. To illustrate this, let's use a basic three-step argument:

  1. If Socrates is a man, then Socrates is mortal.

  2. Socrates is a man.

  3. Therefore, Socrates is mortal.

We use the word "therefore" for the conclusion. In order to better make out the logical structure of this argument, we need to convert it to symbols, or symbolize it. First, let's convert the propositions to symbols using letters. You can use any letters to represent the propositions. For the proposition, "Socrates is a man," let's use the letter S. For the proposition, "Socrates is mortal," let's use the letter M. (Notice that S and M are two different proposition that are both part of the first premise and that there is some connection between S and M in the first premise. We'll discuss this later.)

For the first premise, notice that the statement in the first premise is conditional. Conditional statements are a staple of logical reasoning. All conditional statements have the structure: "If ..., then ...." Because of this, any conditional statement includes the claim that the truth of one proposition has some impact on the truth of another proposition. When we symbolize a conditional statement, an arrow (->) represents the fact that the statement is conditional. Therefore, premise 1 can be written like this: S ->M. We can read this in one of two ways, either as, "S implies M," or as, "If S, then M."

Let's bring all of this together and symbolize the argument above:

  1. S ->M

  2. S

  3. Therefore, M.

By symbolizing the argument, we can see the argument independently of the content of the propositions, which helps us to determine its logical validity. In this case, the argument is valid. Let's call this argument "the Socratic argument." We'll use it to discuss our two rules of inference.

In deductive logic, the rules of inference are the strict logical rules by which conclusions must be derived in deductive logic. If a deductive argument doesn't follow one of these rules, then it is invalid. Therefore, when we see a valid deductive argument, we must be able to name the rule or rules that it follows. For each rule of inference, there is a particular form or structure to the argument, in which generic propositions are represented with the letters p and q.

The first rule of inference is called modus ponens. The form of the Socratic argument above is modus ponens, which is why the argument is valid. Here is the structure:

Modus Ponens

  1. p->q

  2. p

  3. Therefore, q.

As you can see, the form of the Socratic argument matches the form of modus ponens exactly, which is why we symbolize deductive arguments to analyze their form.

The second rule of inference is called modus tollens. Here is the structure:

Modus Tollens

  1. p->q

  2. not-q

  3. Therefore, not-p

In logic, we read the phrase "not-p" as the negation of p or the statement, "p is not true." Here, we can start modifying the Socratic argument in order to derive different conclusions. Starting with the first premise, let's modify the argument to come to the conclusion that Socrates is not a man:

  1. S ->M

  2. not-M

  3. Therefore, not-S.

For clarification, I'll translate this argument to plain English, so the content of the propositions remains clear:

  1. If Socrates is a man, then Socrates is mortal.

  2. Socrates is not mortal.

  3. Therefore, Socrates is not a man.

Hopefully, by this point it becomes clear how intuitive this reasoning is. Just imagine that you were told both of the premises. You would know intuitively that if Socrates is not mortal, then whatever he is, he must not be a man, since men are mortal. Perhaps Socrates is the name for a god or some other non-mortal or immortal being, but he cannot be a man. At this point, we're learning basic arithmetic, which is perfectly okay, since that is most of what we need in the real world. As with mathematics, we're simply learning that which will help us in the real world.

At this point, people often make two basic logical mistakes. We tend to intuitively grasp the rules of inference, but it is easy to get them mixed up. For example, let's take the Socratic argument again. Assume that you only know the first premise. Now, consider what you'd conclude if I were to tell you that Socrates is not a man. Would you conclude that he is therefore not mortal? Many almost immediately come to this conclusion, but it is mistaken. We cannot conclude that Socrates is not a mortal because his not being a man has no bearing on whether or not he is mortal. Socrates could be a cat, for instance, and cats are mortal beings. These logical mistakes are called fallacies, and I'll cover two that are helpful to watch out for.

The example above is an example of a common logical fallacy called negating the antecedent. This term can be confusing, but let me explain. In any conditional statement, there is an antecedent and a consequent. This distinction has to do with the effect one proposition has on another in a conditional statement. As long as the statement has the structure of "if" before "then," then the former proposition is always the antecedent, and the latter is always the consequent. To negate a proposition means that the proposition is now false. So, to negate the antecedent is to render the former proposition as false, as I did above when I proposed that Socrates was not a man. Formal logical fallacies also have forms, so here is the form of negating the antecedent:

Negating the Antecedent

  1. p->q

  2. not-p

  3. Therefore, not-q.

A similar logical fallacy can be found when we propose that Socrates is mortal. If we know that Socrates is mortal, can we therefore conclude that he is a man? Obviously not. He could be some other kind of mortal being, such as a cat. This logical fallacy is called affirming the consequent. Again, the consequent is the latter proposition in a conditional statement. When a statement is affirmed, it is true. Here is the form of affirming the consequent:

Affirming the Consequent

  1. p->q

  2. q

  3. Therefore, p.

If you want to make this into an exercise, then take out a sheet of paper and write out the Socratic argument. Put a heading on the paper for each rule of inference and fallacy that we've discussed and modify the argument according to the form of each rule and fallacy. Then, translate each symbolized argument into plain English, and see if each rule and fallacy makes sense. Logic is similar to mathematics in that it takes practice to get better at it.

Almost all of the arguments for God's existence use one of these rules. As we discuss these arguments, I will explain when a new rule is used or when a fallacy comes up. Again, if you want practice, symbolize each argument on your own on a sheet of paper. I'll symbolize them as well, but this practice will really help you sharpen your skills in logic.

A final note on conditional statements: I noted above that the distinction between the antecedent and the consequent is based on the logical relationship between the two propositions in a conditional statement. Let me explain that in more detail, since it will benefit us in our exploration of these arguments to understand this distinction. As the structure of a conditional statement implies, there is an effect shared between the antecedent and consequent, but that effect is not the same. The antecedent affects the consequent in a way different from the consequent's effect on the antecedent.

How is this so? Notice the connection between both propositions in the statement, "If Socrates is a man, then Socrates is mortal." As we discussed above, showing that Socrates is mortal doesn't necessarily imply that he is a man, since he could be a cat. Yet showing that Socrates is a man necessarily implies that he is mortal, since men are mortal beings. This shows that being a man, though sufficient for being mortal, is not necessary, since more than one kind of being is a mortal being. Since, however, being mortal necessarily follows from being a man, then being mortal is necessary to being a man, yet it isn't sufficient. How so? For the same reason, that there is more than one kind of mortal being.

Therefore, we can see that the antecedent in any true conditional statement is a sufficient condition for the consequent. Likewise, the consequent is a necessary condition for the antecedent. Here is a helpful way to remember it:

Sufficient condition -> Necessary condition

As we discuss these arguments, remembering necessary and sufficient conditions will help.

What have we learned in this post? First, we learned what logic is and what makes a logical argument good. Second, we learned the difference between a deductive and inductive argument. Third, we learned two of the rules of logical inference. Fourth, we learned two fallacies. Finally, we learned about sufficient and necessary conditions. This may seem like a lot of material. I suggest that if you really want to grasp it, read this post again more closely with a pen and paper to take notes. These skills have to be honed through practice. If you are finding it difficult to grasp the material or have any other questions, feel free to send me an email to my address on the homepage. I'd be glad to help.

That's it for this week's post. Next week, we will discuss the famous argument from the beginning of the universe, also known as the Kalam Cosmological Argument. This is one of the most influential and discussed arguments for God's existence today, so I'm excited to discuss it.

As always, if you find yourself continuing to come back to the blog, please consider subscribing to it, so that you will get notified of each new post. With a subscription, you can also comment below and start a discussion! Finally, if you have any questions or comments about the blog, then you can send me a message from the bottom of the homepage. I look forward to receiving messages from you. Thank you for reading!

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